Fermat’s Last Theorem

Henri Darmon (darmon@math.mcgill.ca) Department of Mathematics

McGill University Montreal, QC Canada H3A 2K6

Fred Diamond (fdiamond@pmms.cam.ac.uk) D.P.M.M.S. Cambridge University Cambridge, CB2 1SB United Kingdom

Richard Taylor (taylorr@maths.ox.ac.uk) Mathematics Institute Oxford University 24-29 St. Giles Oxford, OX1 3LB United Kingdom

May 24, 2000

The authors would like to give special thanks to N. Boston, K. Buzzard, and B. Conrad for providing so much valuable feedback on earlier versions of this paper. They are also grateful to A. Agboola, M. Bertolini, B. Edixhoven, J. Fearnley, R. Gross, L. Guo, F. Jarvis, H. Kisilevsky, E. Liverance, J. Manoharmayum, K. Ribet, D. Rohrlich, M. Rosen, R. Schoof, J.-P. Serre, C. Skinner, D. Thakur, J. Tilouine, J. Tunnell, A. Van der Poorten, and L. Washington for their helpful comments.

Darmon thanks the members of CICMA and of the Quebec-Vermont Number Theory Seminar for many stimulating conversations on the topics of this paper, particularly in the Spring of 1995. For the same reason Diamond is grateful to the participants in an informal seminar at Columbia University in 1993-94, and Taylor thanks those attending the Oxford Number Theory Seminar in the Fall of 1995.

Parts of this paper were written while the authors held positions at other institutions: Darmon at Princeton University, Diamond at the Institute for Advanced Study, and Taylor at Cambridge University. During some of the period, Diamond enjoyed the hospitality of Princeton University, and Taylor that of Harvard University and MIT. The writing of this paper was also supported by research grants from NSERC (Darmon), NSF # DMS 9304580 (Diamond) and by an advanced fellowship from EPSRC (Taylor).

This article owes everything to the ideas of Wiles, and the arguments presented here are fundamentally his [W3], though they include both the work [TW] and several simplifications to the original arguments, most notably that of Faltings. In the hope of increasing clarity, we have not always stated theorems in the greatest known generality, concentrating instead on what is needed for the proof of the Shimura-Taniyama conjecture for semi-stable elliptic curves. This article can serve as an introduction to the fundamental papers [W3] and [TW], which the reader is encouraged to consult for a different, and often more in-depth, perspective on the topics considered. Another useful more advanced reference is the article [Di2] which strengthens the methods of [W3] and [TW] to prove that every elliptic curve that is semistable at 3 and 5 is modular.

Introduction

Fermat’s Last Theorem

Fermat’s Last Theorem states that the equation

x^{n} + y^{n} = z^{n}, xyz ≠ 0

has no integer solutions when n is greater than or equal to 3. Around 1630, Pierre de Fermat claimed that he had found a “truly wonderful” proof of this theorem, but that the margin of his copy of Diophantus’ Arithmetica was too small to contain it:

“Cubum autem in duos cubos, aut quadrato quadratum in duos quadrato quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere; cujus rei demonstrationem mirabile sane detexi. Hanc marginis exiguitas non caperet.”

Among the many challenges that Fermat left for posterity, this was to prove the most vexing. A tantalizingly simple problem about whole numbers, it stood unsolved for more than 350 years, until in 1994 Andrew Wiles finally laid it to rest.

Prehistory: The only case of Fermat’s Last Theorem for which Fermat actually wrote down a proof is for the case n = 4. To do this, Fermat introduced the idea of infinite descent which is still one the main tools in the study of Diophantine equations, and was to play a central role in the proof of Fermat’s Last Theorem 350 years later. To prove his Last Theorem for exponent 4, Fermat showed something slightly stronger, namely that the equation x^{4}+y^{4} = z^{2} has no solutions in relatively prime integers with xyz ≠ 0. Solutions to such an equation correspond to rational points on the elliptic curve v^{2} = u^{3} − 4u. Since every integer n ≥ 3 is divisible either by an odd prime or by 4, the result of Fermat allowed one to reduce the study of Fermat’s equation to the case where n = ℓ is an odd prime.

In 1753, Leonhard Euler wrote down a proof of Fermat’s Last Theorem for the exponent ℓ = 3, by performing what in modern language we would call a 3-descent on the curve x^{3} + y^{3} = 1 which is also an elliptic curve. Euler’s argument (which seems to have contained a gap) is explained in [Edw], ch. 2, and [Dic1], p. 545.

It took mathematicians almost 100 years after Euler’s achievement to handle the case ℓ = 5; this was settled, more or less simultaneously, by Gustav Peter Lejeune Dirichlet [Dir] and Adrien Marie Legendre [Leg] in 1825. Their elementary arguments are quite involved. (Cf. [Edw], sec. 3.3.)

In 1839, Fermat’s equation for exponent 7 also yielded to elementary methods, through the heroic efforts of Gabriel Lamé. Lamé’s proof was even more intricate than the proof for exponent 5, and suggested that to go further, new theoretical insights would be needed.

The work of Sophie Germain: Around 1820, in a letter to Gauss, Sophie Germain proved that if ℓ is a prime and q = 2ℓ+1 is also prime, then Fermat’s equation x^{ℓ} + y^{ℓ} = z^{ℓ} with exponent ℓ has no solutions (x, y, z) with xyz ≠ 0 (mod ℓ). Germain’s theorem was the first really general proposition on Fermat’s Last Theorem, unlike the previous results which considered the Fermat equation one exponent at a time.

The case where the solution (x, y, z) to x^{ℓ} + y^{ℓ} = z^{ℓ} satisfies xyz ≠ 0 (mod ℓ) was called the first case of Fermat’s Last Theorem, and the case where ℓ divides xyz, the second case. It was realized at that time that the first case was generally easier to handle: Germain’s theorem was extended, using similar ideas, to cases where kℓ+1 is prime and k is small, and this led to a proof that there were no first case solutions to Fermat’s equation with prime exponents ℓ ≤ 100, which in 1830 represented a significant advance. The division between first and second case remained fundamental in much of the later work on the subject. In 1977, Terjanian [Te] proved that if the equation x^{2ℓ} + y^{2ℓ} = z^{2ℓ} has a solution (x, y, z), then 2ℓ divides either x or y, i.e., “the first case of Fermat’s Last Theorem is true for even exponents”. His simple and elegant proof used only techniques that were available to Germain and her contemporaries.

The work of Kummer: The work of Ernst Eduard Kummer marked the beginning of a new era in the study of Fermat’s Last Theorem. For the first time, sophisticated concepts of algebraic number theory and the theory of L-functions were brought to bear on a question that had until then been addressed only with elementary methods. While he fell short of providing a complete solution, Kummer made substantial progress. He showed how Fermat’s Last Theorem is intimately tied to deep questions on class numbers of cyclotomic fields which are still an active subject of research. Kummer’s approach relied on the factorization

(x + y)(x + ζ_{ℓ}y) ⋯ (x + ζ_{ℓ}^{ℓ−1}y) = z^{ℓ}

of Fermat’s equation over the ring ℤ[ζ_{ℓ}] generated by the ℓth roots of unity. One observes that the greatest common divisor of any two factors in the product on the left divides the element (1 − ζ_{ℓ}), which is an element of norm ℓ.

Since the product of these numbers is a perfect ℓ-th power, one is tempted to conclude that (x + y), …, (x + ζ_{ℓ}^{ℓ−1}y) are each ℓ-th powers in the ring ℤ[ζ_{ℓ}] up to units in this ring, and up to powers of (1 − ζ_{ℓ}). Such an inference would be valid if one were to replace ℤ[ζ_{ℓ}] by ℤ, and is a direct consequence of unique factorization of integers into products of primes. We say that a ring R has property UF if every non-zero element of R is uniquely a product of primes, up to units. Mathematicians such as Lamé made attempts at proving Fermat’s Last Theorem based on the mistaken assumption that the rings ℤ[ζ_{ℓ}] had property UF. Legend even has it that Kummer fell into this trap, although this story now has been discredited; see for example [Edw], sec. 4.1. In fact, property UF is far from being satisfied in general: one now knows that the rings ℤ[ζ_{ℓ}] have property UF only for ℓ < 23 (cf. [Wa], ch. 1).

It turns out that the full force of property UF is not really needed in the applications to Fermat’s Last Theorem. Say that a ring R has property UF_{ℓ} if the following inference is valid:

ab = z^{ℓ}, and gcd(a, b) = 1 ⇒ a and b are ℓth powers up to units of R.

If a ring R has property UF, then it also has property UF_{ℓ}, but the converse need not be true. Kummer showed that Fermat’s last theorem was true for exponent ℓ if ℤ[ζ_{ℓ}] satisfied the property UF_{ℓ} (cf. [Wa]). The proof is far from trivial, because of difficulties arising from the units in ℤ[ζ_{ℓ}] as well as from the possible failure of property UF. (A number of Kummer’s contemporaries, such as Cauchy and Lamé, seem to have overlooked both of these difficulties in their attempts to prove Fermat’s Last Theorem.)

Kummer then launched a systematic study of the property UF_{ℓ} for the rings ℤ[ζ_{ℓ}]. He showed that even if ℤ[ζ_{ℓ}] failed to have unique factorization, it still possessed unique factorization into prime ideals. He defined the ideal class group as the quotient of the group of fractional ideals by its subgroup consisting of principal ideals, and was able to establish the finiteness of this class group. The order of the class group of ℤ[ζ_{ℓ}], denoted h_{ℓ}, could be taken as a measure of the failure of the ring ℤ[ζ_{ℓ}] to satisfy UF. It was rather straightforward to show that if ℓ did not divide h_{ℓ}, then ℤ[ζ_{ℓ}] satisfied the property UF_{ℓ}. In this case, one called ℓ a regular prime. Kummer thus showed that Fermat’s last theorem is true for exponent ℓ if ℓ is a regular prime.

He did not stop here. For it remained to give an efficient means of computing h_{ℓ}, or at least an efficient way of checking when ℓ divides h_{ℓ}. The class number h_{ℓ} can be factorized as a product

h_{ℓ} = h_{ℓ}^{+}h_{ℓ}^{−},

where h_{ℓ}^{+} is the class number of the real subfield ℚ(ζ_{ℓ})^{+}, and h_{ℓ}^{−} is defined as h_{ℓ}/h_{ℓ}^{+}. Essentially because of the units in ℚ(ζ_{ℓ})^{+}, the factor h_{ℓ}^{+} is somewhat difficult to compute, while, because the units in ℚ(ζ_{ℓ})^{+} generate the group of units in ℚ(ζ_{ℓ}) up to finite index, the term h_{ℓ}^{−} can be expressed in a simple closed form. Kummer showed that if ℓ divides h_{ℓ}^{+}, then ℓ divides h_{ℓ}^{−}. Hence, ℓ divides h_{ℓ} if and only if ℓ divides h_{ℓ}^{−}. This allowed one to avoid the difficulties inherent in the calculation of h_{ℓ}^{+}. Kummer then gave an elegant formula for h_{ℓ}^{−} by considering the Bernoulli numbers B_{n}, which are rational numbers defined by the formula

\frac{x}{e^{x} − 1} = ∑ \frac{B_{n}}{n!} x^{n}.

He produced an explicit formula for the class number h_{ℓ}^{−}, and concluded that if ℓ does not divide the numerator of B_{2i}, for 1 ≤ i ≤ (ℓ − 3)/2, then ℓ is regular, and conversely.

The conceptual explanation for Kummer’s formula for h_{ℓ}^{−} lies in the work of Dirichlet on the analytic class number formula, where it is shown that h_{ℓ}^{−} can be expressed as a product of special values of certain (abelian) L-series

L(s, χ) = ∑_{n=1}^{∞} χ(n)n^{−s}

associated to odd Dirichlet characters. Such special values in turn can be expressed in terms of certain generalized Bernoulli numbers B_{1,χ}, which are related to the Bernoulli numbers B_{i} via congruences mod ℓ. (For more details, see [Wa].)

These considerations led Kummer to initiate a deep study relating congruence properties of special values of L-functions and of class numbers, which was to emerge as a central concern of modern algebraic number theory, and was to reappear – in a surprisingly different guise – at the heart of Wiles’ strategy for proving the Shimura-Taniyama conjecture.

Later developments: Kummer’s work had multiple ramifications, and led to a very active line of enquiry pursued by many people. His formulae relating Bernoulli numbers to class numbers of cyclotomic fields were refined by Kenneth Ribet [R1], Barry Mazur and Andrew Wiles [MW], using new methods from the theory of modular curves which also play a central role in Wiles’ more recent work. (Later Francisco Thaine [Th] reproved some of the results of Mazur and Wiles using techniques inspired directly from a reading of Kummer.) In a development more directly related to Fermat’s Last Theorem, Wieferich proved that if ℓ^{2} does not divide 2^{ℓ−1} − 1, then the first case of Fermat’s Last Theorem is true for exponent ℓ. (Cf. [Ri], lecture VIII.)

There were many other refinements of similar criteria for Fermat’s Last theorem to be true. Computer calculations based on these criteria led to a verification that Fermat’s Last theorem is true for all odd prime exponents less than four million [BCEM], and that the first case is true for all ℓ ≤ 8.858 · 10^{20} [Su].

The condition that ℓ is a regular prime seems to hold heuristically for about 61% of the primes. (See the discussion on p. 63, and also p. 108, of [Wa], for example.) In spite of the convincing numerical evidence, it is still not known if there are infinitely many regular primes. Ironically, it is not too difficult to show that there are infinitely many irregular primes. (Cf. [Wa].)

Thus the methods introduced by Kummer, after leading to very strong results in the direction of Fermat’s Last theorem, seemed to become mired in difficulties, and ultimately fell short of solving Fermat’s conundrum^{1}.

Faltings’ proof of the Mordell conjecture: In 1985, Gerd Faltings [Fa] proved the very general statement (which had previously been conjectured by Mordell) that any equation in two variables corresponding to a curve of genus strictly greater than one had (at most) finitely many rational solutions. In the context of Fermat’s Last Theorem, this led to the proof that for each exponent n ≥ 3, the Fermat equation x^{n} + y^{n} = z^{n} has at most finitely many integer solutions (up to the obvious rescaling). Andrew Granville [Gra] and Roger Heath-Brown [HB] remarked that Faltings’ result implies Fermat’s Last Theorem for a set of exponents of density one.

However, Fermat’s Last Theorem was still not known to be true for an infinite set of prime exponents. In fact, the theorem of Faltings seemed illequipped for dealing with the finer questions raised by Fermat in his margin, namely of finding a complete list of rational points on all of the Fermat curves x^{n} + y^{n} = 1 simultaneously, and showing that there are no solutions on these curves when n ≥ 3 except the obvious ones.

Mazur’s work on Diophantine properties of modular curves: Although it was not realized at the time, the chain of ideas that was to lead to a proof of Fermat’s Last theorem had already been set in motion by Barry Mazur in the mid seventies. The modular curves X_{0}(ℓ) and X_{1}(ℓ) introduced in section 1.2 and 1.5 give rise to another naturally occurring infinite family of Diophantine equations. These equations have certain systematic rational solutions corresponding to the cusps that are defined over ℚ, and are analogous to the so-called “trivial solutions” of Fermat’s equation. Replacing Fermat curves by modular curves, one could ask for a complete list of all the rational points on the curves X_{0}(ℓ) and X_{1}(ℓ). This problem is perhaps even more compelling than Fermat’s Last Theorem: rational points on modular curves correspond to objects with natural geometric and arithmetic interest, namely, elliptic curves with cyclic subgroups or points of order ℓ. In [Maz1] and [Maz2], B. Mazur gave essentially a complete answer to the analogue of Fermat’s Last Theorem for modular curves. More precisely, he showed that if ℓ ≠ 2, 3, 5 and 7, (i.e., X_{1}(ℓ) has genus > 0) then the curve X_{1}(ℓ) has no rational points other than the “trivial” ones, namely cusps. He proved analogous results for the curves X_{0}(ℓ) in [Maz2], which implied, in particular, that an elliptic curve over ℚ with square-free conductor has no rational cyclic subgroup of order ℓ over ℚ if ℓ is a prime which is strictly greater than 7. This result appeared a full ten years before Faltings’ proof of the Mordell conjecture.

1However, W. McCallum has recently introduced a technique, based on the method of Chabauty and Coleman, which suggests new directions for approaching Fermat’s Last Theorem via the cyclotomic theory. An application of McCallum’s method to showing the second case of Fermat’s Last Theorem for regular primes is explained in [Mc].

Frey’s strategy: In 1986, Gerhard Frey had the insight that these constructions might provide a precise link between Fermat’s Last Theorem and deep questions in the theory of elliptic curves, most notably the Shimura Taniyama conjecture. Given a solution a^{ℓ} + b^{ℓ} = c^{ℓ} to the Fermat equation of prime degree ℓ, we may assume without loss of generality that a^{ℓ} ≡ −1 (mod 4) and that b^{ℓ} ≡ 0 (mod 32). Frey considered (following Hellegouarch, [He], p. 262; cf. also Kubert-Lang [KL], ch. 8, §2) the elliptic curve

E : y^{2} = x(x − a^{ℓ})(x + b^{ℓ}).

This curve is semistable, i.e., it has square-free conductor. Let E[ℓ] denote the group of points of order ℓ on E defined over some (fixed) algebraic closure ℚ̄ of ℚ, and let L denote the smallest number field over which these points are defined. This extension appears as a natural generalization of the cyclotomic fields ℚ(ζ_{ℓ}) studied by Kummer. What singles out the field L for special attention is that it has very little ramification: using Tate’s analytic description of E at the primes dividing abc, it could be shown that L was ramified only at 2 and ℓ, and that the ramification of L at these two primes was rather restricted. (See theorem 2.15 of section 2.2 for a precise statement.) Moreover, the results of Mazur on the curve X_{0}(ℓ) could be used to show that L is large, in the following precise sense. The space E[ℓ] is a vector space of dimension 2 over the finite field 𝔽_{ℓ} with ℓ elements, and the absolute Galois group G_{ℚ} = Gal (ℚ̄/ℚ) acts 𝔽_{ℓ}-linearly on E[ℓ]. Choosing an 𝔽_{ℓ}-basis for E[ℓ], the action is described by a representation

ρ̄_{E,ℓ} : Gal (L/ℚ) ↪ GL_{2}(𝔽_{ℓ}).

Mazur’s results in [Maz1] and [Maz2] imply that ρ̄_{E,ℓ} is irreducible if ℓ > 7 (using the fact that E is semi-stable). In fact, combined with earlier results of Serre [Se6], Mazur’s results imply that for ℓ > 7, the representation ρ̄_{E,ℓ} is surjective, so that Gal (L/ℚ) is actually isomorphic to GL_{2}(𝔽_{ℓ}) in this case.

Serre’s conjectures: In [Se7], Jean-Pierre Serre made a careful study of mod ℓ Galois representations ρ̄ : G_{ℚ} ⟶ GL_{2}(𝔽_{ℓ}) (and, more generally, of representations into GL_{2}(k), where k is any finite field). He was able to make very precise conjectures (see section 3.2) relating these representations to modular forms mod ℓ. In the context of the representations ρ̄_{E,ℓ} that occur in Frey’s construction, Serre’s conjecture predicted that they arose from modular forms (mod ℓ) of weight two and level two. Such modular forms, which correspond to differentials on the modular curve X_{0}(2), do not exist because X_{0}(2) has genus 0. Thus Serre’s conjecture implied Fermat’s Last Theorem. The link between fields with Galois groups contained in GL_{2}(𝔽_{ℓ}) and modular forms mod ℓ still appears to be very deep, and Serre’s conjecture remains a tantalizing open problem.

Ribet’s work: lowering the level: The conjecture of Shimura and Taniyama (cf. section 1.8) provides a direct link between elliptic curves and modular forms. It predicts that the representation ρ̄_{E,ℓ} obtained from the ℓ-division points of the Frey curve arises from a modular form of weight 2, albeit a form whose level is quite large. (It is the product of all the primes dividing abc, where a^{ℓ} + b^{ℓ} = c^{ℓ} is the putative solution to Fermat’s equation.) Ribet [R5] proved that, if this were the case, then ρ̄_{E,ℓ} would also be associated with a modular form mod ℓ of weight 2 and level 2, in the way predicted by Serre’s conjecture. This deep result allowed him to reduce Fermat’s Last Theorem to the Shimura-Taniyama conjecture.

Wiles’ work: proof of the Shimura-Taniyama conjecture: In [W3] Wiles proves the Shimura-Taniyama conjecture for semi-stable elliptic curves, providing the final missing step and proving Fermat’s Last Theorem. After more than 350 years, the saga of Fermat’s Last theorem has come to a spectacular end.

The relation between Wiles’ work and Fermat’s Last Theorem has been very well documented (see, for example, [R8], and the references contained therein). Hence this article will focus primarily on the breakthrough of Wiles [W3] and Taylor-Wiles [TW] which leads to the proof of the Shimura-Taniyama conjecture for semi-stable elliptic curves.

From elliptic curves to ℓ-adic representations: Wiles’ opening gambit for proving the Shimura-Taniyama conjecture is to view it as part of the more general problem of relating two-dimensional Galois representations and modular forms. The Shimura-Taniyama conjecture states that if E is an elliptic curve over ℚ, then E is modular. One of several equivalent definitions of modularity is that for some integer N there is an eigenform f = ∑ a_{n}q^{n} of weight two on Γ_{0}(N) such that

#E(𝔽_{p}) = p + 1 − a_{p}

for all but finitely primes p. (By an eigenform, here we mean a cusp form which is a normalized eigenform for the Hecke operators; see section 1 for definitions.)

This conjecture acquires a more Galois theoretic flavour when one considers the two dimensional ℓ-adic representation

ρ_{E,ℓ} : G_{ℚ} ⟶ GL_{2}(ℤ_{ℓ})

obtained from the action of G_{ℚ} on the ℓ-adic Tate module of E: T_{ℓ}E = lim_{←} E[l^{n}](ℚ̄). An ℓ-adic representation ρ of G_{ℚ} is said to arise from an eigenform f = ∑ a_{n}q^{n} with integer coefficients a_{n} if

tr (ρ(Frob_{p})) = a_{p},

for all but finitely many primes p at which ρ is unramified. Here Frob_{p} is a Frobenius element at p (see section 2), and its image under ρ is a well-defined conjugacy class.

A direct computation shows that #E(𝔽_{p}) = p + 1 − tr (ρ_{E,ℓ}(Frob_{p})) for all primes p at which ρ_{E,ℓ} is unramified, so that E is modular (in the sense defined above) if and only if for some ℓ, ρ_{E,ℓ} arises from an eigenform. In fact the Shimura-Taniyama conjecture can be generalized to a conjecture that every ℓ-adic representation, satisfying suitable local conditions, arises from a modular form. Such a conjecture was proposed by Fontaine and Mazur [FM].

Galois groups and modular forms

Viewed in this way, the Shimura-Taniyama conjecture becomes part of a much larger picture: the emerging, partly conjectural and partly proven correspondence between certain modular forms and two dimensional representations of G_{ℚ}. This correspondence, which encompasses the Serre conjectures, the Fontaine-Mazur conjecture, and the Langlands program for GL_{2}, represents a first step toward a higher dimensional, non-abelian generalization of class field theory.

Two-dimensional representations of G_{ℚ}: In the first part of this century, class field theory gave a complete description of G_{ℚ}^{ab}, the maximal (continuous) abelian quotient of G_{ℚ}. In fact the Kronecker-Weber theorem asserts that G_{ℚ}^{ab} ≅ ∏_{p} ℤ_{p}^{×}, and one obtains a complete description of all one-dimensional representations of G_{ℚ}. In the second half of this century much attention has focused on attempts to understand the whole group G_{ℚ}, or more precisely to describe all its representations. Although there has been a fair degree of success in using modular forms to construct representations of G_{ℚ}, less is known about how exhaustive these constructions are. The major results in the latter direction along these lines are the work of Langlands [Ll2] and the recent work of Wiles ([W3] completed by [TW]). Both concern two-dimensional representations of G_{ℚ} and give significant evidence that these representations are parametrised (in a very precise sense) by certain modular forms. The purpose of this article is to describe both the proven and conjectural parts of this theory, give a fairly detailed exposition of Wiles’ recent contribution and explain the application to Fermat’s Last theorem. To make this description somewhat more precise let us distinguish three types of representation.

Artin representations and the Langlands-Tunnell theorem: Continuous representations ρ : G_{ℚ} → GL_{2}(ℂ) are called (two-dimensional) Artin representations. Such representations necessarily have finite image, and are therefore semi-simple. We restrict our attention to those which are irreducible. They are conjectured to be in bijection (in a precise way) with certain newforms (a special class of eigenforms). Those ρ which are odd (i.e. the determinant of complex conjugation is −1), should correspond to weight 1 holomorphic newforms. Those which are even should correspond to certain nonholomorphic (Maass) newforms. Two partial but deep results are known.

(a) (Deligne-Serre) If f is a holomorphic weight one newform then the corresponding Artin representation can be constructed ([DS]).

(b) (Langlands-Tunnell) If ρ is a two dimensional Artin representation with soluble image then the corresponding modular form exists ([Ll2] and [Tu]).

The proof of the latter result is analytic in nature, invoking the trace formula and the theory of L-functions.

ℓ-adic representations and the Fontaine-Mazur conjecture: By an ℓadic representation we shall mean any continuous representation ρ : G_{ℚ} → GL_{2}(K) which is unramified outside a finite set of primes and where K is a finite extension of ℚ_{ℓ} (generalizing slightly the notion of ℓ-adic representation that was introduced before). Given a holomorphic newform f one can attach to f a system of ℓ-adic representations, following Eichler, Shimura, Deligne and Serre. These ℓ-adic representations are called modular. The Fontaine-Mazur conjecture (see [FM]) predicts if ρ is an odd, irreducible, ℓ-adic representation whose restriction to the decomposition group at ℓ is well enough behaved, then ρ is modular. (The restriction on the behaviour of the representation on the decomposition group at ℓ is essential in this conjecture; it is not true that all odd, irreducible two dimensional ℓ-adic representation are modular.) Before Wiles’ work almost nothing was known about this conjecture, except that certain very special cases could be deduced from the work of Hecke, Langlands and Tunnell.

Mod ℓ representations and Serre’s conjecture: A mod ℓ representation is a continuous representation ρ̄: G_{ℚ} ⟶ GL_{2}(𝔽̄_{ℓ}). For example if E/ℚ is an elliptic curve then the action of G_{ℚ} on the ℓ-division points of E gives rise to a mod ℓ representation ρ̄_{E,ℓ} which is just the reduction modulo ℓ of ρ_{E,ℓ}. One can use the work of Eichler, Shimura, Deligne and Serre to associate to each mod ℓ eigenform a mod ℓ representation of G_{ℚ}. The mod ℓ representations which arise in this way are called modular. Serre has conjectured [Se7] that every odd (absolutely) irreducible mod ℓ representation is modular and should arise from a mod ℓ eigenform with certain very specific properties. This conjecture can be thought of as having two parts.

The first asserts that every odd irreducible mod ℓ representation is modular. About this very little is known. It is known for ρ̄ : G_{ℚ} → GL_{2}(𝔽_{2}) by work of Hecke. It is also known for ρ̄ : G_{ℚ} → GL_{2}(𝔽_{3}). This latter result is an application of the Langlands-Tunnell theorem using the two accidents that there is a section to the homomorphism GL_{2}(ℤ[\sqrt{−2}]) → → GL_{2}(𝔽_{3}) and that GL_{2}(𝔽_{3}) is soluble. Partial results for ρ̄ : G_{ℚ} → GL_{2}(𝔽_{5}) follow from Wiles’ work.

Given a mod ℓ representation arising from a mod ℓ eigenform, the second part of Serre’s conjecture predicts the minimal weight and level for that mod ℓ eigenform. Here the situation is much better. There has been a lot of work over the last decade (including ideas from Mazur, Ribet, Carayol and Gross) and the problem is nearly completely resolved (see [Di1]). As was pointed out earlier, Ribet’s contribution [R5] implies that, if one can show that the Galois representation ρ̄_{E,ℓ} arising from the (semi-stable) Frey curve attached to a solution of Fermat’s equation with exponent ℓ is modular, then one can show that this representation does not exist—because it would be modular of weight two and level two— and hence one can deduce Fermat’s Last Theorem.

However we have seen that to show ρ̄_{E,ℓ} is modular it suffices to show that for some ℓ_{0}, the ℓ_{0}-adic representation ρ_{E,ℓ_{0}} is modular. In particular it suffices to verify that either ρ_{E,3} or ρ_{E,5} is modular. Hence the Shimura-Taniyama conjecture can be reduced to (part of) the Fontaine-Mazur conjecture for ℓ = 3 and 5. We have seen that for these primes part of Serre’s conjecture is known, so it turns out it suffices to prove results of the form “Serre’s conjecture for ℓ implies the Fontaine-Mazur conjecture for ℓ”. This is the direction of Wiles’ work, although nothing quite this general has been proven yet.

Deformation theory: Thus the problem Wiles faces is to show that if ρ is an odd ℓ-adic representation which has irreducible modular reduction ρ̄ and which is sufficiently well behaved when restricted to the decomposition group at ℓ, then ρ is modular. In fact he only proves a weakened version of such a result, but one which is sufficient to conclude that all semistable elliptic curves are modular.

Wiles approaches the problem by putting it in a more general setting. On the one hand he considers lifts of ρ̄ to representations over complete noetherian local ℤ_{ℓ}-algebras R. For each finite set of primes Σ, one can consider lifts of type Σ; these are lifts which are well-behaved on a decomposition group at ℓ, and whose ramification at primes not in Σ is rather restricted. In particular, such a lift is unramified outside Σ ∪ S where S is the set of ramified primes of ρ̄. A method of Mazur (see [Maz3]) can then be used to show that if ρ̄ is absolutely irreducible, then there is a representation

ρ_{Σ}^{univ} : G_{ℚ} ⟶ GL_{2}(R_{Σ})

which is universal in the following sense. If ρ : G_{ℚ} → GL_{2}(R) is a lift of ρ̄ of type Σ, then there is a unique local homomorphism R_{Σ} ⟶ R such that ρ is equivalent to the pushforward of ρ_{Σ}^{univ}. Thus the equivalence classes of type Σ lifts to GL_{2}(R) can be identified with Hom(R_{Σ}, R). The local ring R_{Σ} is called the universal deformation ring for representations of type Σ.

On the other hand Wiles constructs a candidate for a universal modular lifting of type Σ

ρ_{Σ}^{mod} : G_{ℚ} ⟶ GL_{2}(𝕋_{Σ}).

The ring 𝕋_{Σ} is constructed from the algebra of Hecke operators acting on a certain space of modular forms. The universal property of R_{Σ} gives a map R_{Σ} → 𝕋_{Σ}. The problem thus becomes: to show that this map is an isomorphism^{2}. In fact, it can be shown to be a surjection without great difficulty, and the real challenge is to prove injectivity, i.e., to show, in essence, that R_{Σ} is not larger than 𝕋_{Σ}.

2Maps of this kind were already considered in [Maz3] and [BM], and it is conjectured in [MT] that these maps are isomorphisms in certain cases, though not in exactly the situations considered by Wiles.

By an ingenious piece of commutative algebra, Wiles found a numerical criterion for this map to be an isomorphism, and for the ring 𝕋_{Σ} to be a local complete intersection. This numerical criterion seems to be very close to a special case of the Bloch-Kato conjecture [BK]. Wiles further showed (by combining arguments from Galois cohomology and from the theory of congruences between modular forms) that this numerical criterion was satisfied if the minimal version 𝕋_{∅} of this Hecke algebra (obtained by taking Σ = ∅, i.e., allowing the least possible amount of ramification in the deformations) was a complete intersection. Finally in [TW] it was proved that 𝕋_{∅} is a complete intersection.

Outline of the paper

Chapter 1 recalls some basic notions from the classical theory of elliptic curves and modular forms, such as modular forms and modular curves over ℂ and ℚ, Hecke operators and q-expansions, and Eichler-Shimura theory. The Shimura-Taniyama conjecture is stated precisely in section 1.8.

Chapter 2 introduces the basic theory of representations of G_{ℚ}. We describe Mazur’s deformation theory and begin our study of the universal deformation rings using techniques from Galois cohomology and from the theory of finite flat group schemes. We also recall some basic properties of elliptic curves, both to explain Frey’s argument precisely and illustrate the uses of ℓ-adic representations.

Chapter 3 explains how to associate Galois representations to modular forms. We then describe what was known and conjectured about associating modular forms to Galois representations before Wiles’ work. After introducing the universal modular lifts of certain mod ℓ representations, we give the proof of Wiles’ main theorems, taking for granted certain results of a more technical nature that are proved in the last two chapters.

Chapter 4 explains how to prove the necessary results concerning the structure of Hecke algebras: the generalization by Taylor and Wiles of a result of de Shalit, and the generalization by Wiles of a result of Ribet.

Chapter 5 establishes the fundamental results from commutative algebra discovered by Wiles, following modifications of the approach of Wiles and Taylor-Wiles proposed by Faltings and Lenstra.

Contents

1 Elliptic curves and modular forms 16 1.1 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2 Modular curves and modular forms over ℂ . . . . . . . . . . . 22 1.3 Hecke operators and Hecke theory . . . . . . . . . . . . . . . . 28 1.4 The L-function associated to a cusp form . . . . . . . . . . . . 33 1.5 Modular curves and modular forms over ℚ . . . . . . . . . . . 34 1.6 The Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . 39 1.7 The Shimura construction . . . . . . . . . . . . . . . . . . . . 44 1.8 The Shimura-Taniyama conjecture . . . . . . . . . . . . . . . 47

2 Galois theory 50 2.1 Galois representations . . . . . . . . . . . . . . . . . . . . . . 50 2.2 Representations associated to elliptic curves . . . . . . . . . . 55 2.3 Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . 59 2.4 Representations of G_{ℚ_{ℓ}} . . . . . . . . . . . . . . . . . . . . . . 62 2.5 The theory of Fontaine and Laffaille . . . . . . . . . . . . . . . 69 2.6 Deformations of representations . . . . . . . . . . . . . . . . . 73 2.7 Deformations of Galois representations . . . . . . . . . . . . . 76 2.8 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3 Modular forms and Galois representations 84 3.1 From modular forms to Galois representations . . . . . . . . . 84 3.2 From Galois representations to modular forms . . . . . . . . . 88 3.3 Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.4 Isomorphism criteria . . . . . . . . . . . . . . . . . . . . . . . 98 3.5 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . 100 3.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4 Hecke algebras 106 4.1 Full Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . 106 4.2 Reduced Hecke algebras . . . . . . . . . . . . . . . . . . . . . 111 4.3 Proof of theorem 3.31 . . . . . . . . . . . . . . . . . . . . . . . 119 4.4 Proof of theorem 3.36 . . . . . . . . . . . . . . . . . . . . . . . 124 4.5 Homological results . . . . . . . . . . . . . . . . . . . . . . . . 133

5 Commutative algebra 137 5.1 Wiles’ numerical criterion . . . . . . . . . . . . . . . . . . . . 138 5.2 Basic properties of Φ_{A} and η_{A} . . . . . . . . . . . . . . . . . . 140 5.3 Complete intersections and the Gorenstein condition . . . . . 143 5.4 The Congruence ideal for complete intersections . . . . . . . . 148 5.5 Isomorphism theorems . . . . . . . . . . . . . . . . . . . . . . 149 5.6 A resolution lemma . . . . . . . . . . . . . . . . . . . . . . . . 152 5.7 A criterion for complete intersections . . . . . . . . . . . . . . 153 5.8 Proof of Wiles’ numerical criterion . . . . . . . . . . . . . . . 153 5.9 A reduction to characteristic ℓ . . . . . . . . . . . . . . . . . . 154 5.10 J-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

1 Elliptic curves and modular forms

1.1 Elliptic curves

We begin with a brief review of elliptic curves. A general reference for the results discussed in this section is [Si1] and [Si2].

An elliptic curve E over a field F is a proper smooth curve over F of genus one with a distinguished F -rational point. If E/F is an elliptic curve and if ω is a non-zero holomorphic differential on E/F then E can be realised in the projective plane by an equation (called a Weierstrass equation) of the form

(W) Y^{2}Z + a_{1}XY Z + a_{3}Y Z^{2} = X^{3} + a_{2}X^{2}Z + a_{4}XZ^{2} + a_{6}Z^{3}

such that the distinguished point is (0 : 1 : 0) (sometimes denoted ∞ because it corresponds to the “point at infinity” in the affine model obtained by setting Z = 1) and ω = \frac{dx}{2y+a_{1}x+a_{3}} . We also define the following quantities associated to (W):

b_{2} = a_{1}^{2} + 4a_{2} b_{4} = 2a_{4} + a_{1}a_{3} b_{6} = a_{3}^{2} + 4a_{6}
b_{8} = a_{1}^{2}a_{6} + 4a_{2}a_{6} − a_{1}a_{3}a_{4} + a_{2}a_{3}^{2} − a_{4}^{2}
Δ = 9b_{2}b_{4}b_{6} − b_{2}^{2}b_{8} − 8b_{4}^{3} − 27b_{6}^{2}
j = (b_{2}^{2} − 24b_{4})^{3}/Δ.

One can check that the equation (W) defines an elliptic curve if and only if Δ is nonzero. One can also check that such equations define elliptic curves which are isomorphic over F̄ if and only if they give the same quantity j. Thus j only depends on E so we will denote it j_{E}. The quantity Δ depends only on the pair (E, ω) so we shall denote it Δ(E, ω). If u belongs to F^{×} then u^{12}Δ(E, uω) = Δ(E, ω).

An elliptic curve E/F has a natural structure of a commutative algebraic group with the distinguished F -rational point as the identity element.

An algebraic map between two elliptic curves which sends the distinguished point of one to the distinguished point of the other is automatically a morphism of algebraic groups. A map between elliptic curves which has finite kernel (and hence, is generically surjective) is called an isogeny.

Elliptic curves over ℂ: If F = ℂ, then the curve E is isomorphic as a complex analytic manifold to the complex torus ℂ/Λ, where Λ is a lattice in ℂ, i.e., a discrete ℤ-submodule of ℂ of rank 2. The group law on E(ℂ) corresponds to the usual addition in ℂ/Λ. In terms of Λ, an affine equation for E in A^{2}(ℂ) is given by

y^{2} = 4x^{3} + g_{2}x + g_{3},

where

g_{2} = −60 ∑_{λ∈Λ−\{0\}} \frac{1}{z^{4}} , g_{3} = −140 ∑_{λ∈Λ−\{0\}} \frac{1}{z^{6}} .

In terms of this equation, the map from ℂ/Λ to E(ℂ) sends z to (x, y) = (℘(z), ℘′(z)), where ℘(z) is the Weierstrass ℘-function associated to the lattice Λ. (Cf. [Si1], ch. VI.) The inverse map is given by integrating the holomorphic differential ω, i.e., sending P ∈ E(ℂ) to the image of ∫_{γ} ω in ℂ/Λ, where γ is any path on E(ℂ) from ∞ to P, and Λ is the lattice of periods ∫_{γ} ω, where γ ranges over the integral homology H_{1}(E(ℂ), ℤ). Replacing ω by uω changes Λ to uΛ, so that Λ is determined by E only up to homotheties. We scale Λ so that one of its ℤ-generators is 1, and another, τ, has strictly positive imaginary part. This gives the analytic isomorphism:

E(ℂ) ≃ ℂ/⟨1, τ⟩.

The complex number τ in the complex upper half plane H is well defined, modulo the natural action of SL_{2}(ℤ) on H by Möbius transformations. (Thus the set of isomorphism classes of elliptic curves over ℂ can be identified with the quotient H/SL_{2}(ℤ).)

The map z ↦ e^{2πiz} identifies ℂ/⟨1, τ⟩ with ℂ^{×}/q^{ℤ}, where q = e^{2πiτ} is the multiplicative Tate period. The analytic isomorphism

E(ℂ) ≃ ℂ^{×}/q^{ℤ}

has the virtue of generalizing to the p-adic setting in certain cases, as we will see shortly.

Note that |q| < 1. The invariant j can be expressed in terms of q by a convergent power series with integer coefficients:

j = q^{−1} + 744 + 196884q + · · · .  (1.1.1)

The following basic facts are a direct consequence of the analytic theory:

Proposition 1.1 The subgroup E[n](ℂ) of points of order n on E(ℂ) is isomorphic (non-canonically) to ℤ/nℤ × ℤ/nℤ. More generally, if F is any field of characteristic zero, the subgroup E[n](F) is contained in ℤ/nℤ × ℤ/nℤ.

Proof: The analytic theory shows that E(ℂ) is isomorphic as an abstract group to a product of two circle groups, and the first statement follows. The second statement follows from the Lefschetz principle (cf. [Si1], ch. VI, §6). □

Proposition 1.2 The endomorphism ring End_{ℂ}(E) of an elliptic curve over ℂ is isomorphic either to ℤ or to an order in a quadratic imaginary field. The same is true if one replaces ℂ by any field of characteristic 0.

Proof: An endomorphism of E(ℂ) ≃ ℂ/Λ induces multiplication by complex number α on the tangent space. Hence End_{ℂ}(E) is isomorphic to the ring of α ∈ ℂ satisfying αΛ ⊂ Λ. Such a ring is isomorphic either to ℤ or to a quadratic imaginary order. The corresponding statement for fields of characteristic 0 follows as in the proof of proposition 1.1. □

If End_{ℂ}(E) ⊗ ℚ is a quadratic imaginary field, we say that E has complex multiplication.

Remark 1.3 It follows from the arithmetic theory of complex multiplication (cf. [Si2], ch. 1) that any elliptic curve E with complex multiplication is defined over an abelian extension of the quadratic imaginary field K = End_{ℂ}(E) ⊗ ℚ. If E is defined over ℚ, then K has class number one. There are only finitely many elliptic curves over ℚ with complex multiplication, up to “twists” (i.e., ℂ-isomorphism).

Elliptic curves over ℚ_{p}: Now suppose that E is an elliptic curve defined over the p-adic field ℚ_{p}. There is an equation

(W^{min}) Y^{2}Z + a_{1}XY Z + a_{3}Y Z^{2} = X^{3} + a_{2}X^{2}Z + a_{4}XZ^{2} + a_{6}Z^{3}

for E with the property a_{i} ∈ ℤ_{p} for all i and |Δ| is minimal amongst all such equations for E. Although (W^{min}) is not unique, the associated discriminant depends only on E and is denoted Δ_{E}^{min}. Moreover the reduction of (W^{min}) modulo the uniformizer p defines a projective curve Ē, which is independent of the particular minimal equation chosen. If (W) is any equation for E with coefficients in ℤ_{p} and with discriminant Δ, then Δ_{E}^{min} divides Δ.

If Ē is a smooth curve we say that E has good reduction at p. If Ē has a unique singular point which is a node we say that E has multiplicative reduction at p. Otherwise Ē has a unique singular point which is a cusp and we say that E has additive reduction at p. If E has good or multiplicative reduction we say that it has semi-stable reduction at p, or simply that E is semi-stable.

If (W) defines a smooth curve mod p then E has good reduction at p and (W) is a minimal equation. If Δ ≡ 0 mod p but b_{2}^{2} ≠ 24b_{4} mod p, then modulo p the equation (W) defines a curve with a node. In this case E has multiplicative reduction at p and (W) is a minimal equation.

Curves with good reduction: In that case p does not divide Δ_{E}^{min}, and the reduction Ē is an elliptic curve over 𝔽_{p}.

If q is any power of p, and 𝔽_{q} is the field with q elements, we define the integer N_{q} to be the number of solutions to the equation (W^{min}) in the projective plane ℙ^{2}(𝔽_{q}). Thus N_{q} is the order of the finite group Ē(𝔽_{q}). We define the integer a_{q} by the formula

a_{q} = q + 1 − N_{q}.

The integers a_{q} are completely determined by a_{p}: more precisely, we have

(1 − a_{p}p^{−s} + p^{1−2s})^{−1} = 1 + a_{p}p^{−s} + a_{p^{2}}p^{−2s} + a_{p^{3}}p^{−3s} + · · · . (1.1.2)

We call the expression on the left the (local) L-function associated to E over ℚ_{p}, and denote it by L(E/ℚ_{p}, s). Concerning the size of a_{p} we have the following fundamental result of Hasse, whose proof can be found in [Si1], ch. V, §1:

Theorem 1.4 |a_{p}| ≤ 2\sqrt{p}.

A further division among curves of good reduction plays a significant role in our later discussion. We say that E has (good) ordinary reduction if p does not divide a_{p}, and that it has supersingular reduction if p divides a_{p}.

When E has good reduction at p, we define its local conductor at p to be m_{p}(E) = 0.

Curves of multiplicative reduction: Elliptic curves over ℚ_{p} which have multiplicative reduction at p can be understood by using the p-adic analytic description discovered by Tate. More precisely, we can formally invert the power series (1.1.1) expressing j in terms of q, to obtain to a power series for q in j^{−1}, having integer coefficients:

q = j^{−1} + 744j^{−2} + 750420j^{−3} + 872769632j^{−4} + · · · .  (1.1.3)

If E has multiplicative reduction, then j ∈ ℚ_{p} is non-integral, and hence the power series (1.1.3) converges, yielding a well-defined value of q in pℤ_{p}. This is called Tate’s p-adic period associated to E over ℚ_{p}. Note that we have v_{p}(q) = −v_{p}(j) = v_{p}(Δ_{E}^{min}).

We say that E has split (resp. non-split) multiplicative reduction at p if the two tangent lines to the node on Ē(𝔽_{p}) have slopes defined over 𝔽_{p} (resp. 𝔽_{p^{2}}).

Proposition 1.5 (Tate) There is a p-adic analytic isomorphism

Φ : ℚ̄_{p}^{×}/q^{ℤ} ⟶ E(ℚ̄_{p}),

which has the property that

σ(Φ(x)) = Φ(σx^{δ(σ)}), ∀σ ∈ G_{ℚ_{p}},

where δ : G_{ℚ_{p}} ⟶ ±1 is

• the trivial character, if E has split multiplicative reduction;

• the unique unramified quadratic character of G_{ℚ_{p}}, if E has non-split multiplicative reduction.

The proof of this proposition is explained in [Si2], ch. V, for example. We define the L-function L(E/ℚ_{p}, s) to be

L(E/ℚ_{p}, s) = \{ \begin{matrix}(1 − p^{−s})^{−1} if E has split reduction, \\ (1 + p^{−s})^{−1} if E has non-split reduction. \end{matrix} (1.1.4)

In both cases the conductor m_{p}(E) is defined to be 1.

Curves of additive reduction: If E has additive reduction at p, we simply define

L(E/ℚ_{p}, s) = 1.  (1.1.5)

The conductor m_{p}(E) is defined to be 2, if p > 3. When p = 2 or 3, it is determined by a somewhat more complicated recipe, given in [Ta].

Elliptic curves over ℚ: Let E be an elliptic curve defined over ℚ. In particular E may be viewed as a curve over ℚ_{p} for every p, and we define its (global) conductor by

N_{E} = ∏_{p} p^{m_{p}(E)}.

The curve E is said to be semi-stable if it is semi-stable over all p-adic fields ℚ_{p}. Note that E is semi-stable if and only if its conductor N_{E} is square-free.

Using the fact that ℚ has class number 1, one can show that E has a global minimal Weierstrass model (W^{min}) which gives the equation of a minimal Weierstrass model over each ℚ_{p}. The associated discriminant, denoted Δ_{E}^{min}, depends only on E. The associated differential, denoted ω_{E}^{Neron}, is called the Néron differential. It is well-defined up to sign.

The following, known as the Mordell-Weil theorem, is the fundamental result about the structure of the group of rational points E(ℚ). (Cf. For example [Si1].)

Theorem 1.6 The group E(ℚ) is a finitely generated abelian group. Hence

E(ℚ) ≃ T ⊕ ℤ^{r},

where T is the (finite) torsion subgroup of E(ℚ), and r ≥ 0 is the rank of E over ℚ.

Concerning the possible structure of T, there is the following deep result of Mazur, a variant of which also plays a crucial role in the proof of Fermat’s Last Theorem:

Theorem 1.7 If E/ℚ is an elliptic curve, then its torsion subgroup is isomorphic to one of the following possibilities:

ℤ/nℤ, 1 ≤ n ≤ 10, n = 12, ℤ/2nℤ × ℤ/2ℤ, 1 ≤ n ≤ 4.

The proof is given in [Maz1] (see also [Maz2]). Thanks to this result, the structure of the torsion subgroup T is well understood. (Recently, the techniques of Mazur have been extended by Kamienny [Kam] and Merel [Mer] to prove uniform boundedness results on the torsion of elliptic curves over general number fields.)

Much more mysterious is the behaviour of the rank r. It is not known if r can be arbitrarily large, although results of Mestre [Mes] and Nagao [Na] show that it is greater or equal to 13 for infinitely many elliptic curves over ℚ. It turns out that many of the deep results on E(ℚ) and on r are based on the relation with L-functions.

We define the global L-function of the complex variable s by:

L(E/ℚ, s) = ∏_{p} L(E/ℚ_{p}, s).  (1.1.6)

Exercise 1.8 Using theorem 1.4, show that the infinite product defining the L-function L(E/ℚ, s) converges absolutely on the right half plane Real(s) > 3/2.

Conjecture 1.9 (Birch-Swinnerton-Dyer) The L-function L(E/ℚ, s) has an analytic continuation to the entire complex plane,and in particular is analytic at s = 1. Furthermore:

ord_{s=1}L(E/ℚ, s) = r.

There is also a more precise form of this conjecture, which expresses the leading coefficient of L(E/ℚ, s) at s = 1 in terms of certain arithmetic invariants of E/ℚ. For more details, see [Si1], conj. 16.5.

As we will explain in more detail in section 1.8, the analytic continuation of L(E/ℚ, s) now follows from the work of Wiles and Taylor-Wiles and a strengthening by Diamond [Di2], for a very large class of elliptic curves over ℚ, which includes all the semi-stable ones.

Abelian varieties: Elliptic curves admit higher-dimensional analogues, called abelian varieties, which also play a role in our discussion. Analytically, the set of complex points on an abelian variety is isomorphic to a quotient ℂ^{g}/Λ, where Λ is a lattice in ℂ^{g} of rank 2g, satisfying the so-called Riemann period relations. A good introduction to the basic theory of abelian varieties can be found in [CS] and [We1].

1.2 Modular curves and modular forms over ℂ

Modular curves: The group SL_{2}(ℤ) of two by two integer matrices of determinant one acts by fractional linear (Möbius) transformations on the complex upper half plane

H = {z ∈ ℂ | Im (z) > 0},

equipped with its standard complex analytic structure. The principal congruence group Γ(N) of level N is the subgroup of matrices in SL_{2}(ℤ) which reduce to the identity matrix modulo the positive integer N. A subgroup Γ of SL_{2}(ℤ) is called a congruence group if it contains Γ(N) for some N. The level of Γ is the smallest N for which this is true. The most important examples of congruence groups are:

• The group Γ_{0}(N) consisting of all matrices that reduce modulo N to an upper triangular matrix.

• The group Γ_{1}(N) consisting of all matrices that reduce modulo N to a

matrix of the form ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ).

• The principal congruence group Γ(N) of level N consisting of all matrices that reduce modulo N to the identity.

Notice the natural inclusions of normal subgroups Γ(N) ⊂ Γ_{1}(N) ⊂ Γ_{0}(N). The quotient Γ_{0}(N)/Γ_{1}(N) is canonically isomorphic to (ℤ/Nℤ)^{×} via

( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ d mod N.

For any subgroup H of (ℤ/Nℤ)^{×}, we denote by Γ_{H}(N) the group of matrices in Γ_{0}(N) whose image in Γ_{0}(N)/Γ_{1}(N) belongs to H.

If Γ is a congruence subgroup of SL_{2}(ℤ), define Y_{Γ} to be the quotient of the upper half plane H by the action of Γ. One equips Y_{Γ} with the analytic structure coming from the projection map π : H ⟶ Y_{Γ}. (More precisely, if y = π(τ), and G_{τ} ⊂ Γ is the stabilizer of τ in Γ, then the local ring O_{Y_{Γ},y} is identified with the local ring of germs of holomorphic functions at τ which are invariant under the action of G_{τ}.) This makes Y_{Γ} into a connected complex analytic manifold of dimension one, i.e., a Riemann surface. If Γ is Γ_{0}(N) (resp. Γ_{1}(N), or Γ(N)), we will also denote Y_{Γ} by Y_{0}(N) (resp. Y_{1}(N), or Y (N)). One compactifies Y_{Γ} by adjoining a finite set of cusps which correspond to orbits of ℙ^{1}(ℚ) = ℚ ∪ {∞} under Γ. Call X_{Γ} the corresponding compact Riemann surface. (For more details, notably on the definition of the analytic structure on X_{Γ} at the cusps, see for example [Kn], p. 311, or [Shi2], ch. 1.) It follows from the definition of this analytic structure that the field K_{Γ} of meromorphic functions on X_{Γ} is equal to the set of meromorphic functions on H satisfying

• (Transformation property): f(γτ) = f(τ), for all γ ∈ Γ;

• (Behaviour at the cusps): For all γ ∈ SL_{2}(ℤ), the function f(γτ) has a Puiseux series expansion ∑_{−m}^{∞} a_{n}q^{n/h} in fractional powers of q = e^{2πiτ}.

Riemann’s existence theorem (cf. for example [For], ch. 2) asserts that the analytic structure on X_{Γ} comes from an algebraic one, i.e., the field K_{Γ} is a finitely generated extension of ℂ of transcendence degree 1. Thus we can, and will, view X_{Γ} as a complex algebraic curve over ℂ. If Γ is Γ_{0}(N) (resp. Γ_{1}(N), or Γ(N)), we will also denote X_{Γ} by X_{0}(N) (resp. X_{1}(N), or X(N)).

Examples and exercises:

1. For N = 1, the curve X_{0}(N) = X_{1}(N) = X(N) is a curve of genus 0, and its field of functions is the ring ℂ(j), where j is the classical modular function,

j(τ) = q^{−1} + 744 + 196884q + ⋯, q = e^{2πiτ}.

(Cf., for example, [Se4], ch. 7.)

2. For τ ∈ H ∪ ℙ^{1}(ℚ) = H̄, define G_{τ} to be the stabilizer of τ in P SL_{2}(ℤ), and let e_{τ} = #(G_{τ}/(G_{τ} ∩ Γ)). Show that e_{τ} depends only on the Γ-orbit of τ in H̄, and that e_{τ} = 1 for all but finitely many τ in H̄/Γ. Using the Riemann-Hurwitz formula (cf. [Ki], sec. 4.3) show that the genus of X_{Γ} is given by

g(Γ) = 1 − [P SL_{2}(ℤ) : Γ] + \frac{1}{2} ∑_{τ∈H̄/Γ}(e_{τ} − 1).

Use this to compute the genus of X_{0}(p), X_{1}(p), and X(p) for p prime. For details, see [Shi2], sec. 1.6 or [Ogg].

3. For Γ = Γ(2), show that X_{Γ} is isomorphic to ℙ^{1}, and that Y_{Γ} is isomorphic to ℙ^{1} − {0, 1, ∞}. Show that Γ/⟨±1⟩ is the free group on the two generators

g_{1} = (\begin{matrix}1 & 2 \\ 0 & 1 \end{matrix}) and g_{2} = (\begin{matrix}1 & 0 \\ 2 & 1 \end{matrix}).

4. Define a homomorphism Γ(2) ⟶ ℤ/nℤ × ℤ/nℤ, by sending g_{1} to (1, 0) and g_{2} to (0, 1), and let Γ denote its kernel. Show that Γ is not in general a congruence subgroup and that the curve Y_{Γ} := H/Γ is birationally isomorphic to the Fermat curve of degree n with affine equation x^{n} + y^{n} = 1.

Moduli interpretations: The points in Y_{Γ} = H/Γ can be interpreted as elliptic curves over ℂ with some extra “level N” structure. More precisely,

• If Γ = Γ_{0}(N), then the Γ-orbit of τ ∈ H corresponds to the complex torus E = ℂ/⟨1, τ⟩ with the distinguished cyclic subgroup of order N generated by \frac{1}{N}. Thus, points on Y_{0}(N) parametrize isomorphism classes of pairs (E, C) where E is an elliptic curve over ℂ and C is a cyclic subgroup of E of order N.

• If Γ = Γ_{1}(N), then the Γ-orbit of τ corresponds to the complex torus E = ℂ/⟨1, τ⟩ with the distinguished point of order N given by \frac{1}{N}. Hence, points on Y_{1}(N) parametrize isomorphism classes of pairs (E, P) where now P is a point on E of exact order N.

Remark 1.10 One checks that the above rules set up a bijection between points on Y_{Γ} and elliptic curves with the appropriate structures, and that the projection Y_{1}(N) ⟶ Y_{0}(N) sending Γ_{1}(N)τ to Γ_{0}(N)τ becomes the “forgetful” map sending (E, P) to (E, ⟨P ⟩).

Remark 1.11 (This remark will be used in section 1.3 when discussing Hecke operators.) Define an n-isogeny of Γ-structures to be an n-isogeny of the underlying elliptic curves which sends one Γ-structure to the other. If p is a prime not dividing N, then there are exactly p + 1 distinct p-isogenies from (ℂ/⟨τ, 1⟩, \frac{1}{N}), whose images are the pairs:

(ℂ/ ⟨\frac{τ + i}{p}, 1⟩ , \frac{1}{N} ) (i = 0, …, p − 1), (ℂ/⟨pτ, 1⟩, \frac{p}{N} ) .

If p divides N, then there are only p distinct p-isogenies from (ℂ/⟨τ, 1⟩, \frac{1}{N}), since (ℂ/⟨pτ, 1⟩, \frac{p}{N}) is not a Γ_{1}(N)-structure (the point p/N not being of exact order N on the complex torus ℂ/⟨pτ, 1⟩).

Modular forms: Let k be an even positive integer. A modular form of weight k on Γ is a holomorphic function f on H satisfying:

• (Transformation property): f(γτ) = (cτ +d)^{k}f(τ), for all γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ.

• (Behaviour at the cusps): For all γ ∈ P SL_{2}(ℤ), the function (cτ + d)^{−k}f(γτ) has a Puiseux series expansion ∑_{0}^{∞} a_{n}q^{n/h} in fractional powers of q = e^{2πiτ}. We call ∑ a_{n}q^{n/h} the Fourier expansion of f at the cusp γ^{−1}(i∞).

A modular form which satisfies the stronger property that the constant coefficient of its Fourier expansion at each cusp vanishes is called a cusp form. We denote by M_{k}(Γ) the complex vector space of modular forms of weight k on Γ, and by S_{k}(Γ) the space of cusp forms on Γ. (For examples, see [DI], sec. 2.2 and the references therein, especially, [Shi2], ch. 2.)

This article is mainly concerned with modular forms of weight 2, and hence we will focus our attention on these from now on. A pleasant feature of the case k = 2 is that the cusp forms in S_{2}(Γ) admit a direct geometric interpretation as holomorphic differentials on the curve X_{Γ}.

Lemma 1.12 The map f(τ) ↦ ω_{f} := 2πif(τ)dτ is an isomorphism between the space S_{2}(Γ) and the space Ω^{1}(X_{Γ}) of holomorphic differentials on the curve X_{Γ}.

Sketch of proof: One checks that the transformation property satisfied by f(τ) under Γ causes the expression f(τ)dτ to be Γ-invariant, and that the condition of vanishing at the cusps translates into holomorphicity of f(τ)dτ. (Note, for example, that 2πidτ = dq/q, so that ω_{f} is holomorphic at i∞ precisely when f(q) vanishes at q = 0.) □

As a corollary, we find:

Corollary 1.13 The space S_{2}(Γ) is finite-dimensional, and its dimension is equal to the genus g of X_{Γ}.

Proof: This follows directly from the Riemann-Roch theorem, cf. [Ki], sec. 6.3. □

To narrow still further the focus of our interest, we will be mostly concerned with the cases Γ = Γ_{0}(N) and Γ_{1}(N). A slightly more general framework is sometimes convenient, so we suppose from now on that Γ satisfies

Γ_{1}(N) ⊂ Γ ⊂ Γ_{0}(N).

Such a group Γ is necessarily of the form Γ_{H}(N) for some subgroup H of (ℤ/Nℤ)^{×}. Because the transformation τ ↦ τ + 1 belongs to Γ the forms in S_{2}(Γ) are periodic functions on H of period 1, and hence their Fourier expansions at i∞ are of the form

f(τ) = ∑_{n>0} a_{n}q^{n}, q = e^{2πiτ}, a_{n} ∈ ℂ.

The Petersson inner product: The spaces S_{2}(Γ) are also equipped with a natural Hermitian inner product given by

⟨f, g⟩ = \frac{i}{8π^{2}} ∫_{X_{Γ}} ω_{f} ∧ ω̄_{g} = ∫_{H/Γ} f(τ)ḡ(τ)dxdy,

where τ = x + iy. This is called the Petersson inner product.

The diamond operators: Suppose now that Γ = Γ_{1}(N) and let d be an element of (ℤ/Nℤ)^{×}. The map ⟨d⟩ which sends an elliptic curve with Γ-structure (E, P) to the pair (E, dP) gives an automorphism of Y_{Γ} which extends to X_{Γ}. It is called the diamond operator. For τ in H and γ = (\begin{matrix}a & b \\ c & d \end{matrix}) in Γ_{0}(N), we have

⟨d⟩(Γτ) = Γ(γτ).

Hence ⟨d⟩ acts on S_{2}(Γ), identified with the holomorphic differentials on X_{Γ}, by the rule

⟨d⟩f(τ) = (cτ + d)^{−2}f (\frac{aτ + b}{cτ + d}) .

In geometric terms, the diamond operators are the Galois automorphisms of the natural (branched) covering X_{1}(N) ⟶ X_{0}(N) whose Galois group is isomorphic to Γ_{0}(N)/⟨±Γ_{1}(N)⟩ = (ℤ/Nℤ)^{×}/⟨±1⟩. Given an even Dirichlet character χ : (ℤ/Nℤ)^{×} ⟶ ℂ^{×}, say that f is a modular form of level N and character χ if it belongs to the χ-eigenspace in S_{2}(Γ_{1}(N)) under this action. Let S_{2}(N, χ) denote the space of all such forms. Thus a function f in S_{2}(N, χ) is a cusp form on Γ_{1}(N) which satisfies the stronger transformation property:

f (\frac{aτ + b}{cτ + d}) = χ(d)(cτ + d)^{2}f(τ), for all (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(N).

Note that if 1 is the trivial character, then S_{2}(N, 1) is canonically identified with S_{2}(Γ_{0}(N)), which we will also denote by S_{2}(N). Finally note the direct sum decomposition:

S_{2}(Γ_{1}(N)) = ⊕_{χ}S_{2}(N, χ),

where the sum ranges over all the even Dirichlet characters modulo N.

Exercise 1.14 Show that if f(τ) belongs to S_{2}(N), then f(aτ) belongs to S_{2}(mN), for each integer a dividing m.

Jacobians of modular curves: Let V be the dual space

V = S_{2}(Γ)^{∨} := Hom(S_{2}(Γ), ℂ).

It is a complex vector space of dimension g = genus(X_{Γ}). The integral homology Λ = H_{1}(X_{Γ}, ℤ) maps naturally to V by sending a homology cycle c to the functional ϕ_{c} defined by ϕ_{c}(f) = ∫_{c} ω_{f}. The image of Λ is a lattice in V, i.e., a ℤ-module of rank 2g which is discrete (cf. [Mu1], cor. 3.8). Fix a base point τ_{0} ∈ H, and define the Abel-Jacobi map Φ_{AJ} : X_{Γ}(ℂ) ⟶ V/Λ by Φ_{AJ}(P)(f) = ∫_{τ_{0}}^{P} ω_{f}. Note that this is well-defined, i.e., it does not depend on the choice of path on X_{Γ} from τ_{0} to P, up to elements in Λ.

We extend the map Φ_{AJ} by linearity to the group Div (X_{Γ}) of divisors on X_{Γ}, and observe that the restriction of Φ_{AJ} to the group Div^{0}(X_{Γ}) of degree 0 divisors does not depend on the choice of base-point τ_{0}. Moreover we have the Abel-Jacobi theorem:

Theorem 1.15 The map

Φ_{AJ} : Div^{0}(X_{Γ}) ⟶ V/Λ

has a kernel consisting precisely of the group P (X_{Γ}) of principal divisors on X_{Γ}. Hence Φ_{AJ} induces an isomorphism from Pic^{0}(X_{Γ}) := Div^{0}(X_{Γ})/P (X_{Γ}) to V/Λ.

For the proof, see [Mu1], ch. 3. The quotient V/Λ is a complex torus, and is equal to the group of complex points of an abelian variety. We denote this abelian variety by J_{Γ}, the Jacobian variety of X_{Γ} over ℂ. If Γ = Γ_{0}(N) or Γ_{1}(N), we will also write J_{0}(N) or J_{1}(N) respectively for the Jacobian J_{Γ}.

1.3 Hecke operators and Hecke theory

We maintain our running assumption that Γ satisfies

Γ_{1}(N) ⊂ Γ ⊂ Γ_{0}(N).

If p is a prime not dividing the level N, we define the Hecke operator T_{p} on S_{2}(Γ) by the formula

T_{p}(f) = \frac{1}{p} ∑_{i=0}^{p−1} f (\frac{τ + i}{p}) + p⟨p⟩f(pτ).

We give a more conceptual description of T_{p} in terms of remark 1.11 in the case Γ = Γ_{1}(N). We have

ω_{T_{p}(f)} = ∑ ϕ_{i}^{∗}(ω_{f}),

where ϕ_{i}(τ) = \frac{τ+i}{p}, and ϕ_{∞}(τ) = ⟨p⟩pτ represent the p + 1 curves with Γstructure that are images of (ℂ/⟨τ, 1⟩, \frac{1}{N}) by a p-isogeny, and the ϕ_{i}^{∗} are the pull-back maps on differential forms on H. (An isogeny of elliptic curves with Γ-structure is simply an isogeny between the underlying curves which sends one Γ-structure to the other.) Such a description makes it evident that T_{p}(f) belongs to S_{2}(Γ), if f does. In terms of the Fourier expansion of f = ∑ a_{n}q^{n}, the formula for the operator T_{p} on S_{2}(N, χ) is given by:

T_{p}(f) = ∑_{p|n} a_{n}q^{n/p} + pχ(p) ∑ a_{n}q^{pn}.

If p divides N, then we define the Hecke operator U_{p} analogously, by summing again over all the cyclic p-isogenies of Γ-structures. Since there are only p of them, the formula becomes simpler:

U_{p}(f) = \frac{1}{p} ∑_{i=0}^{p−1} f (\frac{τ + i}{p}) = ∑_{p|n} a_{n}q^{n/p}.

The reader is invited to check that the Hecke operators of the form T_{p} or U_{q} commute with each other, and also that they commute with the diamond operators introduced in the previous section.

We extend the definition of the Hecke operators to operators T_{p^{n}}, with n > 1, by the inductive formulae

T_{p^{n+1}} = T_{p}T_{p^{n}} − ⟨p⟩pT_{p^{n−1}}, if (p, N) = 1,

and T_{p^{n}} = U_{p}^{n} otherwise. We then define the operator T_{n}, where n = ∏ p_{i}^{e_{i}} is written as a product of powers of distinct primes p_{i}, by

T_{n} = ∏_{i} T_{p_{i}^{e_{i}}}.

This definition makes the Hecke operators multiplicative, i.e., T_{m}T_{n} = T_{mn} if (m, n) = 1. (A more conceptual definition of the Hecke operator T_{n} is that T_{n}(f) is obtained by summing the pullback of ω_{f} over the maps describing all the cyclic n-isogenies of Γ-structures.) The relations among the different Hecke operators can be stated succinctly by saying that they obey the following (formal) identity:

∏_{p |̸N}(1 − T_{p}p^{−s} + ⟨p⟩p^{1−2s})^{−1} ∏_{p|N}(1 − U_{p}p^{−s})^{−1} = ∑_{n} T_{n}n^{−s}. (1.3.1)

The reader can consult [DI], sec. 3 and the references therein (especially [Shi2], ch. 3 or [Kn]) for more details and different points of view on Hecke operators. Let 𝕋 be the subring of End_{ℂ}(S_{2}(Γ)) generated over ℂ by all the Hecke operators T_{p} for p |̸N, U_{q} for q|N, and ⟨d⟩ acting on S_{2}(Γ).

Definition 1.16 A modular form f is an eigenform if it is a simultaneous eigenvector for all the Hecke operators in 𝕋, i.e., if there exists a ℂ-algebra homomorphism λ : 𝕋 ⟶ ℂ such that T f = λ(T)f, for all T ∈ 𝕋.

A direct calculation shows that the coefficients a_{n} of an eigenform f can be recovered from the homomorphism λ by the formula:

a_{n}(f) = a_{1}(f)λ(T_{n}).

It follows that the first Fourier coefficient a_{1} of a non-zero eigenform is always non-zero, and that the non-trivial eigenspaces for 𝕋 are all one-dimensional:

Proposition 1.17 Given a non-zero algebra homomorphism λ : 𝕋 ⟶ ℂ, there is exactly one eigenform f up to scaling, which satisfies T f = λ(T)f, for all T ∈ 𝕋.

Sketch of proof: The proof of the existence of f is an exercise in commutative algebra (localize S_{2}(Γ) at the kernel of λ), and the uniqueness is clear from the formula above. □

We call an eigenform satisfying a_{1} = 1 a normalized eigenform.

Atkin-Lehner theory: It is natural to ask whether S_{2}(Γ) can be decomposed into a basis consisting of distinct normalized eigenforms. Unfortunately, this is not always possible, as the following exercise illustrates.

Exercise 1.18 Suppose that p^{3} divides N exactly. Let 𝕋′ be the algebra of Hecke operators (generated by the operators T_{q} with q |̸N/p^{3}, and U_{q} with q|N/p^{3}) acting on S_{2}(N/p^{3}). Let f = ∑_{n=1}^{∞} a_{n}q^{n} be a 𝕋′-eigenform of level N/p^{3} in S_{2}(N/p^{3}). Show that the space S_{f} spanned by the forms f(τ), f(pτ), f(p^{2}τ), and f(p^{3}τ) is contained in S_{2}(N), and is stable for the action of the Hecke operators T_{q}, q |̸N, and U_{q}, q|N. Show that S_{f} has no basis of simultaneous eigenforms for the Hecke algebra 𝕋 of level N, so that the action of 𝕋 on S_{f} is not semi-simple.

Let 𝕋^{0} denote the subalgebra of 𝕋 generated only by the “good” Hecke operators T_{q} with q |̸N, and ⟨d⟩.

Proposition 1.19 If q does not divide N, the adjoint of the Hecke operator T_{q} with respect to the Petersson scalar product is the operator ⟨q⟩^{−1}T_{q}, and the adjoint of ⟨q⟩ is ⟨q⟩^{−1}. In particular, the Hecke operators commute with their adjoints.

Proof: See [Kn], th. 9.18 and 8.22, or [Ogg]. □

Proposition 1.19 implies, by the spectral theorem for commuting operators that commute with their adjoints:

Proposition 1.20 The algebra 𝕋^{0} is semi-simple (i.e, it is isomorphic to a product ℂ × ⋯ × ℂ of a certain number of copies of ℂ), and there is a basis of S_{2}(Γ) consisting of simultaneous eigenvectors for the operators T_{q}.

Thus, 𝕋^{0} has the merit of being semi-simple, while 𝕋 is not in general. The cost of replacing 𝕋 by 𝕋^{0}, however, is that one loses “multiplicity one”, i.e., the eigenspaces for 𝕋^{0} need not be one-dimensional. For example, the space S_{f} defined in the previous exercise is contained in a single eigenspace for 𝕋^{0}.

The theory of Atkin-Lehner [AL] gives essentially a complete understanding of the structure of the algebra 𝕋, and the structure of the space of eigenforms. To motivate the main result, observe that the problem in the exercise above seems to be caused by forms of level N that are coming from forms of lower level N/p^{3} by a straightforward operation, and are therefore not “genuinely” of level N. They are the analogues, in the context of modular forms, of nonprimitive Dirichlet characters.

Definition 1.21 We define the old subspace of S_{2}(Γ) to be the space spanned by those functions which are of the form g(az), where g is in S_{2}(Γ_{1}(M)) for some M < N and aM divides N. We define the new subspace of S_{2}(Γ) to be the orthogonal complement of the old subspace with respect to the Petersson scalar product. A normalized eigenform in the new subspace is called a newform of level N.

The following result is the main consequence of the theory of Atkin-Lehner. It gives a complete answer to the question of what is the structure of the algebra 𝕋 acting on S_{2}(Γ).

Theorem 1.22 If f is in the new subspace of S_{2}(Γ) and is an eigenvector for all the operators in 𝕋^{0}, then it is also an eigenform for 𝕋, and hence is unique up to scaling. More generally, if f is a newform of level N_{f}|N, then the space S_{f} defined by

S_{f} = {g ∈ S_{2}(Γ) such that T g = λ_{f}(T)g, for all T ∈ 𝕋^{0}}

is stable under the action of all the Hecke operators in 𝕋. It is spanned by the linearly independent forms f(az) where a ranges over the divisors of N/N_{f}. Furthermore, we have

S_{2}(Γ) = ⊕_{f}S_{f},

where the sum is taken over all newforms f of some level N_{f} dividing N.

See [AL] for the proof in the case Γ = Γ_{0}(N), and [La2], ch. VIII for the general case. (See also [DI], sec. 6 for an overview.)

Exercise 1.23 Consider the case where Γ = Γ_{0}(22). Show that X_{0}(22) is of genus 2, and hence that S_{2}(22) has dimension 2. Show that S_{2}(22) is equal to S_{f}, where f = (η(τ)η(11τ))^{2} is a newform of level 11, so that in particular there are no newforms of level 22 on Γ. Show that 𝕋^{0} is isomorphic to ℂ in this case, and that 𝕋 is isomorphic to the semisimple algebra ℂ × ℂ.

Action on homology and Jacobians: Note that the Hecke operators act on V = S_{2}(Γ)^{∨} by duality. One checks (cf. [Kn], props. 11.23, 11.24) that the sublattice Λ of V is stable under the action of all the Hecke operators T_{n}, and of the diamond operators ⟨d⟩. Therefore the operators T_{n} and ⟨d⟩ give rise to endomorphisms of the torus V/Λ, and hence the Jacobian variety J_{Γ}, in a natural way. The involution τ ↦ −τ̄ gives rise to an involution on X_{Γ}(ℂ) (which is complex conjugation on the model of X_{Γ} over ℝ deduced from the ℚ- model defined in section 1.5). Since complex conjugation is continuous it also acts on the integral homology Λ = H_{1}(X_{Γ}(ℂ), ℤ). Let Λ^{+} and Λ^{−} be the sublattices of Λ on which complex conjugation acts by +1 and −1. These are sublattices of Λ of rank g which are stable under the Hecke operators, since complex conjugation commutes with the Hecke action.

A more algebraic description of the action of T_{p} on J_{Γ} is given via the notion of an algebraic correspondence. A correspondence on a curve X is a divisor C on X × X which is taken modulo divisors of the form {P } × X and X × {Q}. Let π_{1} and π_{2} denote the projections of X × X onto each factor. Then the correspondence C induces a map on divisors of X, by setting

C(D) = π_{2}(π_{1}^{−1}(D) · C).

(For the definition of the intersection D_{1} · D_{2} of two divisors, see [We1].) The map C preserves divisors of degree 0, and sends principal divisors on X to principal divisors. It gives a well defined algebraic endomorphism of the Jacobian variety Jac (X). Given a correspondence C, its transpose C^{∨} is defined to be the divisor of X × X obtained by interchanging the two factors of X × X. One can define a natural notion of composition for correspondences, and the set of correspondences forms a ring. The general theory of correspondences and the proofs of the above facts are given in [We1], particularly the second chapter.

The Hecke correspondence T_{n} is defined as the closure in X_{Γ} × X_{Γ} of the locus of points (A, B) in Y_{Γ} × Y_{Γ} such that there is a degree n cyclic isogeny of elliptic curves with Γ-structure from A to B. For example, if p is a prime not dividing N, then T_{p} is an algebraic curve in X_{1}(N)×X_{1}(N) which is birational to X_{Γ_{1}(N)∩Γ_{0}(p)}. The induced map on divisors in this case satisfies

T_{p}((E, P)) = ∑(E/C, P mod C)

where the sum runs over the subgroups C of E having order p. Note also that if (A, B) belongs to T_{p}, then the isogeny dual to A ⟶ B gives a p-isogeny from B to ⟨p⟩A, so that

T_{p}^{∨} = ⟨p⟩^{−1}T_{p}.

1.4 The L-function associated to a cusp form

For this section, let f in S_{2}(Γ_{1}(N)) be a cusp form with Fourier expansion at i∞ given by ∑_{n} a_{n}q^{n}. One has the following estimate for the size of the Fourier coefficients a_{n}:

Theorem 1.24 The coefficients a_{n} ∈ ℂ satisfy the inequality

|a_{n}| ≤ c(f)σ_{0}(n)\sqrt{n},

where c(f) is a constant depending only on f, and σ_{0}(n) denotes the number of positive divisors of n.

Sketch of proof: This follows from proposition 1.51 of section 1.7 which relates the p-th Fourier coefficients of eigenforms, for p a prime not dividing the level of Γ, to the number of points on certain abelian varieties over the finite field 𝔽_{p}. The estimates of Hasse and Weil for the number of points on abelian varieties over finite fields (stated in theorem 1.4 of section 1.1 for the special case of elliptic curves; see [We1], §IV for the general case) thus translate into asymptotic bounds for the Fourier coefficients of these eigenforms. We note that the cruder estimate |a_{n}| = O(n), which is enough for the purposes of this section, can be derived by a more elementary, purely analytic argument; cf. [Ogg], ch. IV, prop. 16. □

The L-function associated to f is defined by the formula:

L(f, s) = ∑ a_{n}n^{−s}.

As in exercise 1.8, one can show that the infinite sum defining L(f, s) converges absolutely in the right half-plane Re (s) > \frac{3}{2}. A much better insight is gained into the function L(f, s) by noting that it is essentially the Mellin transform of the modular form f. More precisely, if we set Λ(f, s) = N^{s/2}(2π)^{−s}Γ(s)L(f, s), then we have

Λ(f, s) = N^{s/2} ∫_{0}^{∞} f(iy)y^{s}dy/y  (1.4.1)

Exercise 1.25 Check the formula above.

This integral representation for L(f, s) gives the analytic continuation of L(f, s) to the entire complex plane. The modular invariance of f translates into a functional equation for L(f, s): more precisely, let w_{N} be the Atkin-Lehner involution defined by w_{N}(τ) = −1/Nτ. The reader may check that w_{N} induces an involution of X_{Γ}, and hence of Ω^{1}(X_{Γ}) = S_{2}(Γ). One finds that L(f, s) satisfies the functional equation:

Λ(f, s) = −Λ(w_{N}(f), 2 − s).

For a proof of this, see [Ogg], ch. V, lemma 1. Eigenforms for 𝕋 in S_{2}(N, χ) have a great importance in the theory because their associated L-functions have an Euler product expansion, in addition to an analytic continuation and functional equation:

Theorem 1.26 If f = ∑ a_{n}q^{n} is a normalized eigenform in S_{2}(N, χ) for all the Hecke operators, then the L-function L(f, s) = ∑ a_{n}n^{−s} has the Euler product expansion

∏_{p |̸N}(1 − a_{p}p^{−s} + χ(p)p^{1−2s})^{−1} ∏_{p|N}(1 − a_{p}p^{−s})^{−1}.

Proof: This follows directly from equation (1.3.1) of section 1.3. □

If f is a newform of level N, then it is also an eigenform for w_{N}, so that the functional equation may be viewed as relating L(f, s) and L(f, 2 − s). We can also state the following more precise version of theorem 1.24 (see lemma 3.2 of [Hi2] for example for parts (b), (c) and (d)).

Theorem 1.27 Suppose that f is a newform of level N_{f} and let N_{χ} denote the conductor of its character χ.

(a) If p does not divide N_{f} then |a_{p}| ≤ 2\sqrt{p}.

(b) If p||N_{f} and p does not divide N_{χ} then a_{p}^{2} = χ_{0}(p) where χ_{0} is the primitive character associated to χ.

(c) If p divides N_{f} and p does not divide N_{f}/N_{χ} then |a_{p}| = \sqrt{p}.

(d) If p^{2} divides N_{f} and p divides N_{f}/N_{χ} then a_{p} = 0.

1.5 Modular curves and modular forms over ℚ

Modular curves: For Γ between Γ_{0}(N) and Γ_{1}(N), the modular curve X_{Γ} has a model over ℚ. We describe such a model in the case of Γ = Γ_{0}(N); the construction for general Γ follows from similar considerations.

The key remark here is that, as was noted in section 1.2, the complex points on the curve Y_{0}(N) have a natural interpretation as moduli of elliptic curves together with a cyclic subgroup of order N, given by sending the point τ ∈ H/Γ_{0}(N) to the pair (ℂ/⟨1, τ⟩, ⟨1/N⟩).

Consider the “universal elliptic curve”

E_{j} : y^{2} + xy = x^{3} − \frac{36}{j − 1728}x − \frac{1}{j − 1728}.

It is an elliptic curve over the function field ℚ(j), with j-invariant j. Let d be the order of ℙ^{1}(ℤ/Nℤ), and let C_{1}, …, C_{d} denote the set of all cyclic subgroups of E_{j} of order N, defined over \overline{ℚ(j)}, an algebraic closure of ℚ(j). Fix one of these subgroups, C. The Galois group Gal (\overline{ℚ(j)}/ℚ(j)) permutes the C_{i} in a natural way. Let F_{N} be the smallest extension of ℚ(j) (viewed as embedded in \overline{ℚ(j)}) with the property that σ(C) = C, for all σ ∈ Gal (\overline{ℚ(j)}/F_{N}). It can be seen (cf. [Shi2], thm. 6.6) that the Galois action on the C_{i} is transitive so that F_{N}/ℚ(j) is of degree d, and that it is a regular extension, i.e., F_{N} ∩ ℚ̄ = ℚ. Geometrically, F_{N} can be viewed as the function field of a curve X_{/ℚ} over ℚ, with the inclusion of ℚ(j) corresponding to a map from X_{/ℚ} to the projective “j-line” over ℚ. The pair (E_{j}, C) is an elliptic curve over ℚ(j) with a subgroup of order N defined over F_{N}. Using (E_{j}, C), each complex point of X gives an elliptic curve over ℂ with a subgroup of order N, provided the point does not lie over j = 0, 1728 or ∞. The resulting map to X_{0}(N) extends to an isomorphism from X to X_{0}(N). The curve X thus constructed, together with this identification, is the desired model of X_{0}(N) over ℚ.

More concretely, the functions j = j(τ) and j_{N} = j(Nτ) are related by a polynomial equation Φ_{N}(j, j_{N}) = 0 with coefficients in ℚ, of bidegree d. The field F_{N} is the function field of the affine curve Φ_{N}(X, Y) = 0, and the mapping τ ↦ (j(τ), j(Nτ)) gives a birational equivalence between H/Γ_{0}(N) and the complex curve defined by the equation Φ_{N}. In practice it is not feasible to write down the polynomial Φ_{N}, except for certain very small values of N. To study the models over ℚ of X_{0}(N), more indirect methods are needed, which rely crucially on the moduli interpretation of X_{0}(N). Similar remarks hold for X_{1}(N).

Models over ℤ: The work of Igusa [Ig], Deligne-Rapoport [DR], Drinfeld [Dr], and Katz-Mazur [KM] uses the moduli-theoretic interpretation to describe a canonical proper model for X_{Γ} over Spec ℤ. These models allow us to talk about the reduction of X_{Γ} over finite fields 𝔽_{p}, for p prime. The curve has good reduction at primes p not dividing N, with the “non-cuspidal” points of X_{Γ/𝔽_{p}} corresponding to elliptic curves over 𝔽̄_{p} with Γ-structure. The singular fibers at primes p dividing N can also be described precisely; an important special case (see [DR]) is that of Γ_{0}(N) with p exactly dividing N. For further discussion and references, see [DI], sec. 8, 9.

From now on, when we write X_{Γ}, X_{0}(N), or X_{1}(N), we will mean the curve over ℚ which are the models described above for the complex curves defined in section 1.2.

Remark 1.28 When considering q-expansions, it is more convenient to use a different set of models over ℤ for these complex curves. We define X_{1}^{μ}(N) in the case of Γ_{1}(N) as a model over ℤ which parametrizes pairs (E, i) where i is an embedding of μ_{N} in the (generalized) elliptic curve E. (So X_{1}^{μ}(N) is the model denoted X_{μ}(N) in [DI], sec. 9.3, assuming N > 4.) For Γ between Γ_{0}(N) and Γ_{1}(N), we define X_{Γ}^{μ} as the corresponding quotient of X_{1}^{μ}(N). This model has good reduction at primes p not dividing N, but unlike the models mentioned above, its fibers at primes p dividing N are smooth and irreducible, but not proper. In the case of Γ = Γ_{0}(N), the curve X_{Γ,ℚ}^{μ} can be identified with X_{0}(N). However, this is not the case in general: the cusp ∞ is a rational point of X_{Γ,ℚ}^{μ} but not necessarily of X_{Γ}.

Jacobians: Weil’s theory [We1] of the Jacobian shows that the Jacobians J_{Γ} defined in section 1.2 as complex tori also admit models over ℚ. When we speak of J_{Γ}, J_{0}(N) and J_{1}(N) from now on, we will refer to these as abelian varieties defined over ℚ. Thus, the points in J_{Γ}(K), for any ℚ-algebra K, are identified with the divisor classes on X_{Γ} of degree 0, defined over K.

We let J_{Γ/ℤ}, denote the Néron model of the Jacobian J_{Γ} over Spec (ℤ). Using this model we define J_{Γ/A} for arbitrary rings A. In particular we can consider J_{Γ/𝔽_{p}}, the reduction of the Jacobian in characteristic p, which is closely related to the reduction of the integral model of the curve X_{Γ} mentioned above. In particular, if p does not divide the level of Γ, then J_{Γ/𝔽_{p}} can be identified with the Jacobian of X_{Γ/𝔽_{p}}. For a treatment of the case Γ = Γ_{0}(N) with p exactly dividing N, see the appendix of [Maz1]; for more general discussion and references, see [DI], sec. 10, especially sec. 10.3.

Hecke operators: The Hecke operators have a natural moduli interpretation, which was already touched upon in section 1.3. In particular, one finds that the operator T_{n} arises from a correspondence which is defined over ℚ, and gives rise to an endomorphism of the Jacobian J_{Γ} which is defined over ℚ. This in turn gives rise to an endomorphism of the Néron model J_{Γ/ℤ}, and we can then consider the endomorphism T_{n} on the reduction of the Jacobian in characteristic p. Recall that if p is a prime not dividing N, we may identify this reduction with the Jacobian of X_{Γ/𝔽_{p}}. (Cf. [MW], ch. 2, sec. 1, prop. 2.) Furthermore, one can show that the moduli-theoretic interpretation of the Hecke operator remains valid in characteristic p; i.e., the endomorphism T_{n} of J_{Γ/𝔽_{p}} is induced by a map on divisors satisfying, for all ordinary elliptic curves A with Γ-structure:

T_{n}(A) = ∑_{i} i(A),

where the sum is taken over all cyclic isogenies of degree n. (See [DI], sec. 10.2 for further discussion and references.)

This description allows one to analyse, for example, the Hecke operator T_{p} over 𝔽_{p}, when (p, N) = 1. Let us work with Γ = Γ_{1}(N), to illustrate the idea. For a variety X over 𝔽̄_{p}, let ϕ_{X} be the Frobenius morphism on X defined by raising coordinates to the pth power. Thus if (E, P) corresponds to a point of X_{1}(N)_{/𝔽_{p}}, then ϕ_{E} is an isogeny of degree p from (E, P) to the pair (E_{∞}, P_{∞}) = ϕ_{X_{1}(N)}(E, P). The graph of ϕ_{X_{1}(N)} in (X_{1}(N) × X_{1}(N))_{/𝔽_{p}} is a correspondence of degree p, which we call F. Let F′ be the transpose of this correspondence. The endomorphism F of J_{Γ} induced by F is the Frobenius endomorphism ϕ_{J_{Γ}}, and the endomorphism F′ is the dual endomorphism (in the sense of duality of abelian varieties). Now consider the divisor

F′((E, P)) = (E_{1}, P_{1}) + ⋯ + (E_{p}, P_{p}),

where the (E_{i}, P_{i}) are elliptic curves with Γ-structure in characteristic p. Since ϕ_{E_{i}} is an isogeny of degree p from (E_{i}, P_{i}) to (E, P), we also have the dual isogeny from (E, P) to (E_{i}, pP_{i}). If E is ordinary at p, then the points (E_{∞}, P_{∞}), (E_{1}, pP_{1}), …, (E_{p}, pP_{p}) are a complete list of the distinct curves with Γ-structure which are p-isogenous to (E, P). Hence one has the equality of divisors on X_{1}(N)_{/𝔽_{p}}:

T_{p}((E, P)) = (E_{∞}, P_{∞}) + (E_{1}, pP_{1}) + ⋯ + (E_{p}, pP_{p}) = (F + ⟨p⟩F′)((E, P)).

Since the ordinary points are dense on X_{1}(N)_{/𝔽_{p}}, we deduce that T_{p} = (F + ⟨p⟩F′) as endomorphisms of J_{1}(N)_{/𝔽_{p}}. This equation, known as Eichler-Shimura congruence relation, plays a central role in the theory. (For more details, see [DI], sec. 10.2, 10.3.)

Theorem 1.29 If p |̸N then the endomorphism T_{p} of J_{Γ/𝔽_{p}} satisfies

T_{p} = F + ⟨p⟩F′.

Remark 1.30 This was proved by Eichler [Ei] to hold for all but finitely many p in the case of Γ_{0}(N), and by Shimura ([Shi1], see also [Shi2], ch. 7) in the case of Γ_{1}(N). The fact that it holds for all p not dividing N follows from work of Igusa [Ig].

Modular forms: In the same way that modular curves have models over ℚ and over ℤ, the Fourier coefficients of modular forms also have natural rationality and integrality properties. We start by sketching the proof of:

Theorem 1.31 The space S_{2}(Γ) has a basis consisting of modular forms with integer Fourier coefficients.

Proof: The Hecke operators act on the integral homology Λ^{+} in a way that is compatible with the action on S_{2}(Γ) and respects the natural (Poincaré) duality between these two spaces. Hence, if {λ_{n}}_{n∈ℕ} is a system of eigenvalues for the T_{n}, then the λ_{n} are algebraic integers in some finite extension K of ℚ, and the system {λ_{n}^{σ}}_{n∈ℕ} is a system of eigenvalues for the T_{n} for any Galois automorphism σ of ℚ̄/ℚ. Hence, we have shown:

Proposition 1.32 If f ∈ S_{2}(M, χ) is a newform of some level M dividing N, then its Fourier coefficients lie in a finite extension K of ℚ. Moreover, if σ ∈ Gal (ℚ̄/ℚ) is any Galois automorphism, then the Fourier series f^{σ} obtained by applying σ to the Fourier coefficients is a newform in S_{2}(M, χ^{σ}).

The explicit description of S_{2}(Γ) given in section 1.3 implies that S_{2}(Γ) is spanned by forms having Fourier coefficients which are algebraic integers in some finite (Galois) extension K of ℚ, and that the space of forms with Fourier coefficients in K is stable under the natural action of Gal (K/ℚ) on Fourier expansions. An application of Hilbert’s theorem 90 shows that S_{2}(Γ) has a basis consisting of forms with rational Fourier expansions, and the integrality of the Fourier coefficients of eigenforms yields the integrality statement of theorem 1.31. □

We define S_{2}(Γ, ℤ) to be the space of modular forms with integral Fourier coefficients in S_{2}(Γ). Theorem 1.31 states that S_{2}(Γ, ℤ) ⊗ ℂ = S_{2}(Γ). Given any ring A, we define

S_{2}(Γ, A) = S_{2}(Γ, ℤ) ⊗ A,

and define S_{2}(N, A) and S_{2}(N, χ, A) (where χ now is a character with values in A^{×}) in the obvious way. If A is contained in ℂ, the q-expansion principle below allows us to identify S_{2}(Γ, A) with the set of modular forms in S_{2}(Γ) with Fourier coefficients in A.

The q-expansion principle: Because the modular curve X_{0}(N) has a model over ℚ, the space of modular forms S_{2}(N) = Ω^{1}(X_{0}(N)) has a natural rational structure, given by considering the differential forms on X_{0}(N) defined over ℚ. The fundamental q-expansion principle (see [DR], ch. 7, or [Kat], sec. 1.6) says that these algebraic structures are the same as those obtained analytically by considering q-expansions at ∞. More generally, using the model X_{Γ}^{μ}, we obtain the q-expansion principle over ℤ for cusp forms on Γ (cf. [DI], sec. 12.3).

Theorem 1.33 The map S_{2}(Γ) ⟶ Ω^{1}(X_{Γ}) defined by f ↦ ω_{f} induces an isomorphism from S_{2}(Γ, ℤ) to Ω^{1}(X_{Γ}^{μ}). Furthermore, if A is flat over ℤ or N is invertible in A, then the induced map S_{2}(Γ, A) → Ω^{1}(X_{Γ,A}^{μ}) is an isomorphism. Furthermore, if A is a subring of ℂ, then this isomorphism identifies S_{2}(Γ, A) with set of modular forms in S_{2}(Γ) having coefficients in A.

1.6 The Hecke algebra

It follows directly from the formulas for the Hecke operators acting on qexpansions that the T_{n} leave S_{2}(Γ_{0}(N), ℤ) stable, as well as the subspace of S_{2}(N, χ) with coefficients in ℤ[χ]. Using the q-expansion principle (theorem 1.33), one can also show ([DI], sec. 12.4) that the diamond operators preserve the spaces of cusp forms on Γ with integral Fourier expansions, and hence that the space S_{2}(Γ, ℤ) is preserved by all the Hecke operators.

We define 𝕋_{ℤ} to be the ring generated over ℤ by the Hecke operators T_{n} and ⟨d⟩ acting on the space S_{2}(Γ, ℤ). More generally, if A is any ring, we define 𝕋_{A} to be the A-algebra 𝕋_{ℤ} ⊗ A. This Hecke ring acts on the space S_{2}(Γ, A) in a natural way. Before studying the structure of the Hecke rings 𝕋_{A} as we vary the rings A, we note the following general result (Cf. [Shi2], ch. 3.):

Lemma 1.34 The space S_{2}(Γ, A)^{∨} = Hom_{A}(S_{2}(Γ, A), A) is a free 𝕋_{A}-module of rank one.

Sketch of proof: One checks that the pairing 𝕋_{ℤ} × S_{2}(Γ, ℤ) ⟶ ℤ defined by (T, f) ↦ a_{1}(T f) sets up a perfect, 𝕋_{ℤ}-equivariant duality between 𝕋_{ℤ} and S_{2}(Γ, ℤ). The result for arbitrary A follows. □

Hecke rings over ℂ: If A = ℂ, then the structure of the ring 𝕋 = 𝕋_{ℂ} is completely described by theorem 1.22. More precisely, if 𝕋_{ℂ,f} denotes the image of the Hecke algebra acting on the space S_{f} defined in section 1.3, then

𝕋_{ℂ} = ⊕_{f}𝕋_{ℂ,f},

where the direct sum ranges over all distinct newforms f of some level N_{f} dividing N. Furthermore, the algebra 𝕋_{ℂ,f} can be described explicitly. In particular, if f is a newform of level N then 𝕋_{ℂ,f} is isomorphic to ℂ, but if N_{f} is not equal to N then the ring 𝕋_{ℂ,f} need not be a semi-simple algebra over ℂ.

Lemma 1.34 in the case A = ℂ says that V = S_{2}(Γ)^{∨} is a free 𝕋_{ℂ}-module of rank one, but we also have:

Lemma 1.35 The module S_{2}(Γ) is a free 𝕋_{ℂ}-module of rank one.

Proof: Let g_{1}, …, g_{t} be a complete system of newforms of levels N_{1}, …, N_{t} dividing N. One can check that the form

g = g_{1}(N/N_{1}τ) + ⋯ + g_{t}(N/N_{t}τ)

generates S_{2}(Γ) as a 𝕋_{ℂ}-module. The map 𝕋_{ℂ} ⟶ S_{2}(Γ) defined by T ↦ T g gives an isomorphism from 𝕋_{ℂ} to S_{2}(Γ), as 𝕋_{ℂ}-modules. □

Remark 1.36 Lemmas 1.34 and 1.35 imply that 𝕋_{ℂ} is a Gorenstein ℂ-algebra (of finite rank), i.e., Hom_{ℂ}(𝕋_{ℂ}, ℂ) is isomorphic to 𝕋_{ℂ} as a 𝕋_{ℂ}-module.

Hecke rings over ℚ: Let [f] be the Galois orbit (under the action of G_{ℚ} = Gal (ℚ̄/ℚ)) of a normalized newform f of some level N_{f} dividing N, and let K_{f} be the field extension of ℚ generated by the Fourier coefficients of f. The space ⊕_{g∈[f]}S_{g} is a vector space of dimension [K_{f} : ℚ]σ_{0}(N/N_{f}), which is spanned by modular forms with rational Fourier coefficients. Let S_{[f]} be the ℚ-subspace of forms in ⊕_{g∈[f]}S_{g} with rational Fourier coefficients. The space S_{[f]} is stable under the action of 𝕋_{ℚ}, and letting 𝕋_{ℚ,[f]} be the image of 𝕋_{ℚ} acting on S_{[f]}, we obtain the direct sum decomposition

𝕋_{ℚ} = ⊕_{[f]}𝕋_{ℚ,[f]},

where the sum is taken over the distinct G_{ℚ}-orbits of normalized newforms f of some level N_{f} dividing N. If N_{f} is equal to N, then the algebra 𝕋_{ℚ,[f]} is isomorphic to the field K_{f}. If N_{f} is a proper divisor of N, then, as in the complex case, the algebra 𝕋_{ℚ,[f]} is a (not necessarily semi-simple) algebra over ℚ of rank σ_{0}(N/N_{f})[K_{f} : ℚ]. The nature of the fields K_{f}, and in general the structure of 𝕋_{ℚ}, is very poorly understood at this stage; for example, one does not know how to characterize the number fields that occur as a K_{f} for some f (but they are all known to be totally real or CM fields).

The ring 𝕋_{ℚ} acts naturally on the rational homology H_{1}(X_{Γ}, ℚ) = Λ ⊗ ℚ, and we have

Lemma 1.37 The module Λ ⊗ ℚ is free of rank two over 𝕋_{ℚ}.

Sketch of proof: The modules Λ^{+} ⊗ ℂ ≃ V and Λ^{−} ⊗ ℂ ≃ V are free of rank one over 𝕋_{ℂ}, by lemma 1.34. This implies that Λ^{+} ⊗ ℚ and Λ^{−} ⊗ ℚ are both free of rank one over 𝕋_{ℚ}. □

Hecke rings over ℤ: The ring 𝕋_{ℤ} is a certain (not necessarily maximal) order in 𝕋_{ℚ}. One still has an injection

𝕋_{ℤ} ↪ ⊕_{[f]}𝕋_{ℤ,[f]},

where now 𝕋_{ℤ,[f]} denotes the ring generated over ℤ by the Hecke operators acting on S_{[f]}. Of course the structure of 𝕋_{ℤ} is even more mysterious than that of 𝕋_{ℚ}! The ring 𝕋_{ℤ} acts naturally on Λ, but it is not the case in general that Λ is free of rank two over 𝕋_{ℤ}, i.e., that the integral analogue of lemma 1.37 is true. (See remark 1.42 below.)

Hecke rings over ℚ_{ℓ}: The study of the algebras 𝕋_{ℤ_{ℓ}} and 𝕋_{ℚ_{ℓ}} arises naturally because of the Hecke action on the Tate module T_{ℓ}(J_{Γ}),

T_{ℓ}(J_{Γ}) := lim_{←}(J_{Γ})[ℓ^{n}],

where the inverse limit is taken with respect to the multiplication by ℓ maps. The action of 𝕋_{ℤ_{ℓ}} on T_{ℓ}(J_{Γ}) is compatible with that of G_{ℚ}, and it is this pair of actions on the Tate module which is used to associate two-dimensional Galois representations to modular forms.

It will sometimes be more convenient to consider the ring 𝕋_{ℚ_{ℓ}} and its action on

V = T_{ℓ}(J_{Γ}) ⊗_{ℤ_{ℓ}} ℚ_{ℓ}.

We first record a useful duality property enjoyed by the Tate modules. The Weil pairings on the groups J_{Γ}[ℓ^{n}] for n ≥ 1 induce a perfect pairing

⟨ , ⟩ : T_{ℓ}(J_{Γ}) × T_{ℓ}(J_{Γ}) → ℤ_{ℓ}.

Since each Hecke operator T is adjoint to wT w where w = w_{N} is the Atkin-Lehner involution, we have the following lemma.

Lemma 1.38 The map x ↦ ϕ_{x} where ϕ_{x}(y) = ⟨x, wy⟩ defines an isomorphism of 𝕋_{ℤ_{ℓ}}-modules,

T_{ℓ}(J_{Γ}) ≅ T_{ℓ}(J_{Γ})^{∨} = Hom_{ℤ_{ℓ}}(T_{ℓ}(J_{Γ}), ℤ_{ℓ}),

and hence an isomorphism of 𝕋_{ℚ_{ℓ}}-modules

V ≅ V^{∨} = Hom_{ℚ_{ℓ}}(V, ℚ_{ℓ}).

The following lemma allows us to regard 𝕋_{ℚ_{ℓ}} as a coefficient ring for a two-dimensional Galois representation

Gal (ℚ̄/ℚ) → Aut_{𝕋_{ℚ_{ℓ}}}(V) ≅ GL_{2}(𝕋_{ℚ_{ℓ}}).

Lemma 1.39 The module V is free of rank 2 over 𝕋_{ℚ_{ℓ}}.

Proof: Lemma 1.37 implies directly that the module V = Λ⊗ℚ_{ℓ} is free of rank two over 𝕋_{ℚ_{ℓ}}. □

Corollary 1.40 The ring 𝕋_{ℚ_{ℓ}} is a Gorenstein ℚ_{ℓ}-algebra; i.e.,

𝕋_{ℚ_{ℓ}}^{∨} = Hom_{ℚ_{ℓ}}(𝕋_{ℚ_{ℓ}}, ℚ_{ℓ})

is free of rank one over 𝕋_{ℚ_{ℓ}}.

Proof: Choosing a basis for V over 𝕋_{ℚ_{ℓ}}, we obtain an isomorphism

𝕋_{ℚ_{ℓ}} ⊕ 𝕋_{ℚ_{ℓ}} ≅ 𝕋_{ℚ_{ℓ}}^{∨} ⊕ 𝕋_{ℚ_{ℓ}}^{∨}.

Decomposing 𝕋_{ℚ_{ℓ}} as ∏_{i} R_{i} where each factor R_{i} is a finite-dimensional local ℚ_{ℓ}-algebra, we obtain an isomorphism

R_{i} ⊕ R_{i} ≅ R_{i}^{∨} ⊕ R_{i}^{∨}

for each i. At least one of the four maps R_{i}^{∨} ⟶ R_{i} deduced from this isomorphism must be surjective, and by counting dimensions, we see that it must be injective as well. It follows that 𝕋_{ℚ_{ℓ}} is isomorphic to 𝕋_{ℚ_{ℓ}}^{∨}. □

Recall that for primes p not dividing N, the Jacobian J_{Γ} has good reduction mod p, and the Eichler-Shimura relation, theorem 1.29, states that on J_{Γ/𝔽_{p}}, we have

T_{p} = F + ⟨p⟩F′.

For primes p not dividing Nℓ, we may identify T_{ℓ}(J_{Γ}) with the ℓ-adic Tate module of the reduction (see [ST]) and consider the Frobenius endomorphism F on the free rank two 𝕋_{ℚ_{ℓ}}-module V. As a consequence of the Eichler-Shimura relation, we find:

Theorem 1.41 For p not dividing Nℓ, the characteristic polynomial of F on the 𝕋_{ℚ_{ℓ}}-module V is

X^{2} − T_{p}X + ⟨p⟩p.

Proof: (We are grateful to Brian Conrad for showing us this argument.) Since F F′ = p, it follows from the Eichler-Shimura relation that

F^{2} − T_{p}F + ⟨p⟩p = 0.

To conclude that this is in fact the characteristic polynomial, it suffices to compute the trace of F. To do so, we use the 𝕋_{ℚ_{ℓ}} isomorphism

V → V^{∨} defined by the modified pairing ⟨·, w·⟩. Under this modified pairing, F is adjoint to wF′w = ⟨p⟩F′, so the trace of F on V is the same as that of ϕ ↦ ϕ ∘ (⟨p⟩F′) on V^{∨}. Choosing bases for V and 𝕋_{ℚ_{ℓ}}^{∨}, one sees that this is the same as the trace of ⟨p⟩F′. Hence

2 tr F = tr F + tr (⟨p⟩F′) = tr (T_{p}) = 2 T_{p}.
□

Hecke rings over ℤ_{ℓ}: The ring 𝕋_{ℤ_{ℓ}} is free of finite rank over ℤ_{ℓ}. It therefore decomposes as

𝕋_{ℤ_{ℓ}} = ∏ 𝕋_{𝔪},

where the product runs over the maximal ideals 𝔪 of 𝕋_{ℤ_{ℓ}}, and 𝕋_{𝔪} is the localization of 𝕋_{ℤ_{ℓ}} at 𝔪. For each 𝔪, 𝕋_{𝔪} is a complete local ℤ_{ℓ}-algebra, free of finite rank as a ℤ_{ℓ}-module.

Remark 1.42 While the analogue of lemma 1.39 does not always hold for 𝕋_{ℤ_{ℓ}} (see [MRi], sec. 13) we shall see that it holds for certain localizations 𝕋_{𝔪}. Results of this type are much deeper than lemma 1.39 and were first obtained by Mazur [Maz1], sec. 14, 15. We shall return in chapter 4 to explain Mazur’s result and its generalizations, which play a role in the arguments of [W3] and [TW].

Example 1.43 The curve X_{0}(19) has genus 1, and X_{0}(57) has genus 5. By consulting the tables in the Antwerp volume [Ant4] or Cremona’s book [Cr], one finds that there is exactly one newform of level 19, and that there are three newforms of level 57, which all have rational Fourier coefficients. Their Fourier coefficients a_{p}, for the first few primes p, are listed in the following table:

2 3 5 7 11 13 17 19 23 29 31
19A 0 −2 3 −1 3 −4 −3 1 0 6 −4
57A −2 −1 −3 −5 1 2 −1 −1 −4 −2 −6
57B 1 1 −2 0 0 6 −6 −1 4 2 8
57C −2 1 1 3 −3 −6 3 −1 4 −10 2

Setting f = 19A, we find that a basis of simultaneous eigenforms in S_{2}(57, ℂ) for the Hecke operators T_{p}, l ≠ 3, 19, and U_{3}, U_{19} is:

f(τ) + (1 + \sqrt{−2})f(3τ), f(τ) + (1 − \sqrt{−2})f(3τ), 57A, 57B, 57C.

It appears from the table that the Fourier coefficients corresponding to the forms 57B and 57C are congruent modulo 3. This is in fact the case. One finds that the Hecke ring 𝕋_{ℤ_{3}} generated over ℤ_{3} by the Hecke operators acting on S_{2}(57, ℤ_{3}) is isomorphic to the subalgebra of ℤ_{3}^{5}:

{(x, y, z, t, w) such that t ≡ w (mod 3)}.

The isomorphism sends T_{p} (for p ≠ 3, 19) to the element

(a_{p}(19A), a_{p}(19A), a_{p}(57A), a_{p}(57B), a_{p}(57C)).

It sends U_{3} to (−1 + \sqrt{−2}, −1 − \sqrt{−2}, −1, 1, 1) and U_{19} to (1, 1, −1, −1, −1). Thus there are exactly 4 distinct maximal ideals of 𝕋_{ℤ_{3}}, and the localizations at these maximal ideals are the rings ℤ_{3}, ℤ_{3}, ℤ_{3}, and

𝕋_{𝔪} = {(t, w) such that t ≡ w (mod 3)},

where 𝔪 is the ideal generated by 3 and T_{n} − a_{n}(57B) for all n.

1.7 The Shimura construction

Let f = ∑ a_{n}q^{n} be an eigenform on Γ with (not necessarily rational) Fourier coefficients, corresponding to a surjective algebra homomorphism λ_{f} : 𝕋_{ℚ} ⟶ K_{f}, where K_{f} is the field generated over ℚ by the Fourier coefficients of f. We briefly review in this section a construction of Shimura ([Shi2], ch. 7) which associates to f (or rather, to the orbit [f] of f under G_{ℚ}) an abelian variety A_{f} defined over ℚ and of dimension [K_{f} : ℚ].

Let I_{f} ⊂ 𝕋_{ℤ} be the ideal ker (λ_{f}) ∩ 𝕋_{ℤ}. The image I_{f}(J_{Γ}) is a (connected) subabelian variety of J_{Γ} which is stable under 𝕋_{ℤ} and is defined over ℚ.

Definition 1.44 The abelian variety A_{f} associated to f is the quotient

A_{f} = J_{Γ}/I_{f}J_{Γ}

From this definition one sees that A_{f} is defined over ℚ and depends only on [f], and that its endomorphism ring contains 𝕋_{ℤ}/I_{f} which is isomorphic to an order in K_{f}.

Remark 1.45 Using theorem 1.22, one can show that J_{0}(N) is isogenous to ∏_{[f]} A_{[f]}^{σ_{0}(N/N_{f})}.

We now describe the abelian variety A_{f} as a complex torus. Let V_{f} be the subspace of V = S_{2}(Γ)^{∨} on which 𝕋 acts by λ_{f}. Theorem 1.22 and lemma 1.34 show that V_{f} is a one-dimensional complex vector space. Let π_{f} be the orthogonal projection of V to V_{f}, relative to the Petersson scalar product. The projector π_{f} belongs naturally to 𝕋_{K_{f}}.

Let [f] be the set of all eigenforms whose Fourier coefficients are Galois conjugate to those of f. The number of forms in [f] is equal to the degree d of K_{f} over ℚ. Now, we set

V_{[f]} = ⊕_{g∈[f]}V_{g}, π_{[f]} = ∑_{g∈[f]} π_{g}.

Note that π_{[f]} is simply the orthogonal projection of V to V_{[f]}. Note also that π_{[f]} belongs to the Hecke algebra 𝕋_{ℚ}.

Lemma 1.46 The abelian variety A_{f} is isomorphic over ℂ to the complex torus V_{[f]}/π_{[f]}(Λ), with the map π_{[f]} : V/Λ ⟶ V_{[f]}/π_{[f]}(Λ) corresponding to the natural projection from J_{Γ} to A_{f}.

In particular, one sees that A_{f} is an abelian variety of dimension d = [K_{f} : ℚ]. Hence if f has rational Fourier coefficients, then the abelian variety A_{f} is an elliptic curve. This elliptic curve is called the strong modular elliptic curve associated to f if also f is a newform of level N and Γ = Γ_{0}(N).

Example 1.47 If Γ = Γ_{0}(26), one checks that the genus of X_{Γ} is two, and that there are two distinct normalized eigenforms f_{1} and f_{2} in S_{2}(26). From the tables in [Ant4] or [Cr], one sees that f_{1} and f_{2} have integral Fourier coefficients, whose values for the primes ≤ 31 are:

2 3 5 7 11 13 17 19 23 29 31
f_{1} −1 1 −3 −1 6 1 −3 2 0 6 −4
f_{2} 1 −3 −1 1 −2 −1 −3 6 −4 2 4

Hence the abelian varieties A_{f_{1}} and A_{f_{2}} are elliptic curves. The above table suggests (and this can be checked directly by looking at the equations for these curves given in [Ant4] or [Cr], or by using the discussion in [DO], lemma 2.1) that the Fourier coefficients of f_{1} and f_{2} are congruent modulo 2. The natural projection J_{0}(26) ⟶ A_{f_{1}} ⊕ A_{f_{2}} is an isogeny whose kernel is isomorphic to ℤ/2ℤ × ℤ/2ℤ, and J_{0}(26) is not isomorphic to A_{f_{1}} ⊕ A_{f_{2}}. More generally, one knows that a Jacobian can never decompose as a non-trivial direct product of two principally polarized abelian varieties, cf. [Maz1], prop. 10.6. (There are no non-zero homomorphisms from A_{f_{1}} to A_{f_{2}}, and so a non-trivial product decomposition of J_{Γ} would have to induce a decomposition as a product of principally polarized abelian varieties.)

Let T_{ℓ}(A_{f}) be the Tate module of the abelian variety A_{f},

T_{ℓ}(A_{f}) := lim_{←}(A_{f})[ℓ^{n}],

where the inverse limit is taken with respect to the multiplication by ℓ maps. This module is naturally a module for 𝕋_{f} ⊗ ℤ_{ℓ} , and T_{ℓ}(A_{f}) ⊗ ℚ_{ℓ} is a module for K_{f} ⊗ ℚ_{ℓ} as well.

Lemma 1.48 The module T_{ℓ}(A_{f})⊗ℚ_{ℓ} is a free module of rank 2 over K_{f} ⊗ℚ_{ℓ}.

Proof: This follows directly from lemma 1.39. □

Proposition 1.49 The algebra End_{ℚ}(A_{f}) ⊗ ℚ is isomorphic to K_{f}. In particular A_{f} is simple over ℚ.

The proof is given in [R2], cor. 4.2. The main ingredient is the irreducibility of the Galois representation attached to A_{f} as in section 3.1. (Cf. theorem 3.1.)

Properties of A_{f}: good reduction:

Theorem 1.50 If f is an eigenform of level N, and p is a prime that does not divide N, then the abelian variety A_{f} has good reduction at p.

Proof: This follows from the fact that J_{Γ} has good reduction at such primes, which in turn is a consequence of the good reduction of X_{Γ}. □

If p is a prime not dividing N, it then becomes natural to study the number N_{f,p} of points on the abelian variety A_{f} over the finite field 𝔽_{p}. It is given by the following formula:

Proposition 1.51 The number of points N_{f,p} is given by the formula

N_{f,p} = Norm_{K_{f} /ℚ}(λ_{f}(1 − a_{p}(f) + ⟨p⟩p)),

where a_{f}(p) ∈ K_{f} is the p-th Fourier coefficient of f.

Proof: By Weil’s theory [We1], this number is given by the determinant

N_{f,p} = det(1 − F),

where F is the Frobenius endomorphism acting on the ℓ-adic Tate module T_{ℓ}(A_{f}). So theorem 1.41 implies

det(1 − F) = Norm_{K_{f} /ℚ}(λ_{f}(1 − T_{p} + ⟨p⟩p)),

and proposition 1.51 follows. □ Defining the local Hasse-Weil L-function of A_{f} over 𝔽_{p} by the formula

L(A_{f}/𝔽_{p}, s) = det(1 − F p^{−s})^{−1},

the proof above gives the formula:

L(A_{f}/𝔽_{p}, s) = ∏_{σ} L_{p}(f^{σ}, s),

where the product is taken over all complex embeddings σ : K_{f} ↪ ℂ, and L_{p}(f^{σ}, s) is the Euler factor at p in the L-function that was associated to f^{σ} in section 1.4.

In particular, if A_{f} = E is an elliptic curve, i.e., f has rational Fourier coefficients and f is on Γ_{0}(N), then the number of points on E over 𝔽_{p} is given by the formula

#E(𝔽_{p}) = p + 1 − a_{p}(f).  (1.7.1)

Properties of A_{f}: bad reduction: A fundamental quantity associated to any abelian variety A over ℚ is its arithmetic conductor, which measures the amount of bad reduction of A. If we factor this conductor as a product of prime powers, ∏_{p} p^{m_{p}}, then the exponents m_{p} are equal to 0 precisely when A has good reduction at p. The definition of m_{p}(A) was already given in section 1.1 when A is an abelian variety of dimension 1, i.e., an elliptic curve. In general, the exponent m_{p} coincides with m_{p}(ρ_{A,ℓ}) (see section 2.1 below), where ρ_{A,ℓ} is the Galois representation on the ℓ-adic Tate module of A.

Regarding the bad reduction of the abelian variety A_{f}, one has the following consequence of the results of Langlands, Deligne and Carayol [Ca3] discussed below in section 3.1.

Theorem 1.52 If f is a newform of level N, then the conductor of the abelian variety A_{f} is equal to N^{g}.

1.8 The Shimura-Taniyama conjecture

Let E be any elliptic curve defined over ℚ, and let N denote its arithmetic conductor, defined as in section 1.1. Then we have:

Proposition 1.53 The following are equivalent:

(a) The curve E is isogenous over ℚ to A_{f}, for some newform f on some congruence group Γ.

(b) The curve E is isogenous over ℚ to A_{f}, for a newform f on Γ_{0}(N).

(c) There is a non-constant morphism defined over ℚ, from X_{0}(N) to E.

Sketch of proof: The implication (b) ⇒ (a) is immediate, and (a) ⇒ (b) follows from the work of Carayol [Ca3] (cf. proposition 3.20 below). If (b) holds, then there is a surjective map J_{0}(N) ⟶ E. Composing with the Abel-Jacobi map X_{0}(N) ⟶ J_{0}(N) gives a map to E which is not constant, by the definition of the Jacobian, and hence (c) holds. Conversely, a non-constant map of curves π : X_{0}(N) ⟶ E induces, by Albanese (covariant) functoriality, a surjective map of Jacobians π_{∗} : J_{0}(N) ⟶ E, where we have identified E with its own Jacobian in the natural way. Since J_{0}(N) is isogenous to a product (possibly with multiplicities) of abelian varieties A_{f} where f run over newforms of level N_{f} dividing N (cf. remark 1.45), it follows that there is a surjective map A_{f} ⟶ E for some A_{f}. Since A_{f} is simple (proposition 1.49), this map is an isogeny, and part (b) follows. □

We call an elliptic curve over ℚ satisfying the equivalent properties above a modular elliptic curve. A startling conjecture that was first proposed by Taniyama in the 1950’s and made more precise by Shimura predicts that the Shimura construction explained in the previous section is surjective:

Conjecture 1.54 All elliptic curves defined over ℚ are modular.

This conjecture is now known to be true for a very wide class of elliptic curves. The results of Wiles [W3] and Taylor-Wiles [TW] imply it is true for all semistable elliptic curves, and a strengthening of the method, [Di2], implies the conjecture for all elliptic curves which have semi-stable reduction at 3 and 5.

Remark 1.55 See [DI], sec. 13 for a more thorough list of equivalent forms of the conjecture.

Relation with L-functions: Define numbers a_{p} = a_{p}(E) as in section 1.1. Recall the (global) Hasse-Weil L-function defined in section 1.1 by equations (1.1.2), (1.1.4), (1.1.5), and (1.1.6). The following proposition reveals some of the importance of the Shimura-Taniyama conjecture:

Proposition 1.56 If E is modular, i.e., is associated to a newform f by the Shimura construction, then

L(E/ℚ, s) = L(f, s).

In particular, L(E/ℚ, s) has an analytic continuation to the entire complex plane, and a functional equation.

Sketch of proof: If E is isogenous to an elliptic curve A_{f} associated to a newform f on Γ_{0}(N) by the Shimura construction, then the two L-functions L(A_{f}/ℚ, s) and L(E/ℚ, s) are equal. On the other hand, formula (1.7.1) directly implies that L(A_{f}/ℚ, s) is equal to L(f, s), at least up to finitely many Euler factors (corresponding precisely, by the work of Igusa, to the primes dividing N). The equality of all the Euler factors follows from the work of Deligne, Langlands, and Carayol (cf. [Ca1]). □

Knowing the analytic continuation and functional equation of the Hasse-Weil L-function of an elliptic curve is of great importance in the theory. For example, the conjecture of Birch and Swinnerton-Dyer (which was stated in a weak form in section 1.1, conjecture 1.9) relates arithmetic invariants of E such as the rank of its Mordell-Weil group and the order of its Shafarevich-Tate group to the behaviour at s = 1 of L(E/ℚ, s). (Note that s = 1 is outside the domain of absolute convergence of the infinite product used to define L(E/ℚ, s).)

One consequence of this conjecture is that E(ℚ) is finite if L(E/ℚ, 1) is non-zero. This was proved by Coates and Wiles in [CW] for elliptic curves with complex multiplication, and by Kolyvagin [Kol] for modular elliptic curves. (Recently, a different proof for modular elliptic curves has been announced by K. Kato.) Thanks to the breakthroughs of [W3], [TW], and [Di2], the results of Kolyvagin and Kato are now unconditional for a very large class of elliptic curves.

The Shimura-Taniyama conjecture for abelian varieties: We may view conjecture 1.54 as asserting that the map

\begin{cases}Newforms of weight 2 on X_{0}(N) \\ with rational Fourier coefficients \end{cases}\} ⟶ \begin{cases}Isogeny classes of elliptic curves \\ over ℚ of conductor N \end{cases}\}

provided by the Shimura construction is a bijection. It is natural to extend this result to all eigenforms, not just those having rational Fourier expansions. We say that an abelian variety defined over ℚ is modular if it is isogenous to an abelian variety of the form A_{f} for some newform f on Γ_{1}(N). It would not be reasonable to expect that all abelian varieties over ℚ are modular: the abelian varieties arising from the Shimura construction are very special, in many respects. Most importantly, they have a very large ring of endomorphisms. Following Ribet [R7], we make the definition:

Definition 1.57 An abelian variety over ℚ of dimension g is said to be of GL_{2}-type if its endomorphism ring over ℚ contains an order in a field of degree g over ℚ.

Then we have the following generalized Shimura-Taniyama conjecture:

Conjecture 1.58 Every simple abelian variety A over ℚ which is of GL_{2}-type is modular.

This generalization of the Shimura-Taniyama conjecture is far from being proved, and tackling it still seems to require some major new ideas, even after the ideas introduced by Wiles.

2 Galois theory

2.1 Galois representations

If F is a perfect field, we will let F̄ denote an algebraic closure of F and G_{F} = Gal (F̄/F) its absolute Galois group. Recall that G_{F} is a profinite group: more precisely, G_{F} = lim_{←} Gal (L/F) as L runs over finite Galois extensions of

F contained in F̄. Hence G_{F} carries a natural topology. If ℓ is a prime different from the characteristic of F we will let ϵ_{ℓ} : G_{F} → ℤ_{ℓ}^{×} denote the ℓ-adic cyclotomic character, i.e. if ζ is an ℓ-power root of unity in F̄ then σ(ζ) = ζ^{ϵ_{ℓ}(σ)} for all σ ∈ G_{F}. We will simply write ϵ if the choice of ℓ is clear from the context.

The finite fields 𝔽_{p}: The group G_{𝔽_{p}} is isomorphic to ℤ̂, the profinite completion of ℤ. A natural topological generator is provided by the Frobenius element Frob_{p}, the automorphism of 𝔽̄_{p} which raises elements of 𝔽̄_{p} to their p^{th} power. If ℓ is different from p, then ϵ_{ℓ}(Frob_{p}) = p.

The p-adic fields ℚ_{p}: The p-adic valuation v_{p} : ℚ_{p}^{×} → → ℤ extends uniquely to a valuation v_{p} : ℚ̄_{p}^{×} → → ℚ. Let Oℚ̄_{p} denote the ring of integers of ℚ̄_{p} and 𝔪ℚ̄_{p} its maximal ideal. The residue field Oℚ̄_{p}/𝔪ℚ̄_{p} is an algebraic closure of 𝔽_{p}, which we will identify with 𝔽̄_{p}. The valuation v_{p} is compatible with the action of G_{ℚ_{p}}, and hence Oℚ̄_{p} and 𝔪ℚ̄_{p} are stable under G_{ℚ_{p}}. In particular G_{ℚ_{p}} acts on Oℚ̄_{p}/𝔪ℚ̄_{p} and we obtain a map ϱ : G_{ℚ_{p}} → G_{𝔽_{p}}, which is in fact a surjection. We call the kernel the inertia group and denote it I_{p}. It has a unique maximal pro-p subgroup: the wild inertia group, denoted P_{p}. There is a canonical isomorphism

I_{p}/P_{p} ≅ ∏_{ℓ≠p} ℤ_{ℓ}(1),

where the (1) indicates that if f ∈ G_{ℚ_{p}}/P_{p} is a lifting of Frobenius and if σ ∈ I_{p}/P_{p} then fσf^{−1} = σ^{p}.

The inertia group I_{p} is filtered by a series of subgroups called the higher ramification groups. More precisely, there are closed normal subgroups I_{p}^{u} in G_{ℚ_{p}} for any u ∈ [−1, ∞], with the following properties:

• If u ≤ v then I_{p}^{u} ⊃ I_{p}^{v}, i.e., the I_{p}^{u} form a decreasing filtration.

• For u ≤ 0, I_{p}^{u} = I_{p} while I_{p}^{∞} = {1}.

• P_{p} = ⋃_{u>0} I_{p}^{u}.

• I_{p}^{u} = ⋂_{v<u} I_{p}^{v}.

These groups are defined in section 3 of chapter IV of [Se2], where they are denoted G_{ℚ_{p}}^{u} (except that there, G_{ℚ_{p}}^{−1} = G_{ℚ_{p}} whereas we have I_{p}^{−1} = I_{p}).

The local Kronecker-Weber theorem ([Se2], ch. 14, sec. 7, for example) asserts that the map

ϱ × ϵ_{p} : G_{ℚ_{p}}^{ab} ⟶ G_{𝔽_{p}} × ℤ_{p}^{×}.

is an isomorphism. (We will use G^{ab} to denote the abelianisation of a profinite group G, i.e. the unique maximal abelian continuous image.) Under it I_{p} goes to ℤ_{p}^{×} and for u > 0, I_{p}^{u} goes to (1 + p^{⌈u⌉}ℤ_{p}) ⊂ ℤ_{p}^{×}, where ⌈u⌉ denotes the least integer greater than or equal to u. If ℓ ≠ p then ϵ_{ℓ} is trivial on I_{p} and takes Frob_{p} to p ∈ ℤ_{ℓ}^{×}.

The field ℚ: If p is a rational prime, the usual p-adic valuation v_{p} : ℚ^{×} → ℤ extends to a valuation v : ℚ̄^{×} → ℚ. This extension is not unique, but all extensions are permuted transitively by G_{ℚ}. Fix one such extension v_{p} and let G_{p} denote its stabiliser in G_{ℚ}. Then G_{p} acts on the completion (ℚ̄)_{v_{p}} and preserves the subfield of elements algebraic over ℚ_{p}. This field is an algebraic closure of ℚ_{p}, which we shall identify with ℚ̄_{p}. One can check that the resulting map G_{p} → G_{ℚ_{p}} is an isomorphism. Thus we obtain subgroups G_{p} ⊃ I_{p} ⊃ I_{p}^{u} for u ∈ [−1, ∞] and a distinguished element Frob_{p} ∈ G_{p}/I_{p}. These objects depend on the choice of v_{p} and vary by conjugation in G_{ℚ} as this choice is varied. We call an algebraic extension F/ℚ unramified at p if all conjugates of I_{p} lie in G_{F} ⊂ G_{ℚ}; otherwise we say that F/ℚ is ramified. If F/ℚ is Galois and unramified at p, then there is a well-defined conjugacy class [Frob_{p}] ⊂ Gal (F/ℚ).

By replacing the p-adic completion by an Archimedean completion, one has a well-defined conjugacy class [c] in G_{ℚ} consisting of those elements that arise as complex conjugation for some embedding ℚ̄ ↪ ℂ. We will denote by G_{∞} the subgroup {1, c} for one such element c.

We have the following fundamental results concerning the structure of G_{ℚ} (see for instance [La1]).

Theorem 2.1 If F/ℚ is a finite extension then F is only ramified at finitely many primes (those dividing the discriminant of F/ℚ).

Theorem 2.2 (Hermite-Minkowski) If S is a finite set of primes and if d ∈ ℤ_{>0} then there are only finitely many extensions F/ℚ of degree d which are unramified outside S (contained in a fixed algebraic closure ℚ̄).

Theorem 2.3 (Chebotarev) If F/ℚ is a Galois extension unramified outside a finite set S of primes then ⋃_{p∉S}[Frob_{p}] is dense in Gal (F/ℚ).

Theorem 2.4 (Kronecker-Weber) The product of the p-adic cyclotomic characters gives an isomorphism

∏_{p} ϵ_{p} : G_{ℚ}^{ab} →^{∼} ∏_{p} ℤ_{p}^{×}.

We remark that the Chebotarev density theorem (theorem 2.3) applied to the extension of ℚ generated by the m^{th} roots of unity implies Dirichlet’s theorem that if n and m are coprime integers then there are infinitely many primes p with p ≡ n mod m. We also remark that the Kronecker-Weber theorem is equivalent to the fact that the maximal abelian extension of ℚ is obtained by adjoining all roots of unity to ℚ.

Representations: A d-dimensional representation of G_{ℚ} is a homomorphism

G_{ℚ} ⟶ GL_{d}(K),

where K is any field. (In our later discussion, we will also consider representations with coefficients in a ring.) Often K (and hence GL_{d}(K)) comes equipped with a natural topology, and it is then customary to restrict one’s attention to continuous homomorphisms G_{ℚ} ⟶ GL_{d}(K).

Since any one-dimensional representation has abelian image, theorem 2.4 allows one to give a complete description of the one-dimensional representations of G_{ℚ} together with the behaviour of these representations on the decomposition groups at all primes. The aim of this article is to discuss attempts to give a similar theory for the two-dimensional representations of G_{ℚ}.

We will call a representation ρ of G_{ℚ} unramified at p if it is trivial on the inertia group I_{p}. Otherwise we say that it is ramified at p. If ρ is unramified at p, then ρ(Frob_{p}) is well-defined (and its conjugacy class is independent of the choice of v_{p}).

If ρ is a representation of a group G and i is a non-negative integer, then we let ∧^{i}ρ denote the representation of G on the i^{th} exterior power of the underlying module of ρ. If H is a subgroup of G, we will let ρ^{H} (resp. ρ_{H}) denote the H-invariants (resp. H-coinvariants) of the underlying module of ρ. If H is normal in G, then we shall also use ρ^{H} and ρ_{H} to denote the corresponding representation of G/H. In particular if p is a prime and ρ is a representation of G_{ℚ} or G_{ℚ_{p}}, then ρ^{I_{p}}(Frob_{p}) and ρ_{I_{p}}(Frob_{p}) are well-defined.

We will be primarily interested in three types of representation of G_{ℚ}.

• Artin representations, i.e. continuous representations G_{ℚ} → GL_{d}(ℂ). Since all compact totally disconnected subgroups of GL_{d}(ℂ) are finite, Artin representations have finite image. Hence they are semi-simple and are unramified at all but finitely many primes, by theorem 2.1.

• Mod ℓ representations, i.e. continuous representations G_{ℚ} → GL_{d}(k), where k is a finite field of characteristic ℓ. These always have finite image and hence, like Artin representations, are unramified at all but finitely many primes.

• ℓ-adic representations, i.e. continuous representations G_{ℚ} → GL_{d}(K), where K is a finite extension of ℚ_{ℓ}. We require that an ℓ-adic representation be unramified at all but finitely many primes.

Remark 2.5 (a) Continuous representations G_{ℚ} → GL_{d}(K), unlike those to GL_{d}(ℂ) or GL_{d}(k), may be ramified at infinitely many primes. We shall not need to consider such representations. (One rarely if ever does.) However, the term “ℓ-adic representation” is often used in the more general sense elsewhere in the literature.

(b) Note that the image of an ℓ-adic representation can very well be infinite. For instance this is the case if d = 1 and the representation is the cyclotomic character.

(c) Note that in the case of mod ℓ and ℓ-adic representations, the representations need not be semi-simple.

Proposition 2.6 Let S be any finite set of primes.

(a) An Artin representation ρ : G_{ℚ} → GL_{d}(ℂ) is determined by the values of tr ρ(Frob_{p}) on the primes p ∉ S at which ρ is unramified.

(b) A semi-simple mod ℓ representation ρ : G_{ℚ} → GL_{d}(k) is determined by the values of tr ∧^{i} ρ(Frob_{p}) (i = 1, ..., d) on the primes p ∉ S at which ρ is unramified. If ℓ > d then it is determined by tr ρ(Frob_{p}) on the primes p ∉ S at which ρ is unramified.

(c) A semi-simple ℓ-adic representation ρ : G_{ℚ} → GL_{d}(K) is determined by the values of tr ρ(Frob_{p}) on the primes p ∉ S at which ρ is unramified.

Proof: Combining the Chebotarev density theorem (theorem 2.3) with the continuity of ρ we see that we must show that tr ρ (resp. tr ∧^{i} ρ) determines ρ up to conjugacy in the various settings. For the characteristic zero representations, see [Bour], ch. 8, sec. 12.1, prop. 3; for mod ℓ representations, this is the Brauer-Nesbitt theorem ([CR], (30.16)). □

Let K denote a finite extension of ℚ_{ℓ}, let O be its ring of integers, λ the maximal ideal of O and k the residue field. If ρ : G_{ℚ} → GL_{d}(K) is an ℓ-adic representation, then the image of ρ is compact, and hence ρ can be conjugated to a homomorphism G_{ℚ} → GL_{d}(O). Reducing modulo the maximal ideal λ gives a residual representation ρ̄: G_{ℚ} → GL_{d}(k). This representation may depend on the particular GL_{d}(K)-conjugate of ρ chosen, but its semisimplification ρ̄^{ss} (i.e. the unique semi-simple representation with the same Jordan-Hölder factors) is uniquely determined by ρ, by proposition 2.6 (b).

Note that the kernel of the reduction map GL_{d}(O) → GL_{d}(k) is a proℓ-group. In particular we see that if p ≠ ℓ then ρ(P_{p}) is finite and ρ(P_{p}) is isomorphic to ρ̄(P_{p}).

Suppose ρ is a representation of G_{ℚ} of one of the three sorts above. If p ≠ ℓ then we define the conductor, m_{p}(ρ), of ρ at p by

m_{p}(ρ) = ∫_{−1}^{∞} codimρ^{I_{p}^{u}} du = codimρ^{I_{p}} + ∫_{0}^{∞} codimρ^{I_{p}^{u}} du.

This is well-defined as ρ(P_{p}) is a finite group. If ρ is an Artin representation this makes sense also for p = ℓ. (It would even make sense for a mod ℓ representation if p = ℓ, but in this case it does not seem to be a useful notion.) It is known that m_{p}(ρ) is an integer (see chapter VI of [Se2]). Moreover it is easily seen that m_{p}(ρ) = 0 if and only if ρ is unramified at p. We define the conductor N(ρ) of ρ to be

∏_{p} p^{m_{p}(ρ)},

where the product is over all primes p ≠ ℓ in the case of an ℓ-adic or mod ℓ representation and over all primes in the case of an Artin representation. This makes sense because ρ is unramified almost everywhere.

The following lemma is an exercise.

Lemma 2.7 Suppose that ρ : G_{ℚ} → GL_{d}(K) is an ℓ-adic representation and that ρ̄: G_{ℚ} → GL_{d}(k) is a reduction of ρ. Then for each prime p ≠ ℓ we have ∫_{0}^{∞} codimρ^{I_{p}^{u}} du = ∫_{0}^{∞} codimρ̄^{I_{p}^{u}} du. Thus

N(ρ) = N(ρ̄) ∏_{p≠ℓ} p^{(dim ρ̄^{Ip}−dim ρ^{Ip})},

and in particular N(ρ̄) divides N(ρ).

We take this opportunity to introduce some important notation. Let G denote a group and R a ring. If ρ : G → GL_{d}(R) is a representation we shall let M_{ρ} denote the underlying R[G]-module (so that M_{ρ} ≅ R^{d} as an Rmodule). If M is an R[G]-module we shall let End (M) denote the module of R-linear endomorphisms of M. It is also an R[G]-module with the G-action being defined by

(g(ϕ))(m) = g(ϕ(g^{−1}m)).

If M is a finitely generated free R-module we will let End^{0}(M) denote the sub-R[G]-module of End (M) consisting of endomorphisms of trace zero. We will use adρ to denote End (M_{ρ}) and ad^{0}ρ to denote End^{0}(M_{ρ}). Note that if d is invertible in R then adρ ≅ ad^{0}ρ ⊕ R as R[G]-modules, where R has the trivial action of G. Note also then that the kernel of ad^{0}ρ is the same as the kernel of the composite map G →^{ρ} GL_{d}(R) → P GL_{d}(R).

We also remark that if R is an algebraically closed field of characteristic other than 2 and ρ : G → GL_{2}(R) is irreducible, then either ad^{0}ρ is irreducible or there is a subgroup H ⊂ G of index 2 and a character χ : H → R^{×} such that ρ ≅ Ind _{H}^{G}χ. In the latter case ad^{0}ρ ≅ δ⊕Ind _{H}^{G}(χ/χ′), where δ is the nontrivial character of G/H and χ′ is the composite of χ and conjugation by an element of G − H. Moreover, either Ind _{H}^{G}(χ/χ′) is irreducible or ad^{0}ρ ≅ δ_{1} ⊕ δ_{2} ⊕ δ_{3} where the δ_{i} are distinct characters of G of order 2. In the latter case, ρ is induced from a character from each of the subgroups H_{i} = ker δ_{i} of index 2 in G. We leave the verification of these facts as an exercise to the reader.

2.2 Representations associated to elliptic curves

Perhaps the simplest examples of non-abelian ℓ-adic representations arise from elliptic curves defined over ℚ.

We will let E[n](ℚ̄) denote the group of n-torsion points on E(ℚ̄). By proposition 1.1 there is a non-canonical isomorphism E[n](ℚ̄) ≅ (ℤ/nℤ)^{2}. Furthermore, E[n](ℚ̄) carries a natural action of G_{ℚ} and so we get a representation (defined up to conjugation)

ρ̄_{E,n} : G_{ℚ} → GL_{2}(ℤ/nℤ).

If ℓ is a prime different from the characteristic of ℚ then we set T_{ℓ}E = lim_{←} E[ℓ^{n}](ℚ̄), which is non-canonically isomorphic to ℤ_{ℓ}^{2}. Again it has a natural continuous action of G_{ℚ}, and so we get a representation (defined up to conjugation)

ρ_{E,ℓ} : G_{ℚ} → GL_{2}(ℤ_{ℓ}).

Note that ρ̄_{E,ℓ^{n}} ≅ (ρ_{E,ℓ} mod ℓ^{n}).

Global properties: We have the following basic properties of the representations ρ̄_{E,ℓ^{n}} and ρ_{E,ℓ}:

Proposition 2.8 (a) The determinant of ρ_{E,ℓ} is ϵ_{ℓ}.

(b) The representation ρ_{E,ℓ} is absolutely irreducible for all ℓ. For fixed E, ρ̄_{E,ℓ} is absolutely irreducible for all but finitely many ℓ.

(c) If E does not have complex multiplication then ρ_{E,ℓ} (and hence ρ̄_{E,ℓ}) is surjective for all but finitely many ℓ.

Proof: Part (a) follows from the existence of the non-degenerate alternating Galois-equivariant Weil pairing

T_{ℓ}E × T_{ℓ}E ⟶ ℤ_{ℓ}(1) := lim_{←} μ_{ℓ^{n}}.

Part (b) is proved in [Se5], ch. IV. Part (c) is the main result of [Se6]. □

The following deep result, which is stronger than the second assertion of part (b) of proposition 2.8, is due to Mazur [Maz2], thms. 1 and 2. (Parts (b) and (c) can actually be deduced directly from theorem 1.7, using the fact that a semi-stable curve with a rational subgroup of order ℓ is necessarily isogenous to a curve with a rational point of order ℓ.)

Theorem 2.9 Suppose that E/ℚ is an elliptic curve.

(a) If ℓ > 163 is a prime then ρ̄_{E,ℓ} is irreducible.

(b) If E is semistable then ρ̄_{E,ℓ} is irreducible for ℓ > 7.

(c) If E is semistable and ρ̄_{E,2} is trivial then ρ̄_{E,ℓ} is irreducible for ℓ > 3.

Remark 2.10 Combined with the results of Serre [Se6], Mazur’s results imply that if E is semistable everywhere then ρ̄_{E,ℓ} is surjective for ℓ > 7 ([Maz2], thm. 4).

Local behaviour of ρ_{E,ℓ} and ρ̄_{E,ℓ}:

Proposition 2.11 Suppose E has good reduction at p.

(a) If ℓ ≠ p, then ρ_{E,ℓ} is unramified at p, and we have the formula

tr ρ_{E_{,}ℓ}(Frob_{p}) = p + 1 − #Ē_{p}(𝔽_{p}).

In particular tr ρ_{E,ℓ}(Frob_{p}) belongs to ℤ and is independent of ℓ ≠ p.

(b) For all n ≥ 1 there is a finite flat group scheme F_{n}/ℤ_{p} such that

E[p^{n}](ℚ̄_{p}) ≅ F_{n}(ℚ̄_{p})

as G_{p}-modules.

(c) • If E has good ordinary reduction at p (i.e., a_{p} is not divisible by p) then Ē_{p}[p](𝔽̄_{p}) has order p and

ρ_{E,p}|_{I_{p}} ∼ (\begin{matrix}ϵ_{p} & ∗ \\ 0 & 1 \end{matrix}) ;

• If E has supersingular reduction at p, then Ē_{p}[p](𝔽̄_{p}) is trivial and ρ̄_{E,p}|_{G_{p}} is irreducible.

Proofs: The proof of part (a) can be found in chapters V and VII of [Si1]. For part (b) one considers the finite flat group scheme F_{n} = E[p^{n}], where E is the model for E over ℤ_{p} defined by (W^{min}). (See [Sha] for an introduction to theory of finite flat group schemes.) Part (c) follows from the results in chapters V and VII of [Si1] together with basic results on finite flat group schemes.

Proposition 2.12 Suppose that E has multiplicative reduction at p. Let q = q_{E,p} ∈ pℤ_{p} be the multiplicative Tate period attached to E, defined as in section 1.1. Let δ : G_{p}/I_{p} → {±1} be the unique non-trivial unramified quadratic character of G_{p} if E has non-split multiplicative reduction, and let δ be the trivial character if E has split multiplicative reduction. Then

(a)

ρ_{E,ℓ}|_{G_{p}} ∼ (\begin{matrix}ϵ_{ℓ} & ∗ \\ 0 & 1 \end{matrix}) ⊗ δ,

and if ℓ ≠ p then m_{p}(ρ_{E,ℓ}) = 1.

(b)

ρ̄_{E,ℓ}|_{G_{p}} ∼ (\begin{matrix}ϵ_{ℓ} & Ψ \\ 0 & 1 \end{matrix}) ⊗ δ,

and Ψ ∈ H^{1}(G_{ℚ_{p}}, ℤ/ℓℤ(1)) ≅ ℚ_{p}^{×}/(ℚ_{p}^{×})^{ℓ} corresponds to the Tate period

q_{E,p} ∈ ℚ_{p}^{×}/(ℚ_{p}^{×})^{ℓ}.

(c) If ℓ ≠ p then m_{p}(ρ̄_{E,ℓ}) = 0 (i.e., ρ̄_{E,ℓ} is unramified at p) if and only if ℓ|v_{p}(Δ_{E}^{min}) = −v_{p}(j_{E}).

(d) There is a finite flat group scheme F/ℤ_{p} such that E[p](ℚ̄_{p}) ≅ F(ℚ̄_{p}) if and only if p|v_{p}(Δ_{E}^{min}) = −v_{p}(j_{E}).

Sketch of proof: By Tate’s proposition 1.5 of section 1.1, we have

E(ℚ̄_{p}) ≅ (ℚ̄_{p}^{×}/q^{ℤ})(δ)

as G_{p}-modules. (The (δ) indicates that G_{p} acts on (ℚ̄_{p}^{×}/q^{ℤ})(δ) by σ : x ↦ σ(x)^{δ(σ)}.) The proposition follows directly from this, together with [Edi], prop. 8.2. for the last part. (For further details, see ch. VII, prop. 5.1 of [Si1], ch. IV, thm. 10.2. and ch. V of [Si2], and sec. 2.9 of [Se7].) □

Proposition 2.13 Suppose that E has additive reduction at p. Then for any ℓ ≠ p the conductor of ρ_{E,ℓ}|_{G_{p}} is at least 2, and if p > 3 then it is equal to 2.

Remark 2.14 For an elliptic curve E with any type of reduction at p, the conductor of ρ_{E,ℓ}|_{G_{p}} (for ℓ ≠ p) coincides with the local conductor of E at p given by a formula of Ogg (proved by Saito [Sa] in the case p = 2). In particular the conductor of ρ_{E,ℓ}|_{G_{p}} is independent of ℓ ≠ p. For further discussion and references, see ch. IV, secs. 10 and 11 of [Si2].

The Frey curve: We now consider the elliptic curve

E_{A,B} : Y^{2}Z = X(X − AZ)(X + BZ),

where A and B are non-zero coprime integers with A + B ≠ 0. Set C = A + B. This equation has discriminant Δ = 16(ABC)^{2}. If p is an odd prime then this curve has good reduction at p if p |̸(ABC) and has multiplicative reduction at p if p|ABC. Thus Δ_{E_{A,B}}^{min} = 2^{4−12n}(ABC)^{2} for some n ∈ ℤ_{≥0}. If for instance A ≡ −1 mod 4 and B ≡ 0 mod 16 then E_{A,B} has a minimal Weierstrass equation

Y^{2}Z + XY Z = X^{3} + \frac{B − A − 1}{4}X^{2}Z − (AB/16)XZ^{2}.

In this special case E_{A,B} has semi-stable reduction everywhere and Δ_{E_{A,B}}^{min} = 2^{−8}(ABC)^{2}.

The connection with Fermat’s Last Theorem is provided by the elliptic curve

E : Y^{2}Z = X(X − a^{ℓ}Z)(x + b^{ℓ}Z)

considered by Hellegouarch [He] and Frey [Fr], where a^{ℓ} + b^{ℓ} = c^{ℓ} is a hypothetical non-trivial solution to Fermat’s Last theorem, and where we have supposed without loss of generality that a, b and c are pairwise coprime, that b is even and that a ≡ −1 mod 4. Frey suggested that properties of this hypothetical elliptic curve E could lead to a contradiction if the curve were also known to be modular. This suggestion was made precise by Serre [Se7], who observed that the following list of properties would yield the desired contradiction, when combined with his conjectures on Galois representations associated to modular forms (to be discussed in section 3.2).

Theorem 2.15 Suppose that ℓ > 3 is prime. The representation ρ̄_{E,ℓ} has the following properties.

(a) ρ̄_{E,ℓ} is irreducible.

(b) ρ̄_{E,ℓ} is unramified outside ℓ and 2.

(c) There is a finite flat group scheme F/ℤ_{ℓ} such that F(ℚ̄_{ℓ}) ≅ E[ℓ](ℚ̄_{ℓ}) as G_{ℓ}-modules.

(d) #ρ̄_{E,ℓ}(I_{2}) = ℓ.

Proof: Part (a) follows from Mazur’s theorem (theorem 2.9, (c)). To prove (b), (c), and (d), we note the key fact that Δ_{E}^{min} = 2^{−8}(abc)^{2ℓ} is a perfect ℓ-th power, up to powers of 2. The fact that ρ̄_{E,ℓ} is unramified at p ≠ ℓ, 2 (part (b)) follows from this, from proposition 2.11, (a) (for primes p ≠ ℓ of good reduction for E) and from proposition 2.12 (c) (for primes p ≠ ℓ, 2 of multiplicative reduction for E). Part (c) is a consequence of proposition 2.12 (d), and part (d) follows from proposition 2.12 (c). □

Remark 2.16 Part (a) implies that the image of ρ̄_{E,ℓ} is large, while parts (b) and (c) state that the image has very limited ramification. We remark that the conclusion of the theorem ensures that m_{p}(ρ̄_{E,ℓ}) = 0 for p ≠ 2, ℓ and that m_{2}(ρ̄_{E,ℓ}) = 1.

2.3 Galois cohomology

In this section M will denote a continuous discrete G_{ℚ}-module of finite cardinality. If M and N are two such modules we will endow the space Hom(M, N) of homomorphisms of abelian groups with an action of G_{ℚ} via the formula

(g(ϕ))(m) = g(ϕ(g^{−1}m)).

We will use M^{∗} to denote Hom(M, μ_{n}(ℚ̄)), where n is an integer such that nM = (0) and where μ_{n}(ℚ̄) denotes the group of n^{th} roots of unity in ℚ̄. We will use ∨ to denote Pontryagin duals. All cohomology groups of profinite groups will mean continuous cohomology. The general references for this section are Milne [Mi] and Serre [Se1].

We start by recalling Tate’s local duality theorem (see chapter 1, section 2 of [Mi]).

Theorem 2.17 Let v be a place of ℚ (i.e. either a prime number or ∞).

(a) H^{i}(G_{v}, M) is finite for all i.

(b) For all integers n there are compatible embeddings H^{2}(G_{v}, μ_{n}(ℚ̄)) ↪ ℚ/ℤ (i.e. compatible with the maps coming from μ_{n}(ℚ̄) ↪ μ_{m}(ℚ̄) if n|m).

(c) For v ≠ ∞ and i = 0, 1, 2 the cup product and the above embeddings give rise to a perfect pairing

H^{i}(G_{v}, M) × H^{2−i}(G_{v}, M^{∗}) → H^{2}(G_{v}, μ_{n}(ℚ̄)) ↪ ℚ/ℤ.

(This also holds when v = ∞, if one replaces H^{0}(G_{v}, M) = H^{0}(ℝ, M) by Tate’s modified cohomology group: Ĥ^{0}(ℝ, M) := M^{τ=1}/(1 + τ)M, where τ is complex conjugation.)

(d) If v ≠ ∞ is a prime, then H^{i}(G_{v}, M) = (0) for i > 2 and

#H^{1}(G_{v}, M) = #H^{0}(G_{v}, M)#H^{2}(G_{v}, M)#(M ⊗ ℤ_{v}).

(e) If v ≠ ∞ and M ⊗ ℤ_{v} = 0 then H^{1}(G_{v}/I_{v}, M^{I_{v}}) and H^{1}(G_{v}/I_{v}, M^{∗I_{v}}) are the annihilators of each other under the pairing

H^{1}(G_{v}, M) × H^{1}(G_{v}, M^{∗}) → ℚ/ℤ.

By a collection of local conditions for M we shall mean a collection L = {L_{v}} of subgroups L_{v} ⊂ H^{1}(G_{v}, M) as v runs over all places of ℚ, such that L_{v} = H^{1}(G_{v}/I_{v}, M^{I_{v}}) for all but finitely many v. Note that if L is a collection of local conditions for M then L^{∗} = {L_{v}^{⊥}} is a collection of local conditions for M^{∗}. If L is a collection of local conditions for M we define the corresponding Selmer group, H_{L}^{1}(ℚ, M), to be the subgroup of x ∈ H^{1}(G_{ℚ}, M) such that for all places v of ℚ we have

res_{v}x ∈ L_{v} ⊂ H^{1}(G_{v}, M).

Suppose that v is a finite place of ℚ. We have an exact sequence

(0) → H^{0}(G_{v}, M) → M^{I_{v}} ⟶^{Frob_{v}−1} M^{I_{v}} → H^{1}(G_{v}/I_{v}, M^{I_{v}}) → (0),

and hence we see that #H^{1}(G_{v}/I_{v}, M^{I_{v}}) = #H^{0}(G_{v}, M).

We have the following important observation of Wiles [W3], inspired by a formula of Greenberg [Gre]. An important theme in number theory has been the calculation of Selmer groups. This result, although it does not allow the absolute calculation of Selmer groups, allows the comparison of two (dual) Selmer groups. In the applications in this paper we shall apply the theorem where the various data have been chosen to make one of the Selmer groups trivial. In such a situation it allows the exact calculation of the order of the non-trivial Selmer group.

Theorem 2.18 If L is a collection of local conditions for M then the Selmer group H_{L}^{1}(ℚ, M) is finite. Moreover we have the formula

\frac{#H_{L}^{1}(ℚ, M)}{#H_{L^{∗}}^{1}(ℚ, M^{∗})} = \frac{#H^{0}(G_{ℚ}, M)}{#H^{0}(G_{ℚ}, M^{∗})} ∏_{v} \frac{#L_{v}}{#H^{0}(G_{v}, M)}.

We note that all but finitely many terms in the product are 1 so that it makes sense. Proof of theorem 2.18: Choose a finite set, S, of places of ℚ, which contains ∞, all the places whose residue characteristic divides the order of M, all places at which M is ramified and all places at which L_{v} ≠ H^{1}(G_{v}/I_{v}, M^{I_{v}}). Let ℚ_{S} denote the maximal extension of ℚ unramified outside S and let G_{S} = Gal (ℚ_{S}/ℚ). Then we have an exact sequence

(0) → H_{L}^{1}(ℚ, M) → H^{1}(G_{S}, M) → ⨁_{v∈S} H^{1}(G_{v}, M)/L_{v}.

Because H^{1}(G_{S}, M) is finite (see cor. 4.15 of ch. 1 of [Mi]), we see that H_{L}^{1}(ℚ, M) is finite.

Dualising this sequence for M^{∗} and using theorem 2.17, we see that we have an exact sequence

⨁_{v∈S} L_{v} → H^{1}(G_{S}, M^{∗})^{∨} → H_{L^{∗}}^{1}(ℚ, M^{∗})^{∨} → (0).

On the other hand we have the following part of the Poitou-Tate nine term sequence (see thm. 4.10 of ch. 1 of [Mi]):

(0) → H^{0}(G_{S}, M) → (⨁_{v∈S} H^{0}(G_{v}, M)) /(1 + c)M → H^{2}(G_{S}, M^{∗})^{∨}
↓
H^{1}(G_{S}, M^{∗})^{∨} ←  ⨁_{v∈S} H^{1}(G_{v}, M)  ← H^{1}(G_{S}, M),

where we regard (1 + c)M as contained in H^{0}(G_{∞}, M). Replacing H^{1}(G_{v}, M) by L_{v} for v ∈ S and combining this with the previous exact sequence we get

(0)  → H^{0}(G_{S}, M) → (⨁_{v∈S} H^{0}(G_{v}, M)) /(1 + c)M
↓
⨁_{v∈S} L_{v} ← H_{L}^{1}(ℚ, M) ←  H^{2}(G_{S}, M^{∗})^{∨}
↓
H^{1}(G_{S}, M^{∗})^{∨} → H_{L^{∗}}^{1}(ℚ, M^{∗})^{∨} →  (0).

Theorem 2.18 follows from this and the global Euler characteristic formula (see thm. 5.1 of ch. 1 of [Mi]):

\frac{#H^{1}(G_{S}, M^{∗})}{#H^{0}(G_{S}, M^{∗})#H^{2}(G_{S}, M^{∗})} = \frac{(#M^{∗})}{#H^{0}(G_{∞}, M^{∗})} = #(1 + c)M.

□

We remark, but will not use elsewhere in this article, that the above argument works over any number field:

Theorem 2.19 Let F be a number field and M a discrete G_{F} module of finite cardinality. For each place v of F let G_{v} denote a decomposition group at v and if v is finite let I_{v} ⊂ G_{v} denote the inertia group. Fix subgroups L_{v} ⊂ H^{1}(G_{v}, M), and such that for all but finitely many v, L_{v} = H^{1}(G_{v}/I_{v}, M^{I_{v}}). Let H_{L}^{1}(F, M) (respectively, H_{L^{∗}}^{1}(F, M^{∗})) denote the inverse image of ∏_{v} L_{v} (respectively, ∏_{v} L_{v}^{⊥}) under the map H^{1}(G_{F} , M) → ∏_{v} H^{1}(G_{v}, M) (respectively, H^{1}(G_{F} , M^{∗}) → ∏_{v} H^{1}(G_{v}, M^{∗})). Then H_{L}^{1}(G, M) and H_{L^{∗}}^{1}(G, M^{∗}) are finite and we have the formula

\frac{#H_{L}^{1}(F, M)}{#H_{L^{∗}}^{1}(F, M^{∗})} = \frac{#H^{0}(G_{F} , M)}{#H^{0}(G_{F} , M^{∗})} ∏_{v} \frac{#L_{v}}{#H^{0}(G_{v}, M)}.

2.4 Representations of G_{ℚ_{ℓ}}

In this section we assume that ℓ is an odd prime. If G is any topological group then by a finite O[G]-module we shall mean a discrete O-module of finite cardinality with a continuous action of G. By a profinite O[G]-module we shall mean an inverse limit of finite O[G]-modules.

If M is a profinite O[G_{ℓ}]-module then we will call M

• good if for every discrete quotient M′ of M there is a finite flat group scheme F/ℤ_{ℓ} such that M′ ≅ F (ℚ̄_{ℓ}) as ℤ_{ℓ}[G_{ℓ}]-modules; • ordinary if there is an exact sequence

(0) → M^{(−1)} → M → M^{(0)} → (0)

of profinite O[G_{ℓ}]-modules such that I_{ℓ} acts trivially on M^{(0)} and by ϵ on M^{(−1)} (equivalently, if for all σ, τ ∈ I_{ℓ} we have (σ − ϵ(σ))(τ − 1) = 0 on M);

• semi-stable if M is either good or ordinary.

Suppose that R is a complete Noetherian local O-algebra with residue field k. We will call a continuous representation ρ : G_{ℓ} → GL_{2}(R) good, ordinary or semistable, if

det ρ|_{I_{ℓ}} = ϵ  (2.4.1)

and if the underlying profinite O[G_{ℓ}]-module, M_{ρ} is good, ordinary or semistable. We write ρ̄ for ρ mod 𝔪_{R}. We record the following consequence of Nakayama’s lemma.

Lemma 2.20 If M_{ρ} and ρ̄ are ordinary, then M_{ρ}^{(−1)} and M_{ρ}^{(0)} are each free of rank one over R and ρ is ordinary.

Remark 2.21 (a) These definitions are somewhat ad hoc, but at the moment that is all that seems to be available (though the work of Fontaine and Laffaille [FL] and its generalisations may well provide a more systematic setting).

(b) For part of the motivation for our definitions, see proposition 2.23 and remark 2.24. For further motivation, we shall see in the next chapter that representations arising from certain modular forms are semistable. Moreover the Fontaine-Mazur conjecture predicts that, conversely, any representation of G_{ℚ} with semistable restriction to G_{ℓ} arises from such a modular form. We shall state a weak form of the Fontaine-Mazur conjecture below (conjecture 3.17).

(c) The terminology in [W3] to describe representations of G_{ℓ} is slightly different. In particular, we impose the condition (2.4.1) in our definitions of good and ordinary, as this is all we shall need here. Assuming ρ satisfies this condition, the notion of ordinary in [W3] coincides with the one here, and ρ is flat in the sense of [W3] if and only if it is good but not ordinary. (Note that a representation may be both good and ordinary, for instance ρ_{E,ℓ} for an elliptic curve with good, ordinary reduction; see proposition 2.23.)

The following assertions, in the good case, are consequences of results of Raynaud [Ray] (in particular, see sec. 2.1, prop. 2.3.1 and cor. 3.3.6 of [Ray]). In the ordinary case they are elementary.

Lemma 2.22 (a) Stable profinite O[G_{ℓ}]-modules are closed under taking sub-objects, quotients and direct products. The same is true for ordinary profinite O[G_{ℓ}]-modules.

(b) Suppose that M is a profinite O[G_{ℓ}]-module and that {M_{i}} is a family of sub-objects with trivial intersection, such that each M/M_{i} is good (resp. ordinary). Then M is good (resp. ordinary).

(c) If M is a finite O[G_{ℓ}]-module then M is good if and only if there is a finite flat group scheme F/ℤ_{ℓ} such that M ≅ F(ℚ̄_{ℓ}) as ℤ_{ℓ}[G_{ℓ}]-modules.

(d) If M and M′ are profinite O[G_{ℓ}]-modules with M′ ≅ M as ℤ_{ℓ}[I_{ℓ}]-modules then M is good (resp. ordinary) if and only if M′ is good (resp. ordinary).

(e) Suppose that M is a profinite O[G_{ℓ}]-module which is finite and free over O. Then M is good if and only if there exists an ℓ-divisible group F/ℤ_{ℓ} such that M is isomorphic to the Tate module of F as a ℤ_{ℓ}[G_{ℓ}]-module.

Together with results stated in section 2.2, we have the following.

Proposition 2.23 Suppose that E is an elliptic curve over ℚ and O = ℤ_{ℓ}.

• if E has good (resp. semistable) reduction at ℓ, then ρ_{E,ℓ} and ρ̄_{E,ℓ} are good (resp. semistable);

• if E has semistable reduction at ℓ, then ρ_{E,ℓ} is ordinary if and only if ρ̄_{E,ℓ} is ordinary if and only if either E has multiplicative reduction or E has good ordinary reduction.

Remark 2.24 It is also true that if ρ_{E,ℓ} is good (resp. semistable), then E has good (resp. semistable) reduction at ℓ, but the result is more difficult and we shall not need it.

We will need a few more definitions. We will let ψ : I_{ℓ} → 𝔽̄_{ℓ}^{×} denote the character σ ↦ (σϖ)/ϖ mod ϖ where ϖ = \sqrt[ℓ^{2}−1]{ℓ}. If F is a field of characteristic other than ℓ and if M is a profinite ℤ_{ℓ}[G_{ℓ}]-module then we set M(1) = lim_{←} M ⊗_{ℤ_{ℓ}} μ_{ℓ^{n}}(F̄). If ρ : G_{ℓ} → GL_{2}(R) is ordinary the extension

(0) → R(1) → M_{ρ} → R → (0)

of R[I_{ℓ}]-modules gives rise to a class c_{ρ} ∈ H^{1}(I_{ℓ}, R(1)). Kummer theory and the valuation on (ℚ̄_{ℓ}^{I_{ℓ}})^{×} give rise to a map v : H^{1}(I_{ℓ}, R(1)) → H ⊗_{ℤ_{ℓ}} R → R, where H denotes the ℓ-adic completion of (ℚ̄_{ℓ}^{I_{ℓ}})^{×}. Then we also have the following lemma.

Lemma 2.25 (a) If ρ̄ : G_{ℓ} → GL_{2}(k) is good then either ρ̄ is ordinary or ρ̄|_{I_{ℓ}} ⊗ k̄ = ψ ⊕ ψ^{ℓ}.

(b) If ρ : G_{ℓ} → GL_{2}(R) is such that M_{ρ} is good and ρ̄ is ordinary, then ρ is good and ordinary.

(c) If ρ : G_{ℓ} → GL_{2}(R) is ordinary then ρ is good if and only if v(c_{ρ}) = 0.

Note that we need only consider the case that R has finite cardinality to prove the lemma. Parts (a) and (b) again follow from Raynaud’s results [Ray] (for part (b) consider the connected-étale sequences for M_{ρ} and Mρ̄).

We sketch the proof of part (c) using an argument suggested to us by Edixhoven. As in [Edi], prop. 8.2 it suffices to determine which extensions

(0) ⟶ R(1) ⟶ M ⟶ R ⟶ (0)  (2.4.2)

of R[I_{ℓ}]-modules arise from finite flat group schemes over ℤ_{ℓ}^{nr}, the ring of integers of the maximal unramified extension of ℤ_{ℓ}. By prop. 17.4 of [Oo], the extension (2.4.2) arises from a finite flat group scheme if and only if it corresponds to an extension of sheaves of R-modules in the fpqc topology over ℤ_{ℓ}^{nr}. Therefore we must compute the image of

Ext_{R−mod/fpqc}^{1}(R, R(1)) → Ext_{R[I_{ℓ}]}^{1}(R, R(1)).

Since the sheaf Ext^{1} of R by R(1) vanishes, this is equivalent to computing the image of

H_{fpqc}^{1}(ℤ_{ℓ}^{nr}, R(1)) → H^{1}(I_{ℓ}, R(1)).

Part (c) follows from the fact this is precisely the kernel of v.

Remark 2.26 The authors expect that the theory of Fontaine and Laffaille [FL] discussed in the next section could be used to prove that if ρ : G_{ℓ} → GL_{2}(R) is such that M_{ρ} is good and ρ̄ is good, then det ρ|_{I_{ℓ}} is cyclotomic, i.e., ρ is good.

For semistable representations ρ : G_{ℓ} → GL_{2}(O/λ^{n}) we shall define Osubmodules

H_{f}^{1}(G_{ℓ}, adρ) ⊂ H_{ss}^{1}(G_{ℓ}, adρ) ⊂ H^{1}(G_{ℓ}, adρ).

Before doing so, let us consider more generally a continuous representation ρ of a profinite group G with values in GL_{d}(O/λ^{n}). For each continuous cocycle ξ : G → adρ, we define a representation

\begin{aligned}ρ_{ξ} & : & G & → & GL_{2}(R_{n}) \\ & & g & ↦ & (1 + εξ(g))ρ(g) \end{aligned}

where R_{n} = (O/λ^{n})[ε]/(ε^{2}). We find that the map ξ → ρ_{ξ} induces a bijection between H^{1}(G, adρ) and equivalence classes of representations ρ′ : G → GL_{2}(R_{n}) such that ρ = ρ′ mod εR_{n}, where ρ′_{1} and ρ′_{2} are deemed equivalent if they are conjugate by an element of 1 + εM_{2}(R_{n}). Now let M_{ρ} denote the (O/λ^{n})[G]-module corresponding to ρ, and for each continuous cocycle ξ, let E_{ξ} denote the R_{n}[G]-module corresponding to ρ_{ξ}. Note that multiplication by ε defines an isomorphism from M_{ρ} = E_{ξ}/εE_{ξ} to εE_{ξ} of (O/λ^{n})[G]-modules. We thus obtain an extension

0 → M_{ρ} → E_{ξ} → M_{ρ} → 0

and hence a class in Ext^{1}(M_{ρ}, M_{ρ}) in the category of profinite (O/λ^{n})[G]- modules. Moreover if ρ_{ξ_{1}} and ρ_{ξ_{2}} are equivalent, then E_{ξ_{1}} and E_{ξ_{2}} define the same extension class. We thus obtain a map

H^{1}(G, adρ) → Ext^{1}(M_{ρ}, M_{ρ}),

which the reader can check is an (O/λ^{n})-linear isomorphism. Classes in H^{1}(G, ad^{0}ρ) correspond to the equivalence classes of representations ρ′ satisfying det ρ′ = det ρ, and ϕ in Hom(G, O/λ^{n}) corresponds to the twist of ρ by the character g ↦ 1+εϕ(g). In the case ℓ |̸d where we have the decomposition

H^{1}(G, adρ) = H^{1}(G, ad^{0}ρ) ⊕ Hom(G, O/λ^{n}),

we see that if ρ′ corresponds to a class in H^{1}(G, adρ), then the projection to Hom(G, O/λ^{n}) is the homomorphism ϕ such that det ρ′ = (1 + εdϕ) det ρ.

We now return to the case of a semistable representation

ρ : G_{ℓ} → GL_{2}(O/λ^{n})

and define the cohomology groups H_{f}^{1}(G_{ℓ}, adρ) and H_{ss}^{1}(G_{ℓ}, adρ) as follows. Let H_{ss}^{1}(G_{ℓ}, adρ) denote the natural image in H^{1}(G_{ℓ}, adρ) of Ext^{1}(M_{ρ}, M_{ρ}) taken in the category of semi-stable profinite O/λ^{n}[G_{ℓ}]-modules. If ρ is not good then H_{f}^{1}(G_{ℓ}, adρ) is defined to be H_{ss}^{1}(G_{ℓ}, adρ). If, however, ρ is good then H_{f}^{1}(G_{ℓ}, adρ) will denote the natural image in H^{1}(G_{ℓ}, adρ) of the group Ext^{1}(M_{ρ}, M_{ρ}) taken in the category of good profinite O/λ^{n}[G_{ℓ}]-modules. We define H_{f}^{1}(G_{ℓ}, ad^{0}ρ) (resp. H_{ss}^{1}(G_{ℓ}, ad^{0}ρ)) as the intersection of H_{f}^{1}(G_{ℓ}, adρ) (resp. H_{ss}^{1}(G_{ℓ}, adρ)) and H^{1}(G_{ℓ}, ad^{0}ρ). Note that if ρ is good (resp. semistable) and ξ : G_{ℓ} → adρ is a cocycle, then E_{ξ} is good (resp. semistable) if and only if ρ_{ξ} is.

Suppose that ρ : G_{ℓ} → GL_{2}(O/λ^{n}) is ordinary. Consider the exact sequence

0 → M_{ρ}^{(−1)} → M_{ρ} → M_{ρ}^{(0)} → 0,

where M_{ρ}^{(−1)} denote the maximal submodule of M_{ρ} where I_{ℓ} acts by ϵ. Let ad^{(−1)}ρ denote the submodule Hom(M_{ρ}^{(0)}, M_{ρ}^{(−1)}) of ad^{0}ρ. Then

H_{ss}^{1}(G_{ℓ}, adρ) = ker (H^{1}(G_{ℓ}, adρ) → H^{1}(I_{ℓ}, adρ/ad^{(−1)}ρ)).

The same is true with ad^{0} replacing ad.

If ρ : G_{ℓ} → GL_{2}(O) is semi-stable then we set

H_{f}^{1}(G_{ℓ}, adρ ⊗ K/O) = lim_{→} H_{f}^{1}(G_{ℓ}, adρ ⊗ (λ^{−n}/O)) ⊂ H^{1}(G_{ℓ}, adρ ⊗ K/O).

We define H_{f}^{1}(G_{ℓ}, ad^{0}ρ ⊗ K/O) as

H_{f}^{1}(G_{ℓ}, adρ ⊗ K/O) ∩ H^{1}(G_{ℓ}, ad^{0}ρ ⊗ K/O).

We make similar definitions for H_{ss}^{1}(G_{ℓ}, adρ⊗K/O) and H_{ss}^{1}(G_{ℓ}, ad^{0}ρ⊗K/O). We will need the following calculations.

Proposition 2.27 (a) Suppose that ρ̄: G_{ℓ} → GL_{2}(k) is semi-stable. Then

#H_{f}^{1}(G_{ℓ}, ad^{0}ρ̄) ≤ #H^{0}(G_{ℓ}, ad^{0}ρ̄)#k

and equality holds if ρ̄ is ordinary^{3}.

(b) Suppose that ρ : G_{ℓ} → GL_{2}(O) is both good and ordinary. Let χ_{1} and χ_{2} be the unramified characters such that ρ ∼ (\begin{matrix}χ_{1}ϵ & ∗ \\ 0 & χ_{2} \end{matrix} ). Let

c_{ℓ} = (χ_{1}/χ_{2})(Frob_{ℓ}) − 1.

If c_{ℓ} ≠ 0, then

H_{ss}^{1}(G_{ℓ}, ad^{0}ρ ⊗ K/O)/H_{f}^{1}(G_{ℓ}, ad^{0}ρ ⊗ K/O)

is finite of order #(O/c_{ℓ}O).

3The authors expect equality to hold without this hypothesis; cf. remark 2.26.

(c) If ρ mod λ is either not good or not ordinary, then

H_{ss}^{1}(G_{ℓ}, ad^{0}ρ ⊗ K/O) = H_{f}^{1}(G_{ℓ}, ad^{0}ρ ⊗ K/O).

Proof: Part (c) is clear, and for parts (a) and (b) it suffices to prove the following two results.

Proposition 2.28 Suppose that ρ : G_{ℓ} → GL_{2}(O/λ^{n}) is good. Then

#H_{f}^{1}(G_{ℓ}, adρ) = #H^{0}(G_{ℓ}, ad^{0}ρ)(#O/λ^{n})^{2}.

Lemma 2.29 Suppose that ρ : G_{ℓ} → GL_{2}(O/λ^{n}) is ordinary, so that we have ρ ∼ (\begin{matrix}χ_{1}ϵ & ∗ \\ 0 & χ_{2} \end{matrix} ), for some unramified characters χ_{1} and χ_{2}.

(a) We have

#H_{ss}^{1}(G_{ℓ}, ad^{0}ρ) ≤ #H^{0}(G_{ℓ}, ad^{0}ρ)#(O/λ^{n})#(O/(λ^{n}, c_{ℓ}))

where c_{ℓ} = (χ_{1}/χ_{2})(Frob_{ℓ})−1. Moreover if ρ is good, then equality holds.

(b) If n = 1 and ρ is not good then #H_{ss}^{1}(G_{ℓ}, ad^{0}ρ) = #k.

End of proof of proposition 2.27: To deduce proposition 2.27 (in the good case) from proposition 2.28 and lemma 2.29, note that if ρ : G_{ℓ} → GL_{2}(O/λ^{n}) is good, then

H^{1}(G_{ℓ}/I_{ℓ}, O/λ^{n}) ⊂ H_{f}^{1}(G_{ℓ}, adρ) ∩ H^{1}(G_{ℓ}, O/λ^{n})

by lemma 2.22 (d), and this gives the inequality in proposition 2.27(a). Furthermore if ρ is also ordinary, then the above group is contained in

Im (H_{f}^{1}(G_{ℓ}, adρ) → H^{1}(G_{ℓ}, O/λ^{n})) ⊂ H^{1}(G_{ℓ}/I_{ℓ}, O/λ^{n})

where the last inclusion comes from lemma 2.25 (b). Therefore

#H_{f}^{1}(G_{ℓ}, ad^{0}ρ) = #H^{0}(G_{ℓ}, ad^{0}ρ)#(O/λ^{n}).
□

We will prove proposition 2.28 in the next section using the theory of Fontaine and Laffaille. The proof of lemma 2.29 is a somewhat technical exercise in the Galois cohomology of local fields, for which we refer the reader to ch. 1 of [W3], prop. 1.9, parts (iii) and (iv). We remark that for our purposes, inequalities would suffice in part (b) of proposition 2.27 and part (a) of lemma 2.29 (and this is all that is proved in [W3]). We have included the precise formulas for the sake of completeness, since they are not much more difficult to obtain. The only additional observation required is that if ρ is good, then the composite of the natural maps

H^{1}(G_{ℓ}/I_{ℓ}, (ad^{0}ρ/ad^{(−1)}ρ)^{I_{ℓ}}) → H^{1}(G_{ℓ}, ad^{0}ρ/ad^{(−1)}ρ) → H^{2}(G_{ℓ}, ad^{(−1)}ρ)

is trivial. To prove this, rewrite the composite as

H^{1}(G_{ℓ}/I_{ℓ}, O/λ^{n}) → H^{1}(G_{ℓ}, O/λ^{n}) → H^{2}(G_{ℓ}, ad^{(−1)}ρ)

with the second map given by ∪c_{ρ} where c_{ρ} is the class in H^{1}(G_{ℓ}, ad^{(−1)}ρ) defining the extension M_{ρ} and apply lemma 2.25 (c). (In fact, one only needs the easier half of lemma 2.25 (c): if ρ is good then v(c_{ρ}) = 0.)

2.5 The theory of Fontaine and Laffaille

In this section we again assume that ℓ is an odd prime. We mentioned in the last section the importance of understanding good representations of G_{ℓ}. However the definition of good is somewhat indirect, and this makes computations difficult. The key result we use to address the problem is an equivalence between the category FF_{O} of good finite O[G_{ℓ}]-modules and a category MF_{O} which we define below following Fontaine and Laffaille [FL]. The beauty and utility of the result stems from the elementary algebraic nature of the definition of MF_{O}; we can convert questions about good representations into questions in linear algebra.

Remark 2.30 Let GF_{O} denote the category of finite flat commutative group schemes over ℤ_{ℓ} with an action of O. By results of Raynaud [Ray], taking ℚ̄_{ℓ} points defines an equivalence between the categories GF_{O} and FF_{O}. The equivalence between GF_{O} and a category closely related to MF_{O} was first established by Fontaine [Fo2] and [Fo1]. While Fontaine’s results would suffice for our purposes here, our formulation will be closer to that in [FL], where an equivalence between MF_{O} and FF_{O} is defined as part of a more general construction of representations of G_{ℓ} from linear-algebraic data. We caution however that our formulation is not exactly the same as that of [FL] since we wish to work with covariant functors.

We now turn to the definition of MF_{O}. The objects are O-modules D of finite cardinality together with a distinguished submodule D^{0} and O-linear maps ϕ_{−1} : D → D and ϕ_{0} : D^{0} → D which satisfy:

• ϕ_{−1}|_{D^{0}} = ℓϕ_{0},

• Im ϕ_{−1} + Im ϕ_{0} = D.

The morphisms are O-linear maps compatible with the additional data of the distinguished submodules and maps ϕ.

It is useful to note that if D is an object of MF_{O}, then there is a surjection

ϕ_{−1} ⊕ ϕ_{0} : D/D^{0} ⊕ D^{0}/ℓD^{0} → → D/ℓϕ_{0}(D^{0}),

and on counting orders we see that this is in fact an isomorphism. Thus there is an isomorphism

D/(D^{0} + λD) ⊕ D^{0}/λD^{0} →^{∼} D/λD.

It follows that D^{0}/λD^{0} → D/λD is injective, and hence that D^{0} is (noncanonically) a direct summand of D as an O-module. Note also that ϕ_{0} is injective, and if D = D^{0} ⊕ D′ as O-modules, then also D = ϕ_{0}(D^{0}) ⊕ ϕ_{−1}(D′) as O-modules.

It is then straightforward to check that there is a contravariant functor ·^{∗} from MF_{O} to itself defined by:

• D^{∗} = Hom(D, ℚ_{ℓ}/ℤ_{ℓ});

• (D^{∗})^{0} = Hom(D/D^{0}, ℚ_{ℓ}/ℤ_{ℓ});

• ϕ_{−1}^{∗}(f)(z) = f(ℓx + y), where z = ϕ_{−1}(x) + ϕ_{0}(y);

• ϕ_{0}^{∗}(f)(z) = f(x mod D^{0}), where z ≡ ϕ_{−1}(x) mod (ϕ_{0}D^{0}).

Moreover the canonical isomorphism D ≅ (D^{∗})^{∗} of O-modules defines a natural isomorphism in MF_{O}.

We leave it as an exercise for the reader to use the above observations to define cokernels and then kernels of morphisms in MF_{O} and verify that it is an abelian category (or see [FL], sec. 1).

Theorem 2.31 There is a covariant functor 𝔻 : FF_{O} → MF_{O} which defines an O-additive equivalence of categories. Moreover if M is an object of FF_{O}, then we have

(a) M and 𝔻(M) have the same cardinality;

(b) 𝔻(M) = 𝔻(M)^{0} if and only if M is unramified.

Remark 2.32 (a) It follows on applying part (a) to M/λ^{i}M for each i that M and 𝔻(M) are in fact (non-canonically) isomorphic as O-modules.

(b) As mentioned above, our formulation differs from that of Fontaine and Laffaille in that we are using covariant functors. To deduce theorem 2.31 from the results in [FL], sec. 9, we define 𝔻 as a quasi-inverse of the functor \underline{U}_{S}(·^{∗}[1]), where \underline{U}_{S} is defined in [FL] and [1] indicates a shift by 1 in filtration degrees. In particular, if F is a finite flat group scheme over ℤ_{ℓ} with an action of O, then the underlying O-module of 𝔻(F(ℚ̄_{ℓ})) can be identified with the covariant Dieudonné module of F_{/𝔽_{ℓ}}, and ϕ_{−1} with F.

Suppose now that ρ : G_{ℓ} → GL_{2}(O/λ^{n}) is a good continuous representation. Let D_{ρ} = 𝔻(M_{ρ}) be the corresponding object of MF_{O}. Then D_{ρ} ≅ (O/λ^{n})^{2} as an O-module while D_{ρ}^{0} ≅ O/λ^{n}.

Lemma 2.33 The following are isomorphic.

(a) The group of extensions of M_{ρ} by itself in the category of good finite O/λ^{n}[G_{ℓ}]-modules.

(b) The group of extensions of D_{ρ} by itself in the full subcategory of MF_{O} consisting of objects which are O/λ^{n}-modules.

(c) Pairs (α_{−1}, α_{0}) where α_{−1} ∈ Hom_{O}(D_{ρ}, D_{ρ}), α_{0} ∈ Hom_{O}(D_{ρ}^{0}, D_{ρ}) and ℓα_{0} = α_{−1}|_{D_{ρ}^{0}}, modulo pairs of the form (aϕ_{ρ,−1} −ϕ_{ρ,−1}a, aϕ_{ρ,0} −ϕ_{ρ,0}a|_{D^{0}}) where a ∈ Hom_{O}(D_{ρ}, D_{ρ}) and aD_{ρ}^{0} ⊂ D_{ρ}^{0}.

Proof: The first two groups of extensions are isomorphic by the Fontaine-Laffaille theorem. Following Ramakrishna [Ram] we explain how to calculate the second group of extensions. We will write D for D_{ρ}. If

(0) → D → E → D → (0)

is an extension then we have as O-modules that E ≅ D^{2} by an isomorphism such that E^{0} →^{∼} (D^{0})^{2}. Then E is determined by giving elements α_{E,−1} ∈ Hom(D, D) and α_{E,0} ∈ Hom(D^{0}, D) with ℓα_{E,0} = α_{E,−1}|_{D^{0}}. Explicitly

ϕ_{E,−1} = (\begin{matrix}ϕ_{−1} & α_{E,−1} \\ 0 & ϕ_{−1} \end{matrix} ) and ϕ_{E,0} = (\begin{matrix}ϕ_{0} & α_{E,0} \\ 0 & ϕ_{0} \end{matrix} ) .

Two such extensions E and E′ are isomorphic if there is an element a ∈ End (D) such that aD^{0} ⊂ D^{0},

(\begin{matrix}1 & a \\ 0 & 1 \end{matrix}) (\begin{matrix}ϕ_{−1} & α_{E,−1} \\ 0 & ϕ_{−1} \end{matrix} ) = (\begin{matrix}ϕ_{−1} & α_{E′,−1} \\ 0 & ϕ_{−1} \end{matrix} ) (\begin{matrix}1 & a \\ 0 & 1 \end{matrix}) ,

and

(\begin{matrix}1 & a \\ 0 & 1 \end{matrix}) (\begin{matrix}ϕ_{0} & α_{E,0} \\ 0 & ϕ_{0} \end{matrix} ) = (\begin{matrix}ϕ_{0} & α_{E′,0} \\ 0 & ϕ_{0} \end{matrix} ) (\begin{matrix}1 & a|_{D^{0}} \\ 0 & 1 \end{matrix} );

or equivalently if aD^{0} ⊂ D^{0}, α_{E,−1} − α_{E′,−1} = ϕ_{−1}a − aϕ_{−1} and α_{E,0} − α_{E′,0} = ϕ_{0}a|_{D^{0}} − aϕ_{0}. The lemma follows. □

Corollary 2.34 The group, H_{f}^{1}(G_{ℓ}, adρ), of extensions of M_{ρ} by itself in the category of good, finite O/λ^{n}[G_{ℓ}]-modules is (non-canonically) isomorphic to (O/λ^{n})^{2} ⊕ H^{0}(I_{ℓ}, M_{ρ}).

Proof: Choose generators e_{0}, e_{−1} of D, such that e_{0} is a generator of D^{0}. With respect to this basis let ϕ_{0} have matrix (\begin{matrix}x \\ z \end{matrix}) and ϕ_{−1} have matrix (\begin{matrix}ℓx & y \\ ℓz & w \end{matrix} ). If

a has matrix (\begin{matrix}a_{1} & a_{2} \\ 0 & a_{3} \end{matrix} ) then a direct calculation shows that

\begin{aligned}(aϕ_{−1} − ϕ_{−1}a, & aϕ_{0} − ϕ_{0}a|_{D^{0}}) = \\ & ((\begin{matrix}ℓa_{2}z & (a_{1} − a_{3})y + a_{2}(w − ℓx) \\ (a_{3} − a_{1})ℓz & −ℓza_{2} \end{matrix}) , (\begin{matrix}a_{2}z \\ (a_{3} − a_{1})z \end{matrix} )) . \end{aligned}

Thus the Ext group we want is the quotient of the set of pairs of matrices (α_{−1}, α_{0}) as above by the submodule generated by

((\begin{matrix}ℓz & w − ℓx \\ 0 & −ℓz \end{matrix}) , (\begin{matrix}z \\ 0 \end{matrix})) and ((\begin{matrix}0 & −y \\ ℓz & 0 \end{matrix} ) , (\begin{matrix}0 \\ z \end{matrix})) .

Note that either z or w is a unit in O/λ from which it follows that the Ext group we want is isomorphic to (O/λ^{n})^{2} ⊕ O/(z, λ^{n}). On the other hand H^{0}(I_{ℓ}, M_{ρ}) corresponds to the largest submodule C ⊂ D^{0} such that ϕ_{0}C = C, i.e. C = {d ∈ D^{0} : zd = 0} ≅ O/(z, λ^{n}). □

Corollary 2.35 Suppose that ρ : G_{ℓ} → GL_{2}(O/λ^{n}) is a continuous good representation. Then H_{f}^{1}(G_{ℓ}, adρ) is isomorphic to (O/λ^{n})^{2} ⊕ H^{0}(G_{ℓ}, ad^{0}ρ).

Proof: If ρ is not ordinary then H^{0}(G_{ℓ}, ad^{0}ρ) and H^{0}(I_{ℓ}, M_{ρ}) are both trivial, so suppose that ρ is ordinary. In this case, let ρ′ denote the ordinary representation defined by

M_{ρ′} = Hom(M_{ρ}^{(0)}, M_{ρ}) ⊂ ad^{0}(ρ)

and let M_{ρ′}^{(−1)} = ad^{0}(ρ)^{(−1)}. Then

H^{0}(G_{ℓ}, ad^{0}ρ) = H^{0}(G_{ℓ}, M_{ρ′}) ⊂ H^{0}(I_{ℓ}, M_{ρ′}) ≅ H^{0}(I_{ℓ}, M_{ρ}).

Since H^{0}(I_{ℓ}, M_{ρ′}^{(−1)}) is trivial, the restriction homomorphism

H^{1}(G_{ℓ}, M_{ρ′}^{(−1)}) → H^{1}(I_{ℓ}, M_{ρ′}^{(−1)})

is injective. We deduce from the long exact sequences associated to

(0) → M_{ρ′}^{(−1)} → M_{ρ′} → O/λ^{n} → 0

that H^{0}(G_{ℓ}, M_{ρ′}) = H^{0}(I_{ℓ}, M_{ρ′}). □

2.6 Deformations of representations

In this section we shall review Mazur’s theory of deformations of representations of profinite groups (see [Maz3]).

Let C_{O} denote the category whose objects are complete noetherian local O-algebras with residue field k and whose morphisms are O-algebra homomorphisms which are local (i.e. take maximal ideals into maximal ideals). (The structure maps from O to every object of C_{O} are also assumed to be local.) Let G denote a topologically finitely generated profinite group and let ρ̄ denote an absolutely irreducible representation of G into GL_{d}(k). (In fact all we shall use in the sequel is that k is the centraliser in M_{d}(k) of the image of ρ̄.) Let D_{0} denote the category of profinite O[G]-modules with continuous morphisms. We will let D denote a full subcategory of D_{0} which is closed under taking subobjects, quotients and direct products and which contains Mρ̄. Note that if M is an object of D_{0} and M_{i} is a collection of subobjects which have trivial intersection and such that each M/M_{i} is an object of D, then M is an object of D, since M ⊂ ∏_{i} M/M_{i}.

Let χ : G → O^{×} be a continuous character such that det ρ̄ = χ mod λ. By a lifting of ρ̄ of type D = (O, χ, D) we shall mean an object R of C_{O} and a continuous representation ρ : G → GL_{d}(R) such that:

(a) ρ mod 𝔪_{R} = ρ̄,

(b) det ρ = χ,

(c) M_{ρ} is an object of D.

Note that if ϕ : R → R′ and ρ : G → GL_{d}(R) is of type D so is ϕ ∘ ρ.

Theorem 2.36 There is a lifting ρ_{D}^{univ} : G → GL_{d}(R_{D}) of ρ̄ of type D such that if ρ : G → GL_{d}(R) is any lifting of ρ̄ of type D then there is a unique homomorphism of O-algebras ϕ : R_{D} → R such that ρ is conjugate to ϕ∘ρ_{D}^{univ}.

The representation ρ_{D}^{univ} is referred to as the universal deformation of type D. Mazur [Maz3] proved this theorem for D_{0} and certain other categories D. Ramakrishna [Ram] observed that the arguments work with any category D satisfying the above hypotheses. We will sketch a proof which was suggested by Faltings. (We remark that another explicit construction of the deformation ring has been given by de Smit and Lenstra in [dSL].) Proof of theorem 2.36: Choose a sequence g_{1}, ..., g_{r} of topological generators of G and liftings A_{1}, ..., A_{r} of ρ̄(g_{1}), ..., ρ̄(g_{r}) to M_{d}(O). Define a mapping ι : M_{d}(O) → M_{d}(O)^{r} which sends x to (xA_{1} − A_{1}x, ..., xA_{r} − A_{r}x). Since ι has torsion-free cokernel, so we can decompose

M_{d}(O)^{r} = ι(M_{d}(O)) ⊕ V,

for some submodule V ⊂ M_{d}(O)^{r}. If ρ : G → GL_{d}(R) is a lifting of ρ̄ of type D set v_{ρ} = (ρ(g_{1}) − A_{1}, ..., ρ(g_{r}) − A_{r}) ∈ M_{d}(R)^{r}. Note that v_{ρ} ≡ 0 mod 𝔪_{R} and that v_{ρ} completely determines ρ. We call the lifting ρ well-placed if v_{ρ} belongs to V ⊗_{O} R ⊂ M_{d}(R)^{r}. The crucial observation is the following result.

Lemma 2.37 If ρ : G → GL_{d}(R) is a lifting of ρ̄ of type D then there is a unique conjugate ρ′ of ρ which is well-placed.

The lemma is first proved for algebras R such that 𝔪_{R}^{n} = (0) by induction on n, and then one deduces the general case.

In virtue of the lemma it suffices to find a universal well-placed lifting of type D. Let e_{1}, ..., e_{s} be a basis of V as an O-module. If ρ is a well-placed lifting of ρ̄ of type D then we can write v_{ρ} = ∑_{i=1}^{s} v_{ρ,i}e_{i} for unique elements v_{ρ,i} ∈ 𝔪_{R} and we can define a homomorphism θ_{ρ} : O[[T_{1}, ..., T_{s}]] → R sending T_{i} to v_{ρ,i}. Note that ρ is completely determined by θ_{ρ} (ρ(g_{i}) = A_{i}+∑_{j=1}^{s} θ_{ρ}(T_{j})e_{ji}, where e_{j} = (e_{j1}, ..., e_{jr})). Let I denote the intersection of all ideals J of O[[T_{1}, ..., T_{s}]] such that there is a representation ρ_{J} : G → GL_{d}(O[[T_{1}, ..., T_{s}]]/J) of type D with ρ_{J}(g_{i}) = A_{i} + ∑_{j=1}^{s} T_{j}e_{ji} for all i. Let R_{D} denote the quotient of O[[T_{1}, ..., T_{s}]] by I. Then one can check that there is a representation ρ^{univ} : G → GL_{d}(R_{D}) with ρ^{univ}(g_{i}) = A_{i} + ∑_{j=1}^{s} T_{j}e_{ji} for all i, and that this is the desired universal representation. □

We will need a few elementary properties of these universal deformations. More precisely we will need to know how these universal rings change when we change the base field, we will need to know how to calculate the equicharacteristic tangent space of these rings and more generally how to calculate ℘/℘^{2} for certain prime ideals ℘. The first of these lemmas is a remark of Faltings, the second is due to Mazur [Maz3] and the third to Wiles [W3].

Lemma 2.38 Let K′/K be a finite extension with ring of integers O′ and residue field k′. Let D′ denote the full sub-category of the category of profinite O′[G]-modules such that the underlying object of D_{0} is actually an object of D. Let D′ = (O′, χ, D′). Then R_{D′} = R_{D} ⊗_{O} O′ and ρ_{D′}^{univ} = ρ_{D}^{univ} ⊗ 1.

Proof: Let R̃_{D′} denote the subring of R_{D′} consisting of elements which reduce modulo the maximal ideal to an element of k ⊂ k′. Similarly let R̃_{D} denote the subring of R_{D} ⊗_{O} O′ consisting of elements which reduce modulo the maximal ideal to an element of k ⊂ k′. Then ρ_{D′}^{univ} is in fact valued in GL_{d}(R̃_{D′}) and ρ_{D}^{univ} ⊗ 1 is in fact valued in GL_{d}(R̃_{D}). The universal properties give natural maps α : R_{D} → R̃_{D′} and β : R_{D′} → R_{D} ⊗_{O} O′. Moreover they show that the composite (α ⊗ 1) ∘ β : R_{D′} → R_{D′} is the identity and that β ∘ α : R_{D} → R̃_{D} is the natural embedding. Thus β is an isomorphism. □

We will let D^{(n)} denote the full subcategory of D whose objects are killed by λ^{n}. Suppose that M is an object of D_{0} which is finite and free over O/λ^{n}. Recall from section 2.4 that H^{1}(G, End (M)) may be identified with Ext_{D_{0}^{(n)}}^{1}(M, M). If M is an object of D^{(n)} which is finite and free over O/λ^{n}, then we have a natural inclusion

Ext_{D^{(n)}}^{1}(M, M) ⊂ Ext_{D_{0}^{(n)}}^{1}(M, M) ≅ H^{1}(G, End (M)).

We define H_{D}^{1}(G, End (M)) to be the image of Ext_{D^{(n)}}^{1}(M, M) in the group H^{1}(G, End (M)), and H_{D}^{1}(G, End^{0}(M)) to be the intersection

H^{1}(G, End^{0}(M)) ∩ H_{D}^{1}(G, End (M)).

Lemma 2.39 There is a canonical isomorphism of k-vector spaces

Hom_{k}(𝔪_{R_{D}}/(λ, 𝔪_{R_{D}}^{2}), k) ≅ H_{D}^{1}(G, ad^{0}ρ̄).

Proof: There is a natural bijection between Hom_{k}(𝔪_{R_{D}}/(λ, 𝔪_{R_{D}}^{2}), k) and the set of O-algebra homomorphisms from R_{D} to the algebra k[ε] where ε^{2} = 0 (the correspondence associates ϕ : R_{D} → k[ε] to ϕ|_{𝔪_{R_{D}}}). Hence there is a bijection to the set of liftings ρ : G → GL_{d}(k[ε]) of ρ̄ of type D, modulo conjugation by elements of 1 + εM_{d}(k). On the other hand, recall from section 2.4 that there is a natural bijection between Ext_{D_{0}^{(1)}}^{1}(Mρ̄, Mρ̄) and the set of of all continuous liftings ρ : G → GL_{d}(k[ε]) of ρ̄ modulo conjugation by elements of 1 + εM_{d}(k). Moreover a lifting ρ is type D if and only if det ρ = det ρ̄ and the corresponding extension M_{ρ} is an object of D. The lemma follows on checking linearity. □

Now suppose that θ : R_{D} → O is an O-algebra homomorphism. Let ℘ denote the kernel of θ and let ρ = θ ∘ ρ_{D}^{univ}. We set

H_{D}^{1}(G, ad^{0}ρ ⊗ K/O) = lim_{→} H_{D}^{1}(G, ad^{0}ρ ⊗ λ^{−n}/O) ⊂ H^{1}(G, ad^{0}ρ ⊗ K/O).

Lemma 2.40 There is a canonical O-linear isomorphism

Hom_{O}(℘/℘^{2}, K/O) ≅ H_{D}^{1}(G, ad^{0}ρ ⊗ K/O).

Proof: One shows in a very similar manner to the proof of lemma 2.39 that for all n there is a natural isomorphism.

Hom_{O}(℘/℘^{2}, O/λ^{n}) ≅ H_{D}^{1}(G, ad^{0}ρ ⊗ O/λ^{n}).
□

2.7 Deformations of Galois representations

Again in this section we assume that ℓ is an odd prime. Let ρ̄: G_{ℚ} → GL_{2}(k) denote a continuous absolutely irreducible representation. Suppose moreover that det ρ̄ = ϵ and that ρ̄ is semi-stable in the sense that

• ρ̄|_{G_{ℓ}} is semi-stable,

• and if p ≠ ℓ then #ρ̄(I_{p})|ℓ.

Note that if E/ℚ is a semistable elliptic curve then ρ̄_{E,ℓ} : G_{ℚ} → GL_{2}(𝔽_{ℓ}), satisfies these conditions if it is irreducible. By a theorem of Mazur (theorem 2.9) this will be the case if ℓ > 7.

Let Σ denote a finite set of prime numbers. If R is an object of C_{O} then we say that a continuous lifting ρ : G_{ℚ} → GL_{2}(R) of ρ̄ is of type Σ if the following hold.

• det ρ = ϵ.

• ρ|_{G_{ℓ}} is semi-stable.

• If ℓ ∉ Σ and ρ̄|_{G_{ℓ}} is good then ρ|_{G_{ℓ}} is good.

• If p ∉ Σ ∪ {ℓ} and ρ̄ is unramified at p then ρ is unramified at p.

• If p ∉ Σ ∪ \{ℓ\} then ρ|_{I_{p}} ∼ (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix}).

Roughly speaking we require that at primes p ∉ Σ, ρ is as unramified as could be hoped for and we require that ρ|_{G_{ℓ}} is semi-stable. Note that if Σ ⊂ Σ′ and ρ is a lifting of type Σ then it is also a lifting of type Σ′. Note also that if E/ℚ is an elliptic curve which is semi-stable at ℓ and for which ρ̄_{E,ℓ} is irreducible and semi-stable, then ρ_{E,ℓ} : G_{ℚ} → GL_{2}(ℤ_{ℓ}) is a lifting of type Σ if Σ contains all the primes for which E has bad reduction.

Now suppose that ρ : G_{ℚ} → GL_{2}(O/λ^{n}) is a lifting of ρ̄ of type Σ. We will write H_{Σ}^{1}(ℚ, ad^{0}ρ) for H_{L_{Σ}}^{1}(ℚ, ad^{0}ρ), where • L_{Σ,p} = H^{1}(G_{p}/I_{p}, (ad^{0}ρ)^{I_{p}}) if p ∉ Σ ∪ {ℓ};

• L_{Σ,p} = H^{1}(G_{p}, (ad^{0}ρ)) if p ∈ Σ and p ≠ ℓ;

• L_{Σ,ℓ} = H_{f}^{1}(G_{ℓ}, (ad^{0}ρ)) if ℓ ∉ Σ;

• L_{Σ,ℓ} = H_{ss}^{1}(G_{ℓ}, (ad^{0}ρ)) if ℓ ∈ Σ.

Note that the pairing ad^{0}ρ × ad^{0}ρ → O/λ^{n} given by (a, b) ↦ tr (ab) is perfect and respects the action of G_{ℚ} (i.e. tr ((ad^{0}ρ)(g)a, (ad^{0}ρ)(g)b) = tr (ab) for all g ∈ G_{ℚ}). Choosing a generator for the O-module Hom(O/λ^{n}, ℚ_{ℓ}/ℤ_{ℓ}), we obtain an isomorphism of O[G_{ℚ}]-modules

ad^{0}ρ(1) ≅ Hom_{O}(ad^{0}ρ, O/λ^{n})(1) ⟶^{∼} Hom(ad^{0}ρ, ℚ_{ℓ}/ℤ_{ℓ})(1) = (ad^{0}ρ)^{∗}.

The O-submodule of H^{1}(ℚ, ad^{0}ρ(1)) corresponding to H_{L_{Σ}^{∗}}^{1}(ℚ, (ad^{0}ρ)^{∗}) is independent of the choice of generator and we denote it H_{Σ}^{1}(ℚ, ad^{0}ρ(1)). Thus H_{Σ}^{1}(ℚ, ad^{0}ρ(1)) is defined by the local conditions {L_{Σ,v}^{⊥}} where the orthogonality is with respect to the pairing

H^{1}(G_{v}, ad^{0}ρ) × H^{1}(G_{v}, ad^{0}ρ(1)) → ℚ_{ℓ}/ℤ_{ℓ}

arising from the above isomorphism. We may equivalently regard the orthogonality as being with respect to the natural perfect O-bilinear pairing defined by the composition

H^{1}(G_{v}, ad^{0}ρ) × H^{1}(G_{v}, ad^{0}ρ(1)) → H^{2}(G_{v}, ad^{0}ρ ⊗_{O} ad^{0}ρ(1))
→ H^{2}(G_{v}, O/λ^{n}(1)) ≅ O/λ^{n}.

Note that if p ≠ ℓ we have that

• L_{Σ,p}^{⊥} = H^{1}(G_{p}/I_{p}, (ad^{0}ρ)(1)^{I_{p}}) if p ∉ Σ;

• L_{Σ,p}^{⊥} = (0) if p ∈ Σ.

If ρ : G_{ℚ} → GL_{2}(O) is a lifting of ρ̄ of type Σ, we will write H_{Σ}^{1}(ℚ, ad^{0}ρ ⊗ K/O) for the direct limit lim_{→} H_{Σ}^{1}(ℚ, ad^{0}ρ⊗λ^{−n}/O), and H_{Σ}^{1}(ℚ, ad^{0}ρ(1)⊗K/O)

for the direct limit lim_{→} H_{Σ}^{1}(ℚ, ad^{0}ρ(1) ⊗ λ^{−n}/O).

The main result we need about deformations of ρ̄ is the following.

Theorem 2.41 There is a universal lifting ρ_{Σ}^{univ} : G_{ℚ} → GL_{2}(R_{Σ}) of ρ̄ of type Σ, i.e. ρ_{Σ}^{univ} is a lifting of type Σ and if ρ : G_{ℚ} → GL_{2}(R) is any lifting of type Σ then there is a unique O-algebra homomorphism ϕ : R_{Σ} → R such that ρ ∼ ϕ ∘ ρ_{Σ}^{univ}. Moreover we have the following.

(a) If K′/K is a finite extension and R′_{Σ} is the corresponding deformation ring then R′_{Σ} = R_{Σ} ⊗_{O} O′ and (ρ_{Σ}^{univ})′ = ρ_{Σ}^{univ} ⊗ 1.

(b) R_{Σ} can be topologically generated as an O-algebra by dim_{k} H_{Σ}^{1}(ℚ, ad^{0}ρ̄) elements.

(c) If ϕ : R_{Σ} → → O is a O-algebra homomorphism, if ρ = ϕ ∘ ρ_{Σ} and if ℘ = ker ϕ then Hom(℘/℘^{2}, K/O) ≅ H_{Σ}^{1}(ℚ, ad^{0}ρ ⊗ K/O).

Proof: Let L_{0} denote the fixed field of ρ̄. Let L_{n} (for n ∈ ℤ_{>0}) denote the maximal elementary abelian ℓ-extension of L_{n−1} unramified outside Σ, {ℓ} and the primes where ρ̄ ramifies. Let L_{∞} = ⋃_{n} L_{n} and let G = Gal (L_{∞}/ℚ). Note that any lifting of ρ̄ of type Σ factors through G. Gal (L_{∞}/L_{0}) is a proℓ-group and its maximal elementary abelian quotient, Gal (L_{1}/L_{0}), is finite by theorem 2.2. We deduce from the following lemma that Gal (L_{∞}/L_{0}) and hence G are topologically finitely generated. (See for instance [Koc] Satz 4.10 for a proof of this lemma.)

Lemma 2.42 Let H be a pro-ℓ-group and H̄ its maximal elementary abelian quotient. Suppose h_{1}, ..., h_{r} ∈ H map to a set of generators of H̄, then h_{1}, ..., h_{r} topologically generate H.

Let D denote the category of profinite O[G]-modules M for which

• M is semi-stable as an O[G_{ℓ}]-module,

• if ℓ ∉ Σ and if ρ̄ is good then M is good as an O[G_{ℓ}]-module,

• if p ∉ Σ ∪ {ℓ} and if ρ̄ is ramified at p then there is an exact sequence of

O[I_{p}]-modules

(0) → M^{(−1)} → M → M^{(0)} → (0),

such that I_{p} acts trivially on M^{(−1)} and M^{(0)}.

Then we see that a lifting ρ : G_{ℚ} → GL_{2}(R) of ρ̄ is of type Σ if and only if

• ρ factors through G,

• det ρ = ϵ,

• M_{ρ} is an object of D.

The existence part of theorem 2.41 now follows from theorem 2.36 and part (a) follows from lemma 2.38. Parts (b) and (c) follow from lemmas 2.39 and 2.40, and the following observation: If ρ : G_{ℚ} → GL_{2}(O/λ^{n}) is a lifting of type Σ and if p ∉ Σ ∪ {l} is a prime where ρ̄ ramifies then

ker (H^{1}(G_{p}, ad^{0}ρ) → H^{1}(I_{p}, ad^{0}ρ/(ad^{0}ρ)^{I_{p}})) = H^{1}(G_{p}/I_{p}, (ad^{0}ρ)^{I_{p}}).

(2.7.1)

Equation (2.7.1) follows from the fact that the natural map H^{1}(I_{p}, ad^{0}ρ) → H^{1}(I_{p}, ad^{0}ρ/(ad^{0}ρ)^{I_{p}}) is an injection (in fact an isomorphism). It is nothing other than the map (ad^{0}ρ)_{I_{p}} → (ad^{0}ρ/(ad^{0}ρ)^{I_{p}})_{I_{p}}. This completes the proof of theorem 2.41. □

Corollary 2.43 Suppose that if ℓ = 3 then ρ̄|_{G_{ℚ(\sqrt{−3})}} is absolutely irreducible. Then R_{Σ} can be topologically generated as an O-algebra by

dim_{k} H_{Σ}^{1}(ℚ, ad^{0}ρ̄(1)) + d_{ℓ} + ∑_{p∈Σ−{ℓ}} dim_{k} H^{0}(ℚ_{p}, ad^{0}ρ̄(1))

elements, where d_{ℓ} = dim_{k} H_{ss}^{1}(ℚ_{ℓ}, ad^{0}ρ̄) − dim_{k} H_{f}^{1}(ℚ_{ℓ}, ad^{0}ρ̄) if ℓ ∈ Σ, while d_{ℓ} = 0 if ℓ ∉ Σ.

Proof: Note that H^{0}(ℚ, ad^{0}ρ̄(1)) = 0 unless ℓ = 3 and ρ̄|_{G_{ℚ(\sqrt{−3})}} is not absolutely irreducible. Thus according to theorem 2.18, dim_{k} H_{Σ}^{1}(ℚ, ad^{0}ρ̄) is the sum of the following terms.

• dim_{k} H_{Σ}^{1}(ℚ, ad^{0}ρ̄(1)).

• dim_{k} H_{f}^{1}(ℚ_{ℓ}, ad^{0}ρ̄)−dim_{k} H^{0}(ℚ_{ℓ}, ad^{0}ρ̄)−dim_{k} H^{0}(ℝ, ad^{0}ρ̄) ≤ 0 by proposition 2.27.

• d_{ℓ}.

• For each p ∈ Σ − {l}, dim_{k} H^{1}(ℚ_{p}, ad^{0}ρ̄) − dim_{k} H^{0}(ℚ_{p}, ad^{0}ρ̄) = dim_{k} H^{0}(ℚ_{p}, ad^{0}ρ̄(1)) by the local Euler characteristic formula (see theorem 2.17).

□

2.8 Special cases

In this section we will restrict attention to the deformation problems associated with sets Q of certain special primes q. These sets Q are chosen so that the associated deformation ring R_{Q} enjoys a number of special properties which will be crucial in the sequel. In particular we will only consider sets Q of primes q which satisfy the following two properties

• q ≡ 1 mod ℓ,

• ρ̄ is unramified at q and ρ̄(Frob_{q}) has distinct k-rational eigenvalues.

For such primes q we will let Δ_{q} denote the maximal quotient of (ℤ/qℤ)^{×} of ℓ-power order. Then Δ_{q} is naturally a quotient of both G_{q} and G_{ℚ} via the map

χ_{q} : G_{ℚ} → → Gal (ℚ(ζ_{q})/ℚ) ≅ (ℤ/qℤ)^{×} → → Δ_{q}.

We will let Δ_{Q} = ∏_{q∈Q} Δ_{q}, χ_{Q} = ∏_{q∈Q} χ_{q} : G_{ℚ} → → Δ_{Q} and 𝔞_{Q} denote the augmentation ideal of O[Δ_{Q}]. Note that there is an isomorphism O[Δ_{Q}] ≅ O[[S_{q} : q ∈ Q]]/((1 + S_{q})^{#Δ_{q}} − 1 : q ∈ Q) under which 𝔞_{Q} corresponds to (S_{q} : q ∈ Q).

For each q ∈ Q, we choose an eigenvalue α_{q} of ρ̄(Frob_{q}) and denote the other β_{q}.

Lemma 2.44 If q ∈ Q then ρ_{Q}^{univ}|_{G_{q}} is conjugate to (\begin{matrix}ξ & 0 \\ 0 & ϵξ^{−1} \end{matrix} ) for some char-

acter ξ such that ξ̄(Frob_{q}) = α_{q}.

Proof: Choose a lifting f ∈ G_{q} of Frob_{q}. As ρ̄(f) has distinct k-rational eigenvalues it is a simple application of Hensel’s lemma to see that we can

choose a basis such that ρ_{Q}^{univ}(f) = (\begin{matrix}α̃_{q} & 0 \\ 0 & β̃_{q} \end{matrix} ) where α̃_{q} and β̃_{q} reduce to α_{q}

and β_{q} in k. It suffices to show that for any σ ∈ I_{q}, ρ_{Q}^{univ}(σ) is diagonal in this

basis. Suppose ρ_{Q}^{univ}(σ) = 1_{2} + (\begin{matrix}a & b \\ c & d \end{matrix}) with a, b, c, d ∈ 𝔪_{R_{Q}}. Because ρ_{Q}^{univ} is tamely ramified at q we see that ρ_{Q}^{univ}(f)ρ_{Q}^{univ}(σ)ρ_{Q}^{univ}(f)^{−1} = ρ_{Q}^{univ}(σ)^{q}. Thus (α̃_{q}/β̃_{q} − q)b and (β̃_{q}/α̃_{q} − q)c lie in 𝔪_{R_{Q}}(b, c) so that (b, c) = 𝔪_{R_{Q}}(b, c) and (by Nakayama’s lemma) b = c = 0. □

We let ξ_{q,Q} denote the character ξ : G_{q} → R_{Q}^{×} in the conclusion of the lemma. The restriction of ξ_{q,Q} to I_{q} factors through χ_{q}. We let ξ_{Q} denote the character G_{ℚ} → R_{Q}^{×} which is unramified outside the primes of Q and whose restriction to I_{q} is ξ_{q,Q} for each q in Q. Thus ξ_{Q} factors through χ_{Q}. We wish to regard R_{Q} as an O[Δ_{Q}]-algebra, and it will be most convenient to do so via the map which gives rise to ξ_{Q}^{−2}. Note the following consequence of the lemma 2.44.

Corollary 2.45 The natural map R_{Q} → R_{∅} gives rise to an isomorphism

R_{Q}/𝔞_{Q}R_{Q} →^{∼} R_{∅}.

Lemma 2.46 (a) If q ∈ Q then H^{0}(𝔽_{q}, ad^{0}ρ̄) = H^{0}(𝔽_{q}, ad^{0}ρ̄(1)) = k and H^{1}(𝔽_{q}, ad^{0}ρ̄) = H^{1}(𝔽_{q}, ad^{0}ρ̄(1)) = k.

(b) R_{Q} can be topologically generated as an O-algebra by

#Q + dim_{k} H_{Q}^{1}(ℚ, ad^{0}ρ̄(1))

elements.

(c) If

H_{∅}^{1}(ℚ, ad^{0}ρ̄(1)) →^{∼} ⨁_{q∈Q} H^{1}(𝔽_{q}, ad^{0}ρ̄(1))

then #Q = dim_{k} H_{∅}^{1}(ℚ, ad^{0}ρ̄(1)) and R_{Q} can be topologically generated as an O-algebra by #Q elements.

Proof: The first part is a direct calculation using the fact that Frob_{q} acts semi-simply on ad^{0}ρ̄ with eigenvalues x, 1, x^{−1} for some x ∈ k̄\{0, 1}. The same is true for ad^{0}ρ̄(1) as q ≡ 1 mod ℓ. The second and third parts follow from this and corollary 2.43. □

For the proof of the last theorem of the chapter, we shall need two results on finite groups.

Theorem 2.47 (a) If H is a finite subgroup of P GL_{2}(ℂ) then H is isomorphic to one of the following groups: the cyclic group C_{n} of order n (n ∈ ℤ_{>0}), the dihedral group D_{2n} of order 2n (n ∈ ℤ_{>1}), A_{4}, S_{4} or A_{5}.

(b) If H is a finite subgroup of P GL_{2}(𝔽̄_{ℓ}) then one of the following holds:

• H is conjugate to a subgroup of the upper triangular matrices;

• H is conjugate to P SL_{2}(𝔽_{ℓ^{r}}) or P GL_{2}(𝔽_{ℓ^{r}}) for some r ∈ ℤ_{>0};

• H is isomorphic to A_{4}, S_{4}, A_{5} or the dihedral group D_{2r} of order 2r for some r ∈ ℤ_{>1} not divisible by ℓ.

In fact we shall only need part (b) which is due to Dickson [Dic2], secs. 255, 260 (see also [Hu] II.8.27), but we have included part (a) for later reference.

Lemma 2.48 Let 𝔽 be a finite field of odd characteristic ℓ. If #𝔽 ≠ 5, then

H^{1}(SL_{2}(𝔽), End^{0}(𝔽^{2})) = 0.

Proof: This is a special case of results of [CPS], table 4.5 (assuming #𝔽 ≠ 3). In fact we shall only need it in the case ℓ = 3, but the proof in the general case is no more difficult, and we sketch it here for the reader’s convenience.

We let B (resp. U) denote the group of upper-triangular (resp. unipotent) matrices in G = SL_{2}(𝔽). Since ℓ does not divide the index of B in G, the restriction homomorphism

H^{1}(G, End^{0}(𝔽^{2})) → H^{1}(B, End^{0}(𝔽^{2}))

is injective, so it suffices to prove the latter group vanishes. Since ℓ does not divide the index of U in B, we have

H^{i}(B, M) ≅ H^{i}(U, M)^{B/U}

all integers i ≥ 0 and 𝔽[B]-modules M. If #𝔽 = 3, then one checks directly that for M = End^{0}(𝔽_{3}^{2}),

H^{1}(U, M) ≅ ker N/(σ − 1)M = 0

where σ generates U and N = 1 + σ + σ^{2} on M. If #𝔽 > 5, then one proceeds by writing

(0) = M_{0} ⊂ M_{1} ⊂ M_{2} ⊂ M_{3} = End^{0}(𝔽^{2})

as 𝔽[B]-modules where the successive quotients M_{i}/M_{i−1}, (for i = 1, 2, 3) are one-dimensional over 𝔽. The calculation is then straightforward using long exact sequences of cohomology, except in the case #𝔽 = 9 where one must also check that the one-dimensional space of classes in H^{1}(U, M_{3}/M_{2}) fixed by B/U maps injectively to H^{2}(U, M_{2}/M_{1}) via the connecting homomorphism. □

Theorem 2.49 Keep the assumptions of the last section and suppose moreover that if L = ℚ(\sqrt{(−1)^{(ℓ−1)/2}ℓ}) then ρ̄|_{G_{L}} is absolutely irreducible. Then there exists a non-negative integer r such that for any n ∈ ℤ_{>0} we can find a finite set of primes Q_{n} with the following properties.

(a) If q ∈ Q_{n} then q ≡ 1 mod ℓ^{n}.

(b) If q ∈ Q_{n} then ρ̄ is unramified at q and ρ̄(Frob_{q}) has distinct eigenvalues.

(c) #Q_{n} = r.

(d) R_{Q_{n}} can be topologically generated by r elements as a O-algebra. Proof: Take r = dim_{k} H_{∅}^{1}(ℚ, ad^{0}ρ̄(1)). It suffices to find a set Q_{n} with the following properties.

(a) If q ∈ Q_{n} then q ≡ 1 mod ℓ^{n}.

(b) If q ∈ Q_{n} then ρ̄ is unramified at q and ρ̄(Frob_{q}) has distinct eigenvalues.

(c) H_{∅}^{1}(ℚ, ad^{0}ρ̄(1)) →^{∼} ⨁_{q∈Q_{n}} H^{1}(𝔽_{q}, ad^{0}ρ̄(1)).

As each H^{1}(𝔽_{q}, ad^{0}ρ̄(1)) is one dimensional we may replace the last condition with

H_{∅}^{1}(ℚ, ad^{0}ρ̄(1)) ↪ ⨁_{q∈Q_{n}} H^{1}(𝔽_{q}, ad^{0}ρ̄(1)).

Thus what we need show is that for each non-zero class [ψ] ∈ H_{∅}^{1}(ℚ, ad^{0}ρ̄(1)) there is a prime q (depending on [ψ]) such that

(a) q ≡ 1 mod ℓ^{n},

(b) ρ̄ is unramified at q and ρ̄(Frob_{q}) has distinct eigenvalues,

(c) res_{q}[ψ] ∈ H^{1}(𝔽_{q}, ad^{0}ρ̄(1)) is nontrivial. Using the Chebotarev density theorem we see that it will do to find σ ∈ G_{ℚ} such that

(a) σ|_{ℚ(ζ_{ℓ^{n}})} = 1,

(b) ad^{0}ρ̄(σ) has an eigenvalue other than 1,

(c) ψ(σ) ∉ (σ − 1)ad^{0}ρ̄(1).

For m ≥ 0, let F_{m} denote the extension of ℚ(ζ_{ℓ^{m}}) cut out by ad^{0}ρ̄; i.e., the field fixed by the kernel of the representation ad^{0}ρ̄ restricted to G_{ℚ(ζ_{ℓ^{m}})}. We will show that ψ(G_{F_{n}}) is non-trivial. For this it suffices to prove that H^{1}(Gal (F_{n}/ℚ), ad^{0}ρ̄(1)) = (0). Consider the inflation-restriction exact sequence

(0) → H^{1}(Gal (F_{0}/ℚ), (ad^{0}ρ̄(1))^{G_{F_{0}}}) → H^{1}(Gal (F_{n}/ℚ), ad^{0}ρ̄(1))
→ H^{1}(Gal (F_{n}/F_{0}), ad^{0}ρ̄(1))^{G_{ℚ}}.

Since F_{1}/F_{0} is an extension of degree prime to ℓ, and since G_{ℚ} acts trivially on Gal (F_{n}/F_{1}), we have:

H^{1}(Gal (F_{n}/F_{0}), ad^{0}ρ̄(1))^{G_{ℚ}} ≃ Hom(Gal (F_{n}/F_{1}), ad^{0}ρ̄(1)^{G_{ℚ}}). (2.8.1)

Since ρ̄|_{G_{L}} is absolutely irreducible, the cohomology group in equation (2.8.1) vanishes. On the other hand, G_{F_{0}} acts trivially on ad^{0}ρ̄ so the first term vanishes as well unless Gal (F_{0}/ℚ) has order divisible by ℓ and has Gal (ℚ(ζ_{ℓ})/ℚ) as a quotient. Recall that Gal (F_{0}/ℚ) is isomorphic to the projective image of ρ̄, so by theorem 2.47 we are reduced to the case ℓ = 3 and the map

Gal (F_{0}/ℚ(ζ_{3})) → P SL_{2}(k̄)

has image conjugate to P SL_{2}(𝔽_{3^{r}}) for some r ≥ 1. It suffices to prove that in this case

H^{1}(Gal (F_{0}/ℚ(ζ_{3})), ad^{0}ρ̄ ⊗_{k} k̄)) = (0),

but this follows directly from lemma 2.48.

Now note that since ρ̄|_{G_{L}} is absolutely irreducible, so is ρ̄|_{G_{ℚ(ζ_{ℓ^{n}})}}. Regarding ψ(G_{F_{n}}) as a Gal (F_{n}/ℚ(ζ_{ℓ^{n}}))-submodule of ad^{0}(ρ̄), we find that some g ∈ Gal (F_{n}/ℚ(ζ_{ℓ^{n}})) of order not dividing ℓ fixes a non-zero element of ψ(G_{F_{n}}). Let σ_{0} be a lifting of g to G_{ℚ(ζ_{ℓ^{n}})}. We will look for σ = τσ_{0} with τ ∈ G_{F_{n}}. We only need choose τ so that ψ(τσ_{0}) = ψ(τ) + ψ(σ_{0}) ∉ (σ_{0} − 1)ad^{0}ρ̄(1). This is possible because ψ(G_{F}) ⊄ (g − 1)ad^{0}ρ̄(1). □

3 Modular forms and Galois representations

3.1 From modular forms to Galois representations

We suppose in this section that f = ∑ a_{n}(f)q^{n} is a newform of weight two and level N_{f} (see definition 1.21). Let K_{f} denote the number field in ℂ generated by the Fourier coefficients a_{n}(f). Let ψ_{f} denote the character of f, i.e., the homomorphism (ℤ/N_{f}ℤ)^{×} → K_{f}^{×} defined by mapping d to the eigenvalue of ⟨d⟩ on f.

Recall that a construction of Shimura (section 1.7) associates to f an abelian variety A_{f} of dimension [K_{f} : ℚ]. This abelian variety is a certain quotient of J_{1}(N_{f}), and the action of the Hecke algebra on J_{1}(N_{f}) provides an embedding

K_{f} ↪ End_{ℚ}(A_{f}) ⊗ ℚ.

We saw also that for each prime ℓ the Tate module T_{ℓ}(A_{f}) ⊗_{ℤ_{ℓ}} ℚ_{ℓ} becomes a free module of rank two over K_{f} ⊗ ℚ_{ℓ} (lemma 1.48). The action of the Galois group G_{ℚ} on the Tate module commutes with that of K_{f}, so that a choice of basis for the Tate module provides a representation

G_{ℚ} → GL_{2}(K_{f} ⊗ ℚ_{ℓ}).  (3.1.1)

As K_{f} ⊗ ℚ_{ℓ} can be identified with the product of the completions of K_{f} at its primes over ℓ, we obtain from f certain 2-dimensional ℓ-adic representations of G_{ℚ}.

ℓ-adic representations: In this discussion, we fix a prime ℓ and a finite extension K of ℚ_{ℓ}. We let O denote the ring of integers of K, λ the maximal ideal and k the residue field. We shall consider ℓ-adic representations with coefficients in finite extensions of our fixed field K. We regard K as a subfield of ℚ̄_{ℓ} and fix embeddings ℚ̄ ↪ ℚ̄_{ℓ} and ℚ̄ ↪ ℂ̄. If K′ is a finite extension of K with ring of integers O′, then we say that an ℓ-adic representation G_{ℓ} → GL_{2}(K′) is good (respectively, ordinary, semistable) if it is conjugate over K′ to a representation G_{ℓ} → GL_{2}(O′) which is good (respectively, ordinary, semistable) in the sense of section 2.4.

Let K′_{f} denote the K-algebra in ℚ̄_{ℓ} generated by the Fourier coefficients of f. Thus K′_{f} is a finite extension of K, and it contains the completion of K_{f} at the prime over ℓ determined by our choice of embeddings. We let O′_{f} denote the ring of integers of K′_{f} and write k_{f} for its residue field. We define

ρ_{f} : G_{ℚ} → GL_{2}(K′_{f})

as the pushforward of (3.1.1) by the natural map K_{f} ⊗ ℚ_{ℓ} → K′_{f}. We assume the basis is chosen so that ρ_{f} factors through GL_{2}(O′_{f}). We also let ψ′_{f} denote the finite order ℓ-adic character

G_{ℚ} → → Gal (ℚ(ζ_{N_{f}})/ℚ) → (K′_{f})^{×}

obtained from ψ_{f}.

The following theorem lists several fundamental properties of the ℓ-adic representations ρ_{f} obtained from Shimura’s construction. The result is a combination of the work of many mathematicians. We discuss some of the proofs and provide references below. In the statement we fix f as above and write simply N, a_{n}, ρ, ψ, ψ′ and K′ for N_{f}, a_{n}(f), ρ_{f}, ψ_{f}, ψ′_{f} and K′_{f} respectively.

Theorem 3.1 The ℓ-adic representation

ρ : G_{ℚ} → GL_{2}(K′)

has the following properties.

(a) If p |̸Nℓ then ρ is unramified at p and ρ(Frob_{p}) has characteristic poly-

nomial

X^{2} − a_{p}X + pψ(p).

(b) det(ρ) is the product of ψ′ with the ℓ-adic cyclotomic character ϵ, and ρ(c) is conjugate to (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}).

(c) ρ is absolutely irreducible.

(d) The conductor N(ρ) is the prime-to-ℓ-part of N.

(e) Suppose that p ≠ ℓ and p||N. Let χ denote the unramified character G_{p} → (K′)^{×} satisfying χ(Frob_{p}) = a_{p}. If p does not divide the conductor of ψ, then ρ|_{G_{p}} is of the form

(\begin{matrix}χϵ & ∗ \\ 0 & χ \end{matrix}) .

If p divides the conductor of ψ, then ρ|_{G_{p}} is of the form

χ^{−1}ϵψ′|_{G_{p}} ⊕ χ.

(f) If ℓ |̸2N, then ρ|_{G_{ℓ}} is good. Moreover ρ|_{G_{ℓ}} is ordinary if and only if a_{ℓ} is a unit in the ring of integers of K′, in which case ρ_{I_{ℓ}}(Frob_{ℓ}) is the unit root of the polynomial X^{2} − a_{ℓ}X + ℓψ(ℓ).

(g) If ℓ is odd and ℓ||N, but the conductor of ψ is not divisible by ℓ, then ρ|_{G_{ℓ}} is ordinary and ρ_{I_{ℓ}}(Frob_{ℓ}) = a_{ℓ}.

Proof: Part (a) was established by Shimura ([Shi2], [Shi3]). The key ingredient is the Eichler-Shimura congruence relation, theorem 1.29. Recall that J_{1}(N) has good reduction at primes p not dividing N. So the action of G_{p} on T_{ℓ}(A_{f}) ⊗_{ℤ_{ℓ}} ℚ_{ℓ} is unramified and is in fact described by the action of Frob_{p} ∈ G_{𝔽_{p}} on the Tate module of the reduction. But this is given by the Frobenius endomorphism F whose characteristic polynomial is computed in corollary 1.41.

The first assertion of (b) follows from (a) on applying the Chebotarev density theorem. The second assertion then follows on noting that ψ(−1) = 1.

Part (c) was proved by Ribet (see section 2 of [R3]). Assuming reducibility of the representation, he applies algebraicity results of Lang and Serre to obtain a contradiction to the estimate on the Fourier coefficients stated in theorem 1.24.

Parts (d) and (e) follow from a deep result of Carayol [Ca1], Thm. (A), building on the work of Langlands [Ll1], Deligne and others. In fact, this result and the local Langlands correspondence characterize ρ|_{G_{p}} in terms of ψ|_{G_{p}} and the L- and ϵ-factors at p of twists of f. The descriptions in the case of p||N are based on the analysis of Deligne-Rapoport of the reduction mod p of J_{1}(N) (see [DR], [Ll1]).

The first assertion of (f) follows from the fact that A_{f} has good reduction at ℓ if ℓ does not divide N. The second assertion of (f) (respectively, all of (g)) follows from the Eichler-Shimura congruence relation (respectively, the work of Deligne-Rapoport), and general results on ℓ-divisible groups and the reduction of abelian varieties; see thm. 2.2 of [W2], lemma 2.1.5 of [W1], §12 of [Gro] and thms. 2.5 and 2.6 of [Edi]. The restriction to odd ℓ in (f) and (g) is made primarily out of lack of suitable definitions. □

Mod ℓ representations: We maintain the notation used in the discussion of ℓ-adic representations. Define

ρ̄_{f} : G_{ℚ} → GL_{2}(k_{f})

to be the semi-simplification of the reduction of ρ_{f}. (See the discussion following proposition 2.6.) Assertions analogous to those in theorem 3.1 hold for ρ̄ = ρ̄_{f}, except that

• The representation need not be absolutely irreducible (as in (c)). However if ℓ is odd, one checks using (b) that ρ̄ is irreducible if and only if it is absolutely irreducible.

• In (d) one only has divisibility of the prime-to-ℓ part of N_{f} by N(ρ̄) in general.

The various possibilities for m_{p}(ρ̄) to be strictly less than the exponent of p in N (where p ≠ ℓ) were classified independently by Carayol [Ca2] and Livné [Liv]. We record the following consequence of their results (cf. the introduction of [DT1]):

Proposition 3.2 Suppose that p is a prime such that p|N_{f}, p ≢ 1 mod ℓ and ρ̄_{f} is unramified at p. Then tr (ρ̄_{f}(Frob_{p}))^{2} = (p + 1)^{2} in k_{f}.

Artin representations: The theory of Hecke operators and newforms (see section 1.3) extends to modular forms on Γ_{1}(N) of arbitrary weight. The construction of ℓ-adic representations associated to newforms was generalized to weight greater than 1 by Deligne [De] using étale cohomology. There are also Galois representations associated to newforms of weight 1 by Deligne and Serre [DS], but an essential difference is that these are Artin representations.

Theorem 3.3 If g = ∑ a_{n}(g)q^{n} is a newform of weight one, level N_{g} and character ψ_{g}, then there is an irreducible Artin representation

ρ_{g} : G_{ℚ} ⟶ GL_{2}(ℂ)

of conductor N_{g} with the following property: If p |̸N_{g}, then the characteristic polynomial of ρ_{g}(Frob_{p}) is

X^{2} − a_{p}(g)X + ψ_{g}(p).

Remark 3.4 Note that det(ρ_{g}) is the character of G_{ℚ} corresponding to ψ and ρ_{g}(c) is conjugate to (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) (see theorem 3.1).

Remark 3.5 A basis can be chosen so that the representation ρ_{g} takes values in GL_{2}(K_{g}) (where K_{g} is the number field generated by the a_{n}(g)). Moreover suppose that K is a finite extension of ℚ_{ℓ} in ℚ̄_{ℓ} and we have fixed embeddings of ℚ̄ in ℂ and ℚ̄_{ℓ}). If K_{g} is contained in K, then we can view ρ_{g} as giving rise to an ℓ-adic representation G_{ℚ} → GL_{2}(K) and hence a mod ℓ representation G_{ℚ} → GL_{2}(k).

Remark 3.6 A key idea in the construction of ρ_{g} is to first construct the mod ℓ representations using those already associated to newforms of higher weight. More precisely, suppose that K_{g} ↪ K as in remark 3.5. One can show that for some newform f of weight 2 and level N_{f} dividing Nℓ we have

a_{p}(g) ≡ a_{p}(f), ψ_{g}(p)) ≡ pψ_{f}(p)

for all p |̸Nℓ, the congruence being modulo the maximal ideal of the ring of integers of K′_{f}. Thus ρ̄_{f} is the semisimplification of the desired mod ℓ representation (with scalars extended to k_{f}).

3.2 From Galois representations to modular forms

It is conjectured that certain types of two-dimensional representations of G_{ℚ} always arise from the constructions described in section 3.1. We now state some of the conjectures and the results known prior to [W3] and [TW].

Artin representations:

Conjecture 3.7 Let ρ : G_{ℚ} → GL_{2}(ℂ) be a continuous irreducible representation with det(ρ(c)) = −1. Then ρ is equivalent to ρ_{g} for some newform g of weight one.

Recall that ρ_{g} is the Artin representation associated to g by the Deligne-Serre construction (theorem 3.3).

Remark 3.8 Conjecture 3.7 is equivalent to the statement that the Artin L-functions attached to ρ and to all its twists by one-dimensional characters are entire. (The Artin conjecture predicts that the Artin L-function L(s, ρ) is entire, for an arbitrary irreducible, non-trivial Artin representation ρ : G_{ℚ} ⟶ GL_{d}(ℂ).)

A large part of conjecture 3.7 was proved by Langlands in [Ll2], and the results were extended by Tunnell [Tu].

Theorem 3.9 Let ρ : G_{ℚ} → GL_{2}(ℂ) be a continuous irreducible representation such that ρ(G_{ℚ}) is solvable and det(ρ(c)) = −1. Then ρ is equivalent to ρ_{g} for some newform g of weight one.

Remark 3.10 Note that by theorem 2.47, part (a), the solvability hypothesis excludes only the case where the projective image of ρ is isomorphic to A_{5}.

Remark 3.11 If the projective image of ρ is dihedral, then ρ is induced from a character of a quadratic extension of ℚ. In this case the result can already be deduced from work of Hecke.

Mod ℓ representations: We fix notation as in the discussion of ℓ-adic and mod ℓ representations in section 3.1.

Definition 3.12 We say that a representation ρ̄ : G_{ℚ} → GL_{2}(k) is modular (of level N) if for some newform f of weight 2 (and level N), ρ̄ is equivalent over k_{f} to ρ̄_{f}.

By proposition 1.32 the notion is independent of the choices in section 3.1 of embeddings K ↪ ℚ̄_{ℓ}, ℚ̄ ↪ ℚ̄_{ℓ} and ℚ̄ ↪ ℂ. Moreover if K′ is a finite extension of K with residue field k′, then ρ̄ is modular if and only if ρ̄⊗_{k} k′ is modular.

The following was conjectured by Serre [Se7], (3.2.3). (See also [Da1] for further discussion and references.)

Conjecture 3.13 Let ρ̄: G_{ℚ} → GL_{2}(k) be a continuous absolutely irreducible representation with det(ρ̄(c)) = −1. Then ρ̄ is modular.

Some cases of Serre’s conjecture can be deduced from theorem 3.9.

Theorem 3.14 Let ρ̄ : G_{ℚ} → GL_{2}(k) be a continuous absolutely irreducible representation with det(ρ̄(c)) = −1. Suppose that one of the following holds:

(a) k = 𝔽_{3};

(b) the projective image of ρ̄ is dihedral.

Then ρ̄ is modular.

Sketch of proof: For case (a), we consider the surjection

GL_{2}(ℤ[\sqrt{−2}]) → GL_{2}(𝔽_{3})

defined by reduction mod (1 + \sqrt{−2}). One checks that there is a section s : GL_{2}(𝔽_{3}) → GL_{2}(ℤ[\sqrt{−2}]) and applies theorem 3.9 to s ∘ ρ̄. The resulting representation arises from a weight one newform, and hence its reduction ρ̄ is equivalent to ρ̄_{f} for some f (see remark 3.6).

In case (b), ρ̄ is equivalent to a representation of the form Ind _{G_{F}}^{G_{ℚ}} ξ̄ where F is a quadratic extension of ℚ and ξ̄ is a character G_{F} → k^{×}. (We have here enlarged K if necessary.) Let n be the order of ξ̄, choose an embedding ℚ(e^{2πi/n}) ↪ K and lift ξ̄ to a character ξ : G_{F} → ℤ[e^{2πi/n}]^{×}. We may always choose ξ so that the Artin representation ρ = Ind _{G_{F}}^{G_{ℚ}} ξ is odd, i.e., det(ρ(c)) = −1. (In the case ℓ = 2 and F real quadratic, we may have to multiply ξ by a suitable quadratic character of G_{F}.) We then apply theorem 3.9 to ρ and deduce as in case (a) that ρ̄ is modular. □

Serre also proposed a refinement ([Se7], (3.2.4)) of the conjecture which predicts that ρ̄ is associated to a newform of specified weight, level and character. Through work of Mazur, Ribet [R5], Carayol [Ca2], Gross [Gro] and others, this refinement is now known to be equivalent to conjecture 3.13 if ℓ is odd. (One also needs to impose a mild restriction in the case ℓ = 3.) See [R6] and [Di1] for statements of the results and further references; here we give a variant which applies to newforms of weight two. Before doing so, we assume ℓ is odd and define an integer δ(ρ̄) as follows:

• δ(ρ̄) = 0 if ρ̄|_{G_{ℓ}} is good;

• δ(ρ̄) = 1 if ρ̄|_{G_{ℓ}} is not good and ρ̄|_{I_{ℓ}} ⊗_{k} k̄ is of the form

(\begin{matrix}ϵ^{a} & ∗ \\ 0 & 1 \end{matrix}) , (\begin{matrix}ϵ & ∗ \\ 0 & ϵ^{a} \end{matrix} ) or (\begin{matrix}ψ^{a} & 0 \\ 0 & ψ^{ℓa} \end{matrix} )

for some positive integer a < ℓ. (Recall that ϵ is the cyclotomic character and ψ is the character of I_{ℓ} defined after lemma 2.22.)

• δ(ρ̄) = 2 otherwise.

Theorem 3.15 Suppose that ℓ is odd and ρ̄ is absolutely irreducible and modular. If ℓ = 3, then suppose also that ρ̄|_{G_{ℚ(\sqrt{−3})}} is absolutely irreducible. Then there exists a newform f of weight two such that

• ρ̄ is equivalent over k_{f} to ρ̄_{f};

• N_{f} = N(ρ̄)ℓ^{δ(ρ̄)};

• the order of ψ_{f} is not divisible by ℓ.

Proof: The existence of such an f follows from [Di1] thm. 1.1, thm. 5.1 and lemma 2.1, but with N_{f} dividing N(ρ̄)ℓ^{δ(ρ̄)}. By lemma 2.7 above, we see that N_{f} is divisible by N(ρ̄). The divisibility of N_{f} by δ(ρ̄) follows from results in sec. 8 of [Gro] and sec. 2.4 of [Edi]. □

ℓ-adic representations: We again use the notation of section 3.1. Let ρ : G_{ℚ} → GL_{2}(K) be an ℓ-adic representation.

Definition 3.16 We say that ρ is modular if for some weight 2 newform f, ρ is equivalent over K′_{f} to ρ_{f}.

The notion is independent of the choices of embeddings and well-behaved under extension of scalars by proposition 1.32 (cf. definition 3.12).

The following is a special case of a conjecture of Fontaine and Mazur [FM].

Conjecture 3.17 If ρ : G_{ℚ} → GL_{2}(K) is an absolutely irreducible ℓ-adic representation and ρ|_{G_{ℓ}} is semistable (in the sense of section 2.4), then ρ is modular.

(Recall that for us ℓ-adic representations are defined to be unramified at all but finitely many primes. Recall also that if ρ|_{G_{ℓ}} is semistable, then by definition det ρ|_{I_{ℓ}} is the cyclotomic character ϵ.)

Remark 3.18 Relatively little was known about this conjecture before Wiles’ work [W3]. Wiles proves that under suitable hypotheses, the modularity of ρ̄ implies that of ρ.

Remark 3.19 The conjecture stated in [FM] is stronger than the one here; in particular, the semistability hypothesis could be replaced with a suitable notion of potential semistability. On the other hand, one expects that if ρ|_{G_{ℓ}} is semistable, then it is equivalent to ρ_{f} (over K′_{f}) for some f on Γ_{1}(N(ρ))∩Γ_{0}(ℓ) (and on Γ_{1}(N(ρ)) if ρ|_{G_{ℓ}} is good).

The Shimura-Taniyama conjecture: Conjecture 1.54 can be viewed in the framework of the problem of associating modular forms to Galois representations. Let E be an elliptic curve defined over ℚ. For each prime ℓ, we let ρ_{E,ℓ} denote the ℓ-adic representation G_{ℚ} → GL_{2}(ℚ_{ℓ}) defined by the action of G_{ℚ} on the Tate module of E (see section 2.2).

Proposition 3.20 The following are equivalent:

(a) E is modular.

(b) ρ_{E,ℓ} is modular for all primes ℓ.

(c) ρ_{E,ℓ} is modular for some prime ℓ.

Proof: If E is modular, then E is isogenous to A_{f} for some weight two newform f with K_{f} = ℚ (see section 1.8). It follows that for each prime ℓ, ρ_{E,ℓ} is equivalent to the ℓ-adic representation ρ_{f}. Hence (a) ⇒ (b) ⇒ (c).

To show (c) ⇒ (b), suppose that for some ℓ and some f, the representations ρ_{E,ℓ} and ρ_{f} are equivalent. First observe that for all but finitely primes p, we have

tr (ρ_{f}(Frob_{p})) = tr (ρ_{E,ℓ}(Frob_{p})).

We deduce from proposition 2.11 and theorem 3.1, part (a) that for all but finitely many primes p

a_{p}(f) = p + 1 − #Ē_{p}(𝔽_{p}) ∈ ℤ.  (3.2.1)

Applying proposition 2.6, we find that for each prime ℓ, ρ_{E,ℓ} is equivalent to ρ_{f} and is therefore modular.

We finally show that (b) ⇒ (a). The equality (3.2.1) holds for all primes p not dividing N_{f}, which by theorem 3.1, part (d), is the conductor of E. Since det(ρ_{f}) = det(ρ_{E,ℓ}) = ϵ, we see by 3.1, Part (b) that ψ_{f} is trivial. By theorem 1.27 parts (b) and (d) (or [AL] thm. 3), a_{p} is in {0, ±1} for primes p dividing N_{f}. Thus K_{f} = ℚ and A_{f} is an elliptic curve. Faltings’ isogeny theorem (see [CS], sec. II.5) now tells us that E and A_{f} are isogenous and we conclude that E is modular. □

Remark 3.21 Note that the equivalence (b) ⇔ (c) does not require Faltings’ isogeny theorem.

Proposition 3.22 If the Fontaine-Mazur conjecture (conjecture 3.17) holds for some prime ℓ, then the Shimura-Taniyama conjecture (conjecture 1.54) holds. If Serre’s conjecture (conjecture 3.13) holds for infinitely many ℓ, then conjecture 1.54 holds.

Proof: The first assertion is immediate from proposition 3.20 and the irreducibility of ρ_{E,ℓ}. See [Se7], sec. 4.6 for a proof of the second. (We have implicitly chosen the field K to be ℚ_{ℓ} in the statements of conjectures 3.17 and 3.13, but it may be replaced by a finite extension.) □

Remark 3.23 Note that to prove a given elliptic curve E is modular, it suffices to prove that conjecture 3.17 holds for a single ℓ at which E has semistable reduction. Wiles’ approach is to show that certain cases of conjecture 3.13 imply cases of conjecture 3.17 and hence cases of conjecture 1.54.

3.3 Hecke algebras

In this section fix the following notation. Let ℓ be an odd prime, let K be a finite extension of ℚ_{ℓ}, let O denote the ring of integers of K, let λ denote its maximal ideal and k its residue field. Fix embeddings K ↪ ℚ̄_{ℓ}, ℚ̄ ↪ ℚ̄_{ℓ} and ℚ̄ ↪ ℂ. Let ρ̄ : G_{ℚ} → GL_{2}(k) denote a continuous representation with the following properties

(a) ρ̄ is irreducible,

(b) ρ̄ is modular,

(c) det ρ̄ = ϵ,

(d) ρ̄|_{G_{ℓ}} is semi-stable,

(e) and if p ≠ ℓ then #ρ̄(I_{p})|ℓ.

Let us first record the following lemma.

Lemma 3.24 The representation ρ̄|_{G_{L}} is absolutely irreducible where L = ℚ(\sqrt{(−1)^{(ℓ−1)/2}ℓ}).

Proof: If it were not then we see that ℓ |̸#ρ̄(G_{ℚ}) and so ρ̄ is unramified at all p ≠ ℓ. Moreover we can check that ρ̄|_{I_{ℓ}} ∼ (\begin{matrix}ϵ & 0 \\ 0 & 1 \end{matrix}). If ℓ > 3 we can use theorem 3.15 to deduce that ρ̄ is modular of weight 2 and level 1 and hence obtain a contradiction. If ℓ = 3 we see that the splitting field of ρ̄ is everywhere unramified over ℚ(\sqrt{−3}) and hence must equal ℚ(\sqrt{−3}), a contradiction. □

Let Σ denote a finite set of primes. For the application to modularity of semistable elliptic curves, it suffices to consider sets Σ contained in Σρ̄ where Σρ̄ is defined as follows Definition 3.25 For a representation ρ̄ as above, we let Σρ̄ denote the set of primes p satisfying

• p = ℓ and ρ̄|_{G_{ℓ}} is good and ordinary; or

• p ≠ ℓ and ρ̄ is unramified at p.

We shall sometimes assume that Σ ⊂ Σρ̄ in order to simplify statements and proofs. Let N_{Σ} denote the set of newforms f such that

ρ_{f} : G_{ℚ} → GL_{2}(O′_{f})

is equivalent to a lifting of ρ̄ ⊗_{k} k_{f} of type Σ and ℓ^{2} |̸N_{f} (the last condition is presumably not necessary, cf. remark 3.19). From theorem 3.1 and lemma 2.7 one deduces the following description of N_{Σ}.

Lemma 3.26 The set N_{Σ} consists of newforms f such that

• ρ̄_{f} ≅ ρ̄ ⊗_{k} k_{f},

• ψ_{f} is trivial,

• N_{f} divides ℓ^{δ}N(ρ̄) ∏_{p∈Σ−{ℓ}} p^{dim^{ρ̄^{Ip}}}, where δ = 0 if ρ̄ is good and ℓ ∉ Σ and

δ = 1 otherwise.

As ρ̄ is modular it follows from theorem 3.15 that for all Σ, N_{Σ} ≠ ∅. Set 𝕋̃_{Σ} = ∏_{f∈N_{Σ}} O′_{f}. If p is a prime not in Σ and not dividing ℓN(ρ̄), we let T_{p} denote the element (a_{p}(f))_{f} in 𝕋̃_{Σ}. Then define 𝕋_{Σ} to be the O-subalgebra of 𝕋̃_{Σ} generated by the elements T_{p} for such primes p. Then 𝕋_{Σ} is a complete noetherian local O-algebra with residue field k. Moreover it is reduced and it is a finitely generated free O-module.

Lemma 3.27 There is a continuous representation

ρ_{Σ}^{mod} : G_{ℚ} → GL_{2}(𝕋_{Σ})

such that if p |̸ℓN(ρ̄) and p ∉ Σ then ρ_{Σ}^{mod} is unramified at p and we have tr ρ_{Σ}^{mod}(Frob_{p}) = T_{p}. Moreover we have the following.

(a) ρ_{Σ}^{mod} is a lift of ρ̄ of type Σ and there is a unique surjection ϕ_{Σ} : R_{Σ} → → 𝕋_{Σ} such that ρ_{Σ}^{mod} ∼ ϕ_{Σ} ∘ ρ_{Σ}^{univ}.

(b) If Σ′ ⊃ Σ then there is a unique surjection 𝕋_{Σ′} → → 𝕋_{Σ} such that ρ_{Σ′}^{mod} pushes forward to ρ_{Σ}^{mod} and T_{p} maps to T_{p} for p |̸ℓN(ρ̄) and p ∉ Σ′.

(c) If K′ is a finite extension of K and 𝕋′_{Σ} is constructed in the same way as 𝕋_{Σ} but with K′ replacing K then 𝕋′_{Σ} ≅ 𝕋_{Σ} ⊗_{O} O_{K′}.

Proof: Consider ρ̃_{Σ}^{mod} = ∏ ρ_{f} : G_{ℚ} → GL_{2}(𝕋̃_{Σ}). Choose some complex conjugation, c, and conjugate ρ̄ so that ρ̄(c) = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}). By conjugating ρ̃_{Σ}^{mod} we may assume that modulo every maximal ideal it reduces to ρ̄ and that ρ̃_{Σ}^{mod}(c) = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}). Call the corresponding basis \{e_{+}, e_{−}\}. Note that by continuity and the Chebotarev density theorem tr ρ̃_{Σ}^{mod} is valued in 𝕋_{Σ}. Thus for any g ∈ G_{ℚ} the diagonal entries of ρ̃_{Σ}^{mod}(g) lie in 𝕋_{Σ}, because tr ρ̃_{Σ}^{mod}(g) and tr ρ̃_{Σ}^{mod}(cg) both do.

By irreducibility of ρ̄ we can find some σ ∈ G_{ℚ} such that ρ̄(σ) = (\begin{matrix}∗ & b \\ ∗ & ∗ \end{matrix})

where b ≠ 0. Rescaling e_{+} we may assume that ρ̃_{Σ}^{mod}(σ) = (\begin{matrix}a & 1 \\ c & d \end{matrix}). Then for all g ∈ G_{ℚ} the lower left entry of ρ̃_{Σ}^{mod}(g) lies in 𝕋_{Σ} (look at the upper left entry of ρ̃_{Σ}^{mod}(σg)). Again using the irreducibility of ρ̄ we can find τ ∈ G_{ℚ} such that ρ̃_{Σ}^{mod}(τ) = (\begin{matrix}∗ & ∗ \\ e & ∗ \end{matrix}) where e ∈ 𝕋_{Σ}^{×}. Looking at the lower right entry of ρ̃_{Σ}^{mod}(τg) we see that for any g ∈ G_{ℚ} the upper right entry of ρ̃_{Σ}^{mod}(g) lies in 𝕋_{Σ}. Thus ρ̃_{Σ}^{mod} is now in fact valued in GL_{2}(𝕋_{Σ}) and will be our candidate for ρ_{Σ}^{mod}. We leave the verification of the other properties of ρ_{Σ}^{mod} as an exercise. □

Example 3.28 Let ρ̄ = ρ̄_{f_{57B},3} where f_{57B} is the newform of level 57 discussed in the example of section 1.6. As f_{57B} is not congruent modulo 3 to any form of level 19 or 3 we see by theorem 3.15 that ρ̄ is ramified at 19 and ρ̄|_{G_{3}} is not good. On the other hand ρ̄ is semi-stable. The facts that ρ̄(Frob_{2}) has order 8 (see the table below) and 3|#ρ̄(I_{19}) (from the discussion above) imply that ρ̄: G_{ℚ} → → GL_{2}(𝔽_{3}). The table below also shows that 𝕋_{∅} ≅ {(x, y) ∈ ℤ_{3}^{2} : x ≡ y mod 3}. There is a unique continuous representation ρ_{∅}^{mod} : G_{ℚ} → GL_{2}(𝕋_{∅}) such that if p |̸57 then ρ_{∅}^{mod} is unramified at p and tr ρ_{∅}^{mod}(Frob_{p}) = T_{p}. We have an isomorphism 𝕋_{∅}/3𝕋_{∅} →^{∼} 𝔽_{3}[ε] (where ε^{2} = 0) given by (x, y) ↦ x+ \frac{y−x}{3}ε. Thus we get a representation ρ : G_{ℚ} → GL_{2}(𝔽_{3}[ε]). The following table lists the traces of Frobenius elements for the first few unramified primes under these representations:

\begin{matrix}p & 2 & 5 & 7 & 11 & 13 & 17 & 23 & 29 \\ ρ̄ & 1 & 1 & 0 & 0 & 0 & 0 & 1 & −1 \\ ρ_{∅}^{mod} & (1, −2) & (−2, 1) & (0, 3) & (0, −3) & (6, −6) & (−6, 3) & (4, 4) & (2, −10) \\ ρ & 1 − ε & 1 + ε & ε & −ε & −ε & 0 & 1 & −1 − ε \end{matrix}

Exercise 3.29 Show that there are three algebra homomorphisms 𝕋_{∅} → ℤ/9ℤ and hence show that there are at least three liftings of ρ̄ of type ∅ to ℤ/9ℤ.

Exercise 3.30 What is the image of ρ?

We will need two deeper properties of the Hecke algebras 𝕋_{Σ} (theorem 3.31 and theorem 3.36 below). These results will be proved in the next chapter (sections 4.3 and 4.4).

Theorem 3.31 Let Q be a finite set of primes as in section 2.8. Then 𝕋_{Q} is a free O[Δ_{Q}]-module, where 𝕋_{Q} inherits the structure of an O[Δ_{Q}]-module from R_{Q} via the homomorphism ϕ_{Q} : R_{Q} → → 𝕋_{Q}.

Corollary 3.32 𝕋_{∅} = 𝕋_{Q}/𝔞_{Q}.

Proof: From the definitions and corollary 2.45 we have that

𝕋_{∅} ⊗_{O} K = (𝕋_{Q} ⊗_{O} K)/𝔞_{Q},

and from the theorem we have that 𝕋_{Q}/𝔞_{Q} is torsion free. The corollary follows. □

For the second of these results we will need some additional notation and we restrict our attention to sets of primes contained in Σρ̄. Suppose that Σ ⊂ Σρ̄ and that f is an element of N_{Σ} whose Fourier coefficients are in O. Then O′_{f} = O and projection to the component corresponding to f gives rise to an O-algebra homomorphism

π = π_{f} : 𝕋_{Σ} → O.

The pushforward of ρ_{Σ}^{mod} by π is equivalent to

ρ_{f} : G_{ℚ} → GL_{2}(O).

Remark 3.33 Most of the objects considered in the rest of the section will depend on the choice of newform f. We also remark that to give an O-algebra homomorphism 𝕋_{Σ} → O is equivalent to giving a newform f in N_{Σ} with coefficients in O. Indeed given such a homomorphism, there exists a newform f in N_{Σ} such that the homomorphism is defined by T_{p} ↦ a_{p}(f) for all p ∉ Σ with p not dividing ℓN(ρ̄). The fact that a_{p}(f) ∈ O for such p implies (using for example parts (b) and (d) of theorem 1.27 and lemma 4.1 below) that all the Fourier coefficients of f are in O. The uniqueness of f is a consequence of the theory of newforms, theorem 1.22.

For Σ′ satisfying

Σ ⊂ Σ′ ⊂ Σρ̄,

let π_{Σ′} denote the composite 𝕋_{Σ′} → 𝕋_{Σ} → O. Let

η_{Σ′} = π_{Σ′}(Ann_{𝕋_{Σ′}}(ker π_{Σ′})).  (3.3.1)

Note that because 𝕋_{Σ′} is reduced, η_{Σ′} ≠ (0). Also let ℘_{Σ′} denote the kernel of ϕ_{Σ′} ∘ π_{Σ′}. Recall that

#℘_{Σ′}/℘_{Σ′}^{2} = #H_{Σ′}^{1}(ℚ, ad^{0}ρ_{f} ⊗_{O} K/O).

Remark 3.34 We have not yet shown that these groups are finite, but if either is finite, then so is the other and their cardinality is the same.

Note that if ℓ is in Σρ̄ − Σ then T_{ℓ} := (a_{ℓ}(g))_{g} ∈ 𝕋̃_{Σ} is actually in 𝕋_{Σ}. This follows from theorem 3.1(f), which shows that T_{ℓ} = α_{ℓ} + ℓα_{ℓ}^{−1} where α_{ℓ} is the eigenvalue of Frob_{ℓ} on the free rank one 𝕋_{Σ}-module M^{(0)} where M is the module underlying ρ_{Σ}^{mod} (see lemma 2.20). Now that we have shown that T_{ℓ} ∈ 𝕋_{Σ}, we may characterize α_{ℓ} as the unit root in 𝕋_{Σ} of

X^{2} − T_{ℓ}X + ℓ = 0.  (3.3.2)

For primes p in Σρ̄ − Σ we define an element c_{p} ∈ 𝕋_{Σ} by

c_{p} = (p − 1)(T_{p}^{2} − (p + 1)^{2}).

Note that π(c_{p}) = (p − 1)(a_{p}(f)^{2} − (p + 1)^{2}), which is non-zero by theorem 1.27(a). A calculation using theorem 2.17 and theorem 3.1(a) shows that if p is in Σρ̄ − (Σ ∪ ℓ) then

\begin{matrix}#H^{1}(G_{p}, ad^{0}ρ_{f} ⊗_{O} K/O)/H^{1}(G_{p}/I_{p}, ad^{0}ρ_{f} ⊗_{O} K/O) \\ = #H^{0}(G_{p}, ad^{0}ρ_{f} ⊗_{O} K/O(1)) = #O/π(c_{p}). \end{matrix}

If ℓ is in Σρ̄ − Σ then we see from proposition 2.27 and theorem 3.1(f) that

#H_{ss}^{1}(G_{ℓ}, ad^{0}ρ_{f} ⊗_{O} K/O)/H_{f}^{1}(G_{ℓ}, ad^{0}ρ_{f} ⊗_{O} K/O) = #O/π(c_{ℓ}).

For the next proposition define groups H_{p} for p ∈ Σρ̄ − Σ by

• H_{p} = H^{1}(G_{p}, ad^{0}ρ_{f} ⊗_{O} K/O)/H^{1}(G_{p}/I_{p}, ad^{0}ρ_{f} ⊗_{O} K/O) if p ≠ ℓ

• and H_{ℓ} = H_{ss}^{1}(G_{ℓ}, ad^{0}ρ_{f} ⊗_{O} K/O)/H_{f}^{1}(G_{ℓ}, ad^{0}ρ_{f} ⊗_{O} K/O).

Then we have the following result.

Proposition 3.35 If Σ ⊂ Σ′ ⊂ Σρ̄, then

#H_{Σ′}^{1}(ℚ, ad^{0}ρ_{f} ⊗_{O} K/O)/H_{Σ}^{1}(ℚ, ad^{0}ρ_{f} ⊗_{O} K/O) ≤ #(O/π( ∏_{p∈Σ′−Σ} c_{p})).

Moreover if we have equality then the sequence

(0) → H_{Σ}^{1}(ℚ, ad^{0}ρ_{f} ⊗_{O} K/O) → H_{Σ′}^{1}(ℚ, ad^{0}ρ_{f} ⊗_{O} K/O) → ⨁_{p∈Σ′−Σ} H_{p} → (0)

is exact.

Finally we state the second theorem on Hecke algebras which shall be proved in section 4.4.

Theorem 3.36 If Σ ⊂ Σ′ ⊂ Σρ̄ and f is a newform in N_{Σ} with coefficients in O, then we have

η_{Σ′} ⊂ π( ∏_{p∈Σ′−Σ} c_{p})η_{Σ}.

Corollary 3.37 With the above notation

#H_{Σ′}^{1}(ℚ, ad^{0}ρ_{f} ⊗_{O} K/O)/H_{Σ}^{1}(ℚ, ad^{0}ρ_{f} ⊗_{O} K/O) ≤ #(η_{Σ}/η′_{Σ}).

3.4 Isomorphism criteria

The main thrust of Wiles’ approach is to prove that in many circumstances the map ϕ_{Σ} : R_{Σ} → 𝕋_{Σ} is an isomorphism. For this we will need two criteria from commutative algebra. The first was found by Wiles [W3] (but is presented here in a slightly stronger form due to Lenstra [Len]); the second was developed by Faltings, from the original arguments of [TW]. In both criteria the notion of complete intersection plays a vital part.

Proofs of all the results in this section, together with some background, references and examples, is given in chapter 5.

Definition 3.38 Suppose that A is an object of C_{O} which is finitely generated and free as an O-module. Then we call A a complete intersection if and only if for some r ∈ ℤ_{≥0} and some f_{1}, ..., f_{r} ∈ O[[X_{1}, ..., X_{r}]] we have

A ≅ O[[X_{1}, ..., X_{r}]]/(f_{1}, ..., f_{r})

(i.e. there are the same number of generators as relations).

We first record a lemma about complete intersections.

Lemma 3.39 Suppose that K′/K is a finite extension with ring of integers O′ and that A is an object of C_{O} which is finitely generated and free as an O-module. Then A is a complete intersection if and only if A ⊗_{O} O′ is.

For the proof, see chapter 5, lemma 5.30.

Now fix objects R and T of C_{O} and a surjection of O-algebras ϕ : R → → T. Also assume that T is a finitely generated, free O-module. The first criterion is as follows.

Theorem 3.40 Suppose that π : T → → O. Let ℘ = ker (π ∘ ϕ) ⊲ R and let η = π(Ann_{T} (ker π)) ⊲ O. Suppose also that η ≠ (0). Then the following are equivalent.

(a) The inequality #℘/℘^{2} ≤ #O/η is satisfied.

(b) The equality #℘/℘^{2} = #O/η is satisfied.

(c) The rings R and T are complete intersections, and the map ϕ : R ⟶ T is an isomorphism.

The proof is explained in chapter 5, sections 5.1 to 5.8. (See theorem 5.3.)

For the second criterion let us also fix a non-negative integer r. If J ⊲ O[[S_{1}, ..., S_{r}]] is an ideal contained in (S_{1}, ..., S_{r}), then by a J-structure we mean a commutative diagram in C_{O}

O[[S_{1}, ..., S_{r}]]
↓
O[[X_{1}, ..., X_{r}]] → → R′ → → T′
↓  ↓
R  → → T,

such that

(a) T′/(S_{1}, ..., S_{r}) →^{∼} T and R′/(S_{1}, ..., S_{r}) → → R,

(b) for each ideal I ⊃ J, I = ker (O[[S_{1}, ..., S_{r}]] → T′/I).

Theorem 3.41 Suppose there exist a sequence of ideals J_{n} ⊲ O[[S_{1}, ..., S_{r}]] such that J_{0} = (S_{1}, ..., S_{r}), J_{n} ⊃ J_{n+1}, ⋂_{n} J_{n} = (0) and for each n there exists a J_{n}-structure. Then the map R ⟶ T is an isomorphism and these rings are complete intersections.

The proof of this statement is explained in sections 5.9 and 5.10.

3.5 The main theorem

We are now in a position to deduce the main theorems. We will keep the notation and assumptions from the start of section 3.3.

Theorem 3.42 Keep the notation and assumptions of section 3.3. Then, for all finite sets Σ ⊂ Σρ̄, ϕ_{Σ} : R_{Σ} ⟶ 𝕋_{Σ} is an isomorphism and these rings are complete intersections.

Remark 3.43 There seems to be a deep link between the fact that ϕ_{Σ} is an isomorphism and the fact that 𝕋_{Σ} is a complete intersection. The proof of the theorem divides into two parts. One first proves it in the minimal case where Σ = ∅. One then deduces the full theorem from this special case by a different argument. In both these steps the facts that ϕ_{Σ} is an isomorphism and that 𝕋_{Σ} is a complete intersection are proved simultaneously.

Proof of theorem 3.42: Note that to prove the theorem we may extend scalars if necessary (by lemma 3.39) and hence assume that both of the following hold:

• The eigenvalues of all elements of the image of ρ are rational over k.

• There is a newform f in N_{∅} with coefficients in O, hence an O-algebra homomorphism 𝕋_{∅} → O.

We first prove that ϕ_{∅} : R_{∅} ⟶ 𝕋_{∅} is an isomorphism and that the rings R_{∅} and 𝕋_{∅} are complete intersections.

Note that according to theorem 2.49 and lemma 3.24 we can find an integer r ≥ 0 and for each n ∈ ℤ_{>0} we can find a set Q_{n} of r primes such that

• if q ∈ Q_{n} then q ≡ 1 mod ℓ^{n};

• if q ∈ Q_{n} then ρ̄ is unramified at q and ρ̄(Frob_{q}) has distinct eigenvalues;

• R_{Q_{n}} can be topologically generated by r elements as a O-algebra.

Let J′_{n} = ((S_{i} + 1)^{ℓ^{n_{i}}} − 1 : i = 1, …, r) ⊲ O[[S_{1}, …, S_{r}]], where the n_{i} are chosen such that O[[S_{1}, …, S_{r}]]/J_{n} ≅ O[Δ_{Q_{n}}]. Then for each n we have a diagram

O[[S_{1}, …, S_{r}]]
↓
O[[X_{1}, …, X_{r}]] → →  R_{Q_{n}}  → → 𝕋_{Q_{n}}
↓  ↓
R_{∅}  → → 𝕋_{∅},

where (a) R_{Q_{n}}/(S_{1}, …, S_{r}) →^{∼} R_{∅} (see corollary 2.45),

(b) 𝕋_{Q_{n}} is finite and free over O[[S_{1}, …, S_{r}]]/J′_{n} (see theorem 3.31),

(c) 𝕋_{Q_{n}}/(S_{1}, …, S_{r}) →^{∼} 𝕋_{∅} (see corollary 3.32).

Let J_{n} = ((S_{i} + 1)^{ℓ^{n}} − 1 : i = 1, …, r). Replacing 𝕋_{Q_{n}} and R_{Q_{n}} by 𝕋_{Q_{n}}/J_{n} and R_{Q_{n}}/J_{n} we see that we have a J_{n}-structure for every n. Theorem 3.42 for Σ = ∅ now follows from the criterion of theorem 3.41.

We now turn to the proof of theorem 3.42 in the general case. By theorem 2.41 and theorem 3.40 we see that

#H_{∅}^{1}(ℚ, ad^{0}ρ_{f} ⊗_{O} K/O) = #O/η_{∅},  (3.5.1)

and so applying corollary 3.37 we see that for any Σ ⊂ Σρ̄

#H_{Σ}^{1}(ℚ, ad^{0}ρ_{f} ⊗_{O} K/O) ≤ #O/η_{Σ}.  (3.5.2)

A second application of theorems 2.41 and 3.40 allows us to deduce theorem 3.42. □

Remark 3.44 In certain cases where η_{∅} = (1), the bound (3.5.1) on the order of the Selmer group H_{∅}^{1}(ℚ, ad^{0}ρ_{f} ⊗_{O} K/O) also follows from the previous work of Flach, by a different method. See [Fl1] for details.

Corollary 3.45 Keep the notation of theorem 3.42 and suppose that f is a newform in N_{Σ} with coefficients in O.

(a) We have

#H_{Σ}^{1}(ℚ, ad^{0}ρ_{f} ⊗_{O} K/O) = #O/η_{Σ} < ∞,

where η_{Σ} was defined in equation (3.3.1) after remark 3.33.

(b) If Σ ⊂ Σ′ ⊂ Σρ̄ then

(0) → H_{Σ}^{1}(ℚ, ad^{0}ρ_{f} ⊗_{O} K/O) → H_{Σ′}^{1}(ℚ, ad^{0}ρ_{f} ⊗_{O} K/O)
→ ⨁_{p∈Σ′−Σ} H_{p} → (0)

is exact, where the groups H_{p} and H_{ℓ} were defined before proposition 3.35.

Proof: The first part now follows as a direct consequence of theorem 3.42 and another application of theorems 2.41 and 3.40. The second part follows from the first, together with proposition 3.35 and theorem 3.36. □

Corollary 3.46 Suppose ρ : G_{ℚ} → GL_{2}(K) is a continuous representation and let ρ̄ denote its reduction. Suppose also that

(a) ρ̄ is irreducible and modular,

(b) if p ≠ ℓ then ρ|_{I_{p}} ∼ (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix}),

(c) ρ|_{G_{ℓ}} is semi-stable,

(d) det ρ = ϵ.

Then ρ is modular.

Proof: We let Σ denote the set of primes in Σρ̄ at which ρ is ramified. Then ρ : G_{ℚ} → GL_{2}(O) is a deformation of ρ̄ of type Σ, so there is an O-algebra homomorphism R_{Σ} → O such that ρ = ρ_{Σ}^{univ} ⊗_{R_{Σ}} O. Since ϕ_{Σ} : R_{Σ} → 𝕋_{Σ} is an isomorphism by theorem 3.42, it follows that there is a homomorphism 𝕋_{Σ} → O sending T_{p} to tr (ρ(Frob_{p})). Since such a homomorphism is necessarily of the form T_{p} ↦ a_{p}(f) for some newform f, it follows that ρ is equivalent to ρ_{f} and hence is modular. □

Corollary 3.47 Suppose that E/ℚ is a semi-stable elliptic curve such that ρ̄_{E,3} is irreducible. Then E is modular.

Proof: One need only apply the last corollary with ℓ = 3 and theorem 3.14. □

3.6 Applications

The Shimura-Taniyama conjecture for semi-stable elliptic curves:

Theorem 3.48 If E/ℚ is a semistable elliptic curve, then E is modular.

Proof: By corollary 3.47, it is enough to show that E is modular when its associated mod 3 representation ρ̄_{E,3} is reducible, i.e., when E has a subgroup of order 3 defined over ℚ. Consider the group E[5] of 5-division points of E. The mod 5 Galois representation ρ̄_{E,5} associated to E[5] is irreducible: for otherwise, E would have a subgroup of order 15 defined over ℚ, and would give rise to a (non-cuspidal) rational point on the modular curve X_{0}(15). This curve is of genus one, and is known to have only 4 non-cuspidal rational points, which do not correspond to semi-stable elliptic curves (and, at any rate, are known to correspond to modular elliptic curves). Hence we know that ρ̄_{E,5} satisfies all the assumptions of corollary 3.46, except the (crucial!) modularity property. To show that ρ̄_{E,5} is modular, one starts with Lemma 3.49 There is an auxiliary (semi-stable) elliptic curve A/ℚ which satisfies

(a) A[5] ≃ E[5] as G_{ℚ}-modules.

(b) A[3] is an irreducible G_{ℚ}-module.

Proof: Let Y′(5) be the curve over ℚ which classifies elliptic curves A together with an isomorphism E[5] ≃ A[5] compatible with Weil pairings. Elliptic curves over ℚ satisfying (a) correspond to rational points in Y′(5)(ℚ). Adjoining a finite set of points to Y′(5) yields its compactification X′(5) which is a twist of the modular curve X(5) with full level 5 structure. (I.e., it becomes isomorphic to this curve, over ℚ̄.) As was shown by Klein, the modular curve X(5) over ℂ has genus 0. Since X′(5) has a point x_{0} defined over ℚ corresponding to E, it is isomorphic over ℚ to ℙ^{1}. The rational points of Y′(5) therefore give a plentiful supply of elliptic curves satisfying condition (a). Now consider the curve Y′(5, 3) classifying elliptic curves A with an isomorphism E[5] ≃ A[5] (respecting Weil pairings) and a subgroup of A of order 3. One checks that the compactification of Y′(5, 3) has genus greater than 1, hence has only finitely many rational points by Faltings’ theorem (the Mordell conjecture). It follows that only finitely many points Y′(5)(ℚ) are in the image of Y′(5, 3)(ℚ) under the natural map Y′(5, 3) → Y′(5). Hence for all but finitely many points x in Y′(5)(ℚ), the corresponding elliptic curve A satisfies (b) since it has no rational subgroup of order 3. Choosing x arbitrarily close in the 5-adic topology to x_{0}, we find that the elliptic curve A associated to x is semistable and satisfies the two conditions in the lemma. □

We can now finish the proof of theorem 3.48. Applying corollary 3.47 to the curve A, we find that A is modular. Hence so is the mod 5 representation ρ̄_{A,5} ≃ ρ̄_{E,5}. Now applying corollary 3.46 with ℓ = 5 and ρ the representation of G_{ℚ} acting on the 5-adic Tate module of E, we find that ρ_{E,5} is modular, and hence, so is E, as was to be shown.

Remark 3.50 Wiles’ original argument uses Hilbert’s irreducibility theorem where we have used Faltings’ theorem. The alternative presented here is based on a remark of Karl Rubin.

Remark 3.51 The results of [W3] and [TW] actually apply to a larger class of elliptic curves than those which are semistable. In [Di2], their methods are further strengthened to prove that all elliptic curves which have semi-stable reduction at 3 and 5 are modular.

Remark 3.52 Rubin and Silverberg observed that an elliptic curve of the form y^{2} = x(x − a)(x + b) has a twist with semi-stable reduction at all odd primes, hence is modular by [Di2]. In fact it is shown in [DK] that their observation together with the general results of [W3] and [TW] already imply modularity.

Fermat’s Last Theorem: As was already mentioned in the introduction, the Shimura-Taniyama conjecture for semi-stable elliptic curves (and, more precisely, for the elliptic curves that arise in Frey’s construction explained in section 2.2) implies Fermat’s Last Theorem.

More precisely, suppose that there is a non-trivial solution to the Fermat equation x^{ℓ} + y^{ℓ} = z^{ℓ}, with ℓ > 3. By theorem 2.15 the Frey curve constructed from this solution (cf. section 2.2) is a semistable elliptic curve E/ℚ whose associated mod ℓ representation ρ̄_{E,ℓ} is irreducible, unramified outside 2ℓ and is good at ℓ. Serre’s conjecture predicts that ρ̄_{E,ℓ} arises from a newform of weight 2 and level 2; the “lowering the level” result of Ribet [R5] (cf. theorem 3.15) actually proves this, once we know that E is modular, i.e., ρ̄_{E,ℓ} arises from a modular form of weight 2 and some level. But this is a contradiction, since there are no modular forms of weight two and level two: such forms would correspond to holomorphic differentials on the modular curve X_{0}(2) which is of genus 0. This contradiction completes the proof.

Values of L-functions: Also mentioned in the introduction was the relationship between the calculation of the Selmer group (3.5.1) and certain cases of a conjecture of Bloch-Kato [BK], called the Tamagawa number conjecture. It was in this context that partial results were obtained by Flach in [Fl1] (cf. remark 3.44).

If f is a newform of weight 2, then one can associate to f a certain “symmetric square” L-function L(Symm^{2}f, s). We shall recall the definition in section 4.4 and explain how a method of Hida establishes a relationship between L(Symm^{2}f, 2) and O/η_{Σ} in the setting of corollary 3.45. We may therefore regard part (a) of that corollary as a relationship between L(Symm^{2}f, 2) and the size of a Selmer group. While the result is in the spirit of the Tamagawa number conjecture of [BK], we have not verified that the relevant cases of the conjecture can be deduced from it. We shall however state a partial result in the context of semistable elliptic curves. The reader can consult [Fl1] and [Fl2] for a discussion of the relation to the Tamagawa number conjecture.

Suppose that E is a semistable elliptic curve over ℚ of conductor N_{E} and (minimal) discriminant

Δ_{E} = ∏_{p|N_{E}} p^{d_{p}}.

The symmetric square L-function associated to E is defined by

L(Symm^{2}E, s) = ∏_{p} L_{p}(Symm^{2}E, s)

where the Euler factors L_{p}(Symm^{2}E, s) are defined as follows:

• If p ∤ N_{E}, then

L_{p}(Symm^{2}E, s) = ((1 − α_{p}^{2}p^{−s})(1 − p^{1−s})(1 − β_{p}^{2}p^{−s}))^{−1}

where α_{p} and β_{p} are the roots roots of X^{2} −a_{p}X +p with a_{p} = p+1−N_{p} as in section 1.1).

• If p|N_{E}, then L_{p}(Symm^{2}E, s) = (1 − p^{−s})^{−1}.

Let

Ω_{E} = ∫_{E(ℂ)} ω_{E}^{Neron} ∧ ω̄_{E}^{Neron}

where ω_{E}^{Neron} is the Néron differential defined in section 1.1. Since E is modular by theorem 3.48, a method of Shimura (see [Shi4] and the introduction of [St]) establishes the analytic continuation of L(Symm^{2}E, s) to an entire function and shows that L(Symm^{2}E, 2) is a non-zero rational multiple of iπΩ_{E}. We now explain how to deduce the following theorem from Wiles’ results and a formula of Hida, corollary 4.21.

Theorem 3.53 Suppose that E is a semistable elliptic curve and ℓ is a prime such that

• ρ̄_{E,ℓ} is irreducible, and

• ℓ does not divide 2 ∏_{p|N_{E}} d_{p}.

Then the ℓ-part of

\frac{N_{E}L(Symm^{2}E, 2)}{πiΩ_{E}}

is the order of

H_{∅}^{1}(ℚ, ad^{0}ρ_{E,ℓ} ⊗_{ℤ_{ℓ}} ℚ_{ℓ}/ℤ_{ℓ}).

Sketch of proof: Since E is semistable, it is modular by theorem 3.48. Letting f denote the associated newform, we have N_{f} = N_{E} and L(Symm^{2}E, s) = L(Symm^{2}f, s). Furthermore the representations ρ_{E,ℓ} and

ρ_{f} : G_{ℚ} → GL_{2}(ℤ_{ℓ})

are equivalent, since they are equivalent over ℚ_{ℓ} and ρ̄ = ρ̄_{E,ℓ} is irreducible. The conditions on E and ℓ ensure that ρ̄ and f satisfy the hypotheses of corollary 3.45 with Σ = ∅, so that

#H_{∅}^{1}(ℚ, ad^{0}ρ_{E,ℓ} ⊗_{ℤ_{ℓ}} ℚ_{ℓ}/ℤ_{ℓ}) = #(ℤ_{ℓ}/η_{∅}).

To apply Hida’s formula, corollary 4.21, it remains to relate det A and Ω_{E} where the matrix A ∈ GL_{2}(ℂ) is defined in section 4.4. Using that E is semistable, ℓ is odd and ρ̄_{E,ℓ} is irreducible, one obtains a modular parametrization π : X_{0}(N_{E}) → E such that

• π^{∗} : H^{1}(E, ℤ_{ℓ}) → H^{1}(X_{0}(N_{E}), ℤ_{ℓ}) has torsion-free cokernel;

• the Manin constant for π is not divisible by ℓ ([Maz2], sec. 4(a)).

One can then verify that Ω_{E}^{−1} det A is an ℓ-adic unit and theorem 3.53 follows from corollary 4.21. □

4 Hecke algebras

4.1 Full Hecke algebras

Suppose that K is a finite extension of ℚ_{ℓ} for some prime ℓ. Let O denote its ring of integers and let k = O/λ where λ is the maximal ideal of O. Fix embeddings K ↪ ℚ̄_{ℓ}, ℚ̄ ↪ ℚ̄_{ℓ} and ℚ̄ ↪ ℂ. Recall that we defined “Hecke algebras” over O in two different contexts:

• In section 1.6 as an algebra 𝕋_{O} generated by the full set of Hecke operators acting on a space of modular forms;

• In section 3.3 as a certain subring 𝕋_{Σ} of a product of fields of Fourier coefficients of newforms giving rise to the same mod ℓ representation.

The first of these provides a concrete geometric description useful for establishing properties of the fine structure of the algebra; the second yields a reduced ring which is more easily interpreted as the coefficient ring of a Galois representation. In the next section we shall relate the two notions by identifying the “reduced Hecke algebras” of the form 𝕋_{Σ} as localizations of the “full Hecke algebras” of the form 𝕋_{O}. Before doing so we need to recall some fundamental properties of the algebras 𝕋_{K}, 𝕋_{O} and 𝕋_{k}.

Let Γ = Γ_{H}(N) for some positive integer N and subgroup H of (ℤ/Nℤ)^{×} (see section 1.2). We let 𝕋_{ℤ} denote the subring of End (S_{2}(Γ)) generated by the operators T_{n} for all positive integers n and ⟨d⟩ for all d ∈ (ℤ/Nℤ)^{×}. If R is a ring, then 𝕋_{R} denotes the R-algebra 𝕋_{ℤ}⊗R. Recall that 𝕋_{R} acts faithfully on S_{2}(Γ, R) and is finitely generated and free as an R-module. (This holds for R = ℤ, hence for arbitrary R.)

We first record the following lemma:

Lemma 4.1 (a) 𝕋_{R} is generated as an R-algebra by either of the following sets of elements:

• T_{n} for all positive integers n.

• T_{p} for all primes p and ⟨d⟩ for all d in (ℤ/Nℤ)^{×}.

(b) Suppose that D is a positive integer relatively prime to N. If either D is odd or 2 is invertible in R, then 𝕋_{R} is generated as an R-algebra by either of the following sets of elements:

• T_{n} for all positive integers n relatively prime to D.

• T_{p} for all primes p not dividing D, and ⟨d⟩ for all d in (ℤ/Nℤ)^{×}.

For a proof of (a), see [DI], prop. 3.5.1; for (b), see p. 491 of [W3].

The spectrum of 𝕋_{O}: First note that 𝕋_{K} and 𝕋_{k} are Artinian, hence have only a finite number of prime ideals, all of which are maximal. Since 𝕋_{O} is finitely generated and free as an O-module, its maximal (resp. minimal) prime ideals are those lying over the prime λ (resp. (0)) of O. (This follows from the going-up and going-down theorems, [Mat] thms. 9.4 and 9.5, for example.) It follows that the natural maps

𝕋_{O} ↪ 𝕋_{O} ⊗_{O} K ≅ 𝕋_{K}; and 𝕋_{O} → → 𝕋_{O} ⊗_{O} k ≅ 𝕋_{k}

induce bijections

{ maximal ideals of 𝕋_{K} } ↔ { minimal primes of 𝕋_{O} } and
{ maximal ideals of 𝕋_{k} } ↔ { maximal primes of 𝕋_{O} }.

Moreover, since O is complete we have (by [Mat] thms. 8.7 and 8.15, for example) that the natural map

𝕋_{O} → ∏_{𝔪}𝕋_{𝔪}

is an isomorphism, where the product is over the finite set of maximal ideals 𝔪 of 𝕋_{O} and 𝕋_{𝔪} denotes the localization of 𝕋_{O} at 𝔪. Furthermore each 𝕋_{𝔪} is a complete local O-algebra which is finitely generated and free as an O-module, and each minimal prime P of 𝕋_{O} is contained in a unique 𝔪.

Now suppose that f = ∑ a_{n}q^{n} is a normalized eigenform in S_{2}(Γ, K̄) for the operators T_{n} for all n ≥ 1. Then T_{n} ↦ a_{n} defines a map 𝕋_{ℤ} → K̄ and induces a K-algebra homomorphism θ_{f} : 𝕋_{K} → K̄. The image is the finite extension of K generated by the a_{n}, and the kernel is a maximal ideal of 𝕋_{K} which depends only on the G_{K}-conjugacy class of f. Similarly a G_{k}-conjugacy class of normalized eigenforms in S_{2}(Γ, k̄) gives rise to a maximal ideal of 𝕋_{k}.

Recall also that a normalized eigenform f in S_{2}(Γ, K̄) has coefficients in OK̄, hence gives rise by reduction to a normalized eigenform f̄ in S_{2}(Γ, k̄). Furthermore if f and g are G_{K}-conjugate, then f̄ and ḡ are G_{k}-conjugate.

We have thus constructed a diagram of maps of finite sets whose commutativity is easily verified.

\begin{cases}normalized eigenforms in \\ S_{2}(Γ, K̄) modulo G_{K}-conjugacy \end{cases}\} → \begin{cases}normalized eigenforms in \\ S_{2}(Γ, k̄) modulo G_{k}-conjugacy \end{cases}\}
↓  ↓
{ maximal ideals of 𝕋_{K} }  { maximal ideals of 𝕋_{k} }
↕  ↕
{ minimal primes of 𝕋_{O} }  → → { maximal primes of 𝕋_{O} }.

(4.1.1)

Proposition 4.2 The vertical maps are bijective, and the horizontal maps are surjective.

Proof: For the injectivity of the upper-left vertical arrow, note that if 𝔭 is a maximal ideal of 𝕋_{K}, then all K-algebra homomorphisms 𝕋_{K}/𝔭 ↪ K̄ are obtained from a single one by composing with an element of G_{K}. For the surjectivity, let K′ = 𝕋_{K}/𝔭 and 𝔭′ denote the kernel of the natural K′-algebra homomorphism 𝕋_{K′} → → K′. Since 𝕋_{K′} acts faithfully on S_{2}(Γ, K′), the localization S_{2}(Γ, K′)_{𝔭} is non-zero, hence so is S_{2}(Γ, K′)[𝔭′]. (For an R-module M and an ideal I of R, we write M[I] for the intersection over the elements r in I of the kernels of r : M → M.) It follows that there is a normalized eigenform f in S_{2}(Γ, K′) so that 𝔭′ is the kernel of the θ′_{f} : 𝕋_{K′} → K′, and therefore 𝔭 is the kernel of θ_{f}. To prove that the upper-right vertical arrow is bijective, note that the above arguments carry over with K replaced by k. □

Remark 4.3 The surjectivity of the top arrow is called the Deligne-Serre lifting lemma ([DS], lemma 6.11).

Suppose that 𝔪 is a maximal ideal of 𝕋_{O}. Note that the maximal ideals of 𝕋_{K} mapping to 𝔪 are precisely those 𝔭 for which 𝔭 ∩ 𝕋_{O} is contained in 𝔪. Note also that the natural map

α_{𝔪} : 𝕋_{𝔪} ⊗_{O} K → ∏_{𝔭}𝕋_{𝔭}

is an isomorphism, where the product is over such 𝔭.

It is straightforward to check that the above constructions are well-behaved with respect to replacing the field K by an extension K′. More precisely, for each set S in the above diagram, there is a natural surjective map from ϖ : S′ → S where S′ is defined by replacing K with K′, and these maps are compatible with the maps in the diagram. Furthermore the maps

𝕋_{𝔭} ⊗_{K} K′ → ∏_{𝔭′∈ϖ^{−1}(𝔭)}𝕋_{𝔭′} and 𝕋_{𝔪} ⊗_{O} O′ → ∏_{𝔪′∈ϖ^{−1}(𝔪)} 𝕋_{𝔪′}

are isomorphisms by which α_{𝔪} ⊗_{K} K′ can be identified with ∏_{𝔪′∈ϖ^{−1}(𝔪)} α_{𝔪′}.

Associated Galois representations: Suppose that 𝔭 is a maximal ideal of 𝕋_{K} and 𝔪 is the associated maximal ideal of 𝕋_{O}. By lemma 1.39,

T_{ℓ}(J_{H}(N)) ⊗_{ℤ_{ℓ}} K

is free of rank two over 𝕋_{K}, so reduction mod 𝔭 yields a two-dimensional vector space over the field 𝕋_{K}/𝔭 endowed with an action of G_{ℚ}. The resulting Galois representation

ρ_{𝔭} : G_{ℚ} → GL_{2}(𝕋_{K}/𝔭)

is unramified at all primes p not dividing Nℓ, and for such p the characteristic polynomial of ρ_{𝔭}(Frob_{p}) is

X^{2} − T_{p}X + p⟨p⟩ mod 𝔭.

If ℓ is odd, then ρ̄_{𝔭} is defined over 𝕋_{O}/𝔪 and we write ρ_{𝔪} for its semisimplification

ρ_{𝔪} : G_{ℚ} → GL_{2}(𝕋_{O}/𝔪).

Thus ρ_{𝔪} is unramified at primes p not dividing Nℓ and the characteristic polynomial of Frob_{p} is

X^{2} − T_{p}X + p⟨p⟩ mod 𝔪.

Suppose that g is in the G_{K}-conjugacy class of eigenforms in S_{2}(Γ) corresponding to 𝔭. If g is a newform then 𝕋_{K}/𝔭 is isomorphic to the field denoted K′_{g} in section 3.1 and ρ_{𝔭} can be identified with the representation

ρ_{g} : G_{ℚ} → GL_{2}(K′_{g})

considered in theorem 3.1. If also ℓ is odd then ρ̄_{g} is obtained from ρ_{𝔪} by extending scalars. More generally suppose that g is not necessarily a newform and consider the associated newform f. Let D denote the product of the primes which divide N but not N_{f}. Let 𝕋_{K}^{(D)} denote the K-subalgebra of 𝕋_{K} generated by the operators T_{n} for n relatively prime to D and ⟨d⟩ for d in (ℤ/Nℤ)^{×}. Let Γ′ = Γ_{H′}(N_{f}) where H′ is the image of H in (ℤ/N_{f}ℤ)^{×}, and let 𝕋′_{K} = 𝕋′_{ℤ} ⊗ K where 𝕋′_{ℤ} is the Hecke algebra acting on S_{2}(Γ′). Restriction of operators defines a natural homomorphism 𝕋_{K}^{(D)} → 𝕋′_{K} which is surjective by lemma 4.1. The composite with 𝕋′_{K} → → K′_{f} factors through the field 𝕋_{K}^{(D)}/(𝔭∩𝕋_{K}^{(D)}), so we may identify K′_{f} with a subfield of 𝕋_{K}/𝔭 and ρ_{𝔭} is then equivalent to the extension of scalars of ρ_{f}. If ℓ is odd, then ρ_{𝔪} and ρ̄_{f} are defined and equivalent over a common subfield of 𝕋_{O}/𝔪 and k_{f}.

The structure of 𝕋_{K}: We now give an explicit description of 𝕋_{K} in the case that K contains the coefficients of all eigenforms of level dividing N. Let N_{Γ} denote the set of newforms in S_{2}(Γ), i.e., the set of newforms f of level N_{f} dividing N such that H is contained in the kernel of the character

(ℤ/Nℤ)^{×} → (ℤ/N_{f}ℤ)^{×} →^{ψ_{f}} K^{×}.

By theorem 1.22

S_{2}(Γ, K) = ⨁_{f∈N_{Γ}} S_{K,f},

where S_{K,f} is spanned by the linear independent elements

{ f(aτ) | a divides N/N_{f} }.

For each f = ∑ a_{n}(f)q^{n} in N_{Γ}, let

• 𝕋_{K,f} denote the image of 𝕋_{K} in End_{K}S_{K,f};

• A_{K,f} denote the polynomial ring over K in the variables u_{f,p} indexed by the prime divisors of N/N_{f};

• I_{K,f} denote the ideal in A_{K,f} generated by the polynomials

P_{f,p}(u_{f,p}) = u_{f,p}^{v_{p}(N/N_{f})−1}(u_{f,p}^{2} − a_{p}(f)u_{f,p} + ψ_{f}(p)p),

for primes p dividing N/N_{f} (setting ψ_{f}(p) = 0 if p divides N_{f}).

Consider the K-algebra homomorphism A_{K,f} → → 𝕋_{K,f} defined by mapping u_{f,p} to the operator T_{p}. Since P_{f,p} is the characteristic polynomial of T_{p} on the span of { f(ap^{i}τ) | i = 1, …, v_{p}(N/N_{f}) } for each a dividing N/(N_{f}p^{v_{p}(N/N_{f})}), we see that I_{K,f} is contained in the kernel of A_{K,f} → → 𝕋_{K,f}. Taking the product over f in N_{Γ}, we have a surjective K-algebra homomorphism

∏_{f} A_{K,f}/I_{K,f} → → ∏_{f} 𝕋_{K,f}.

Since the natural map 𝕋_{K} → ∏_{f} 𝕋_{K,f} is injective and

dim_{K} 𝕋_{K} = dim_{ℂ} S_{2}(Γ, ℂ) = ∑_{f} σ_{0}(N/N_{f}) = ∑_{f} dim_{K} A_{K,f}/I_{K,f},

we conclude

Lemma 4.4 There is an isomorphism of K-algebras:

α : 𝕋_{K} → ∏_{f∈N_{Γ}} A_{K,f}/I_{K,f}

defined by

• α(T_{p})_{f} = a_{p}(f) if p is a prime not dividing N/N_{f};

• α(T_{p})_{f} = u_{f,p} mod I_{K,f} if p is a prime dividing N/N_{f};

• α(⟨d⟩)_{f} = ψ_{f}(d) if d is relatively prime to N.

Remark 4.5 It follows that the algebra 𝕋_{K} is a “complete intersection” over K in the sense that it is a finite-dimensional K-algebra of the form

K[X_{1}, …, X_{r}]/(P_{1}, …, P_{r})

for some r.

4.2 Reduced Hecke algebras

As in section 4.1, we suppose K is a finite extension of ℚ_{ℓ} with ring of integers O and residue field k, and fix embeddings K ↪ ℚ̄_{ℓ}, ℚ̄ ↪ ℚ̄_{ℓ} and ℚ̄ ↪ ℂ. We assume in this section that ℓ is odd and we fix a representation

ρ̄: G_{ℚ} → GL_{2}(k)

which is modular (definition 3.12). Thus ρ̄ is equivalent to ρ_{𝔪} (over 𝕋_{O}/𝔪) for some Γ = Γ_{H}(N) and maximal ideal 𝔪 of 𝕋_{O}.

We suppose also that ρ̄ has the properties listed at the beginning of section 3.3 and that Σ is a finite set of primes contained in Σρ̄ (definition 3.25). We shall show that Γ and 𝔪 can be chosen so that the reduced Hecke algebra 𝕋_{Σ} can be identified with a localization 𝕋_{𝔪} of the full Hecke algebra 𝕋_{O}. The main result is due to Wiles ([W3], prop. 2.15), but we also explain an important variant ([TW], lemma 1) which arises when considering special sets of primes Q as in section 2.8 above.

𝕋_{Σ} and the full Hecke algebra: Let Γ = Γ_{0}(N_{Σ}) where

N_{Σ} = ℓ^{δ}N(ρ̄) ∏_{p∈Σ−{ℓ}} p^{dim^{ρ̄^{Ip}}} = ℓ^{δ} ∏_{p|N(ρ̄)} p ∏_{p∈Σ−{ℓ}} p^{2}  (4.2.1)

with δ = 0 if ρ̄ is good and ℓ ∉ Σ and δ = 1 otherwise.

Suppose that f is a newform in N_{Σ}. Recall that N_{Σ}, defined in section 3.3, is the set of newforms f in S_{2}(Γ) such that ρ̄ is equivalent to ρ̄_{f} over k_{f} (lemma 3.26). Note that these representations are equivalent if and only if

a_{p}(f) mod λ′ = tr (ρ̄(Frob_{p})) for all p ∤ N_{Σ}ℓ,

where λ′ is maximal ideal of the ring of integers of K′_{f}. There is then a normalized eigenform g in S_{2}(Γ, K′_{f}) characterized by

• a_{p}(g) = a_{p}(f) if p does not divide N_{Σ}/N_{f};

• a_{p}(g) = 0 if p ≠ ℓ and p divides N_{Σ}/N_{f};

• a_{ℓ}(g) is the unit root of X^{2} − a_{ℓ}(f)X + ℓ if ℓ divides N_{Σ}/N_{f}.

Note that if ℓ divides N_{Σ}/N_{f}, then δ = 1 and ρ̄ is flat, hence ρ̄ is ordinary and a_{ℓ}(f) is a unit by theorem 3.1 (f).

Lemma 4.6 For f and g as above, the reduction ḡ is the normalized eigenform in S_{2}(Γ, k′) characterized by:

• a_{p}(ḡ) = tr ρ̄_{I_{p}}(Frob_{p}) if p = ℓ or p is not in Σ (recall that ρ̄_{I_{p}} denotes the representation on I_{p}-coinvariants);

• a_{p}(ḡ) = 0 otherwise.

Furthermore, if K is as in lemma 4.4, then g is the unique normalized eigenform in S_{K,f} with this reduction.

Proof: If p is not in Σ ∪ {ℓ} then the formula for a_{p}(ḡ) = a_{p}(f̄) follows from theorem 3.1, parts (a) and (e). If p = ℓ, then we use parts (f) and (g) of theorem 3.1. Otherwise, we have a_{p}(ḡ) = 0 by theorem 1.27(d) (if p^{2}|N_{f} and f has trivial character, then a_{p}(f) = 0). The uniqueness follows from the fact that if p||N_{f}, then a_{p}(f) = ±1 (by theorem 1.27(b)). □

Note in particular that ḡ has coefficients in k and is independent of the choice of f in N_{Σ}. We let 𝔪 denote the corresponding maximal ideal of 𝕋_{O}.

Proposition 4.7 There is an isomorphism 𝕋_{Σ} ⟶^{∼} 𝕋_{𝔪} of O-algebras such that T_{p} ↦ T_{p} for all primes p not dividing N_{Σ}ℓ.

Proof: Replacing K by a larger field K′, we have 𝕋_{Σ}⊗_{O}O′ ⟶^{∼} 𝕋′_{Σ} (lemma 3.27) and 𝕋_{𝔪} ⊗_{O} O′ ⟶^{∼} 𝕋′_{𝔪′} (since there is a unique maximal ideal 𝔪′ of 𝕋_{O′} over 𝔪). We are therefore reduced to the case where K is as in lemma 4.4. Note that N_{Σ} is the set of newforms f in N_{Γ} such that

a_{p}(f̄) = tr (ρ̄(Frob_{p})) for all p ∤ N_{Σ}ℓ.

(We have K = K′_{f} and write f̄ for f mod λ.)

We now define an isomorphism of K-algebras

κ : 𝕋_{𝔪} ⊗ K ⟶^{∼} ∏_{f∈N_{Σ}} K,

such that for each f ∈ N_{Σ}:

• κ(T_{p})_{f} = a_{p}(f) if p is not in Σ;

• κ(T_{p})_{f} = 0 if p is in Σ − {ℓ};

• κ(T_{ℓ})_{f} is the unit root of X^{2} − a_{ℓ}(f)X + ψ_{f}(ℓ)ℓ if ℓ is in Σ.

Recall that 𝕋_{𝔪} ⊗ K ≅ ∏_{𝔭} 𝕋_{𝔭} where 𝔭 runs over the primes of 𝕋_{K} whose preimage in 𝕋_{O} is contained in 𝔪. Thus according to lemma 4.4 we have

𝕋_{𝔪} ⊗ K ⟶^{∼} ∏_{f∈N_{Γ}} ∏_{𝔭∈M_{f}}(A_{K,f}/I_{K,f})_{𝔭},

where M_{f} is the set of prime ideals in A_{K,f}/I_{K,f} whose preimage in 𝕋_{O} is contained in 𝔪. If f is not in N_{Σ}, then M_{f} is empty. If f is in N_{Σ}, then M_{f} consists only of the kernel 𝔭_{f} of the map defined by u_{p,f} ↦ a_{p}(g) where g is the eigenform of lemma 4.6. Furthermore the maps u_{p,f} ↦ a_{p}(g) induce isomorphisms (A_{K,f}/I_{K,f})_{𝔭_{f}} ⟶^{∼} K, and taking the product over f ∈ N_{Σ}, we obtain the desired isomorphism κ.

Identifying 𝕋_{Σ} with the O-subalgebra of ∏_{f∈N_{Σ}} K generated by the elements T_{p} = (a_{p}(f))_{f} for all p not dividing N_{Σ}ℓ, it suffices to prove that 𝕋_{Σ} contains κ(T_{p}) for all p dividing N(ρ̄)ℓ. Now observe that for each f in N_{Σ} the representation ρ_{f} is isomorphic to

G_{ℚ} → GL_{2}(𝕋_{Σ}) → GL_{2}(K)

obtained by composing ρ_{Σ}^{mod} with the projection 𝕋_{Σ} → K. It follows that for p dividing N(ρ̄)ℓ^{δ}, the image of (ρ_{Σ}^{mod})_{I_{p}}(Frob_{p}) in K is (ρ_{f})_{I_{p}}(Frob_{p}) = a_{p}(f)_{f} and therefore

(ρ_{Σ}^{mod})_{I_{p}}(Frob_{p}) = κ(T_{p}).

Finally if δ = 0, then we conclude from lemma 4.1 that 𝕋_{Σ} contains κ(T_{ℓ}). □

A twisted variant: Suppose now that Σ is a finite set of primes contained in Σρ̄ − {ℓ} such that if p is in Σ then ρ̄(Frob_{p}) has eigenvalues α_{p} and β_{p} in k satisfying

α_{p}/β_{p} ≠ p^{±1}.  (4.2.2)

Note that (4.2.2) is equivalent to

tr (ρ̄(Frob_{p}))^{2} ≠ (p + 1)^{2}.  (4.2.3)

Let Q denote the set of primes in Σ such that q ≡ 1 mod ℓ. The set of primes Q is therefore as in section 2.8. Choosing an eigenvalue α_{q} of ρ̄(Frob_{q}) for each q ∈ Q as in section 2.8, we regard R_{Q} and hence 𝕋_{Q} as an O[Δ_{Q}]-algebra. (Recall that Δ_{Q} is the the maximal quotient of (ℤ/(∏_{q∈Q} q)ℤ)^{×} of ℓ-power order.)

Instead of working with Γ_{0}(N_{Σ}) as in proposition 4.7, we shall now work with the group

Γ = Γ_{0}(N_{∅}) ∩ Γ_{1}(M),  (4.2.4)

where

M = ∏_{p∈Σ−Q} p^{2} ∏_{q∈Q} q.

Remark 4.8 We are about to relate 𝕋_{Q} to a localization of 𝕋_{O}, where 𝕋_{O} is now defined using the Hecke operators on S_{2}(Γ), with Γ defined by (4.2.4). Recall that 𝕋_{Q} is defined using modular forms with trivial character, but note that the modular forms involved in the definition of 𝕋_{O} may have non-trivial character. The purpose of establishing this relationship is to give a concrete realization of the image of Δ_{Q} in 𝕋_{Q} for the purpose of proving theorem 3.31.

Suppose that f is a newform pair in N_{∅}. For each q in Q, let α̃_{q}(f) be the root of X^{2} − a_{q}(f)X + q = 0 (in K′_{f}) whose image in k_{f} is α_{q}. We let g denote the unique normalized eigenform in S_{2}(Γ, K′_{f}) of trivial character such that

• a_{p}(g) = a_{p}(f) if p is not in Σ;

• a_{p}(g) = α̃_{p}(f) if p is in Q;

• a_{p}(g) = 0 otherwise.

The reduction ḡ is the unique normalized eigenform in S_{2}(Γ, k′) of trivial character such that

• a_{p}(ḡ) = tr ρ̄_{I_{p}}(Frob_{p}) if p is not in Σ;

• a_{p}(ḡ) = α_{p} if p is in Q;

• a_{p}(ḡ) = 0 otherwise.

Thus ḡ has coefficients in k and is independent of the choice of f in N_{∅}. We let 𝔪 denote the corresponding maximal ideal of 𝕋_{O}.

Suppose for the moment that K contains the coefficients of all eigenforms of level dividing N_{Σ}. Let N′ denote the set of newforms g in N_{Γ} such that

• ρ̄_{g} ∼ ρ̄;

• a_{p}(ḡ) = α_{p} for all p dividing N_{g}/N_{∅}.

Suppose we are given a newform g ∈ N′. Let ψ_{g} denote its character and write Q_{g} for the conductor of ψ_{g}. Note that ψ_{g} has trivial reduction and hence ℓ-power order, and that Q_{g} divides Q. By proposition 3.2 we see that if p ∈ Σ − Q, then ρ_{g} is unramified at p and hence N_{g} is not divisible by p. Furthermore, by theorem 3.1 (e) and (4.2.2), N_{g} = N_{∅}Q_{g}. Let ξ_{g} denote the character of (ℤ/Q_{g}ℤ)^{×} of ℓ-power order such that ξ_{g}^{−2} is the primitive character associated to ψ_{g}. Then

g ⊗ ξ_{g} = ∑ ξ_{g}(n)a_{n}(g)q^{n}

is in N_{Q_{g}} ⊂ N_{Q}.

Lemma 4.9 The map g ↦ g ⊗ ξ_{g} defines a bijection between N′ and N_{Q}.

Proof: Suppose we are given a newform f in N_{Q}; i.e., f is in N_{Γ_{0}(Q)} and ρ̄_{f} ∼ ρ̄. For q ∈ Q, we have (by lemma 2.44 for example) that

ρ_{f}|_{G_{q}} ∼ (\begin{matrix}ξ_{f,q} & 0 \\ 0 & ϵξ_{f,q}^{−1} \end{matrix} ) ,

where ξ_{f,q} : G_{q} → K^{×} is a character whose reduction is the unramified character sending Frob_{q} to α_{q}. Note that the characters ξ_{f,q}|_{I_{q}} have ℓ-power order. In particular N_{f}/N_{∅} is Q_{f}^{2} where Q_{f} is the product of the primes q ∈ Q such that ξ_{f,q} is ramified (theorem 3.1 (d)). Note also that there is a unique character

ξ_{f} : (ℤ/Q_{f}ℤ)^{×} → K^{×}

such that

I_{q} → G_{ℚ} → → Gal (ℚ(ζ_{Q_{f}})/ℚ) →^{ξ_{f}} K^{×}

coincides with the restriction of ξ_{f,q} to I_{q} for each q|Q_{f}. (We have written ξ_{f} for the character of G_{ℚ} as well as the corresponding Dirichlet character. We shall also write ξ_{f} for the corresponding character of Δ_{Q}.) Let g denote the newform associated to the eigenform

∑ ξ_{f}^{−1}(n)a_{n}(f)q^{n} ∈ S_{2}(Γ_{0}(N_{∅}) ∩ Γ_{1}(Q_{f}^{2}), K).

By proposition 2.6 and theorem 3.1 we have

• ρ_{g} ∼ ρ_{f} ⊗ ξ_{f}^{−1},

• ψ_{g} = ξ_{f}^{−2},

• N_{g} = N_{∅}Q_{f} and

• a_{q}(g) = (ξ_{f}|_{G_{q}}^{−1}ξ_{f,q})(Frob_{q}) has reduction ξ̄_{f,q}(Frob_{q}) = α_{q} for all q dividing Q_{f}.

Therefore g is in N′ and f = g ⊗ ξ_{g}. In particular g ↦ g ⊗ ξ_{g} is surjective and ξ_{g} = ξ_{g⊗ξ_{g}}. Injectivity follows as well on noting that if g ⊗ ξ_{g} = g′ ⊗ ξ_{g′} then ξ_{g} = ξ_{g′}, hence g = g′. □

Recall that 𝕋_{O} contains the operators ⟨d⟩ for d in (ℤ/Mℤ)^{×}, and note that ⟨d⟩ − 1 ∈ 𝔪 for all d. Let H′ denote the kernel of

(ℤ/Mℤ)^{×} → → (ℤ/(∏_{q∈Q} q)ℤ)^{×} → → Δ_{Q}.

Since the order of H′ is not divisible by ℓ, we find that ∑_{d∈H′} ⟨d⟩ ∉ 𝔪. It follows that if d ∈ H′ then ⟨d⟩ = 1 in 𝕋_{𝔪}. We may therefore regard 𝕋_{𝔪} as an O[Δ_{Q}]-algebra via the map d ↦ ⟨d⟩. Recall that 𝕋_{Q} is considered an O[Δ_{Q}]-algebra via the map

O[Δ_{Q}] → R_{Q} → 𝕋_{Q}.

If p is a prime not in Q, let x_{p} denote the unique element of Δ_{Q} such that x_{p}^{−2} = p̄.

Proposition 4.10 There is an isomorphism 𝕋_{Q} ⟶^{∼} 𝕋_{𝔪} of O[Δ_{Q}]-algebras such that T_{p} ↦ x_{p}T_{p} for all primes p ∉ Σ with p ∤ ℓN(ρ̄).

Proof: We may enlarge K so that we are in the setting of lemma 4.9. We then define an isomorphism of K-algebras

κ : 𝕋_{𝔪} ⊗ K ⟶^{∼} ∏_{g∈N′} K

such that for each g ∈ N′:

• κ(T_{p})_{g} = a_{p}(g) if p is not in Σ or if p divides Q_{g},

• κ(T_{p})_{g} = 0 if p is in Σ − Q,

• κ(T_{p})_{g} is the root of X^{2} − a_{p}(g) + ψ_{g}(p)p with reduction α_{p} if p is in Q but does not divide Q_{g}, and

• κ(d)_{g} = ψ_{g}(d) for all d ∈ Δ_{Q}.

The existence of such an isomorphism follows from lemma 4.4 which gives

𝕋_{𝔪} ⊗ K ⟶^{∼} ∏_{g∈N_{Γ}} ∏_{𝔭∈M_{g}}(A_{K,g}/I_{K,g})_{𝔭},

where M_{g} is the set of prime ideals in A_{K,g}/I_{K,g} whose preimage in 𝕋_{O} is contained in 𝔪. If g is not in N′, then M_{g} is empty. If g is in N′, then M_{g} consists of the prime ideal corresponding to the eigenform whose eigenvalues are prescribed as above, and one checks that (A_{K,g}/I_{K,g})_{𝔭} = K.

Viewing 𝕋_{Q} as a subalgebra of ∏_{f∈N_{Q}} K and matching indices via the bijec-

tion f = g ⊗ ξ_{g} ↔ g, we obtain an injective homomorphism of O[Δ_{Q}]-algebras

κ′ : 𝕋_{Q} ⟶ ∏_{g∈N′} K

such that

κ′(T_{p}) = (ξ_{g}(p)a_{p}(g))_{g} = κ(x_{p}T_{p})

for primes p ∉ Σ such that p ∤ ℓN_{∅}. Since 𝕋_{Q} is generated over O by the set of such T_{p}, we see that κ′(𝕋_{Q}) is contained in κ(𝕋_{𝔪}). On the other hand, the image of κ′ contains the image of Δ_{Q}, hence contains κ(⟨d⟩) for d in (ℤ/Mℤ)^{×} as well as κ(T_{p}) for p ∉ Σ with p ∤ N_{∅}ℓ.

It remains to prove that κ(T_{p}) is in the image of κ′(𝕋_{Q}) for p dividing N_{∅}ℓ ∏_{q∈Q} q. Consider the composite

R_{Q} ⟶^{ϕ_{Q}} 𝕋_{Q} ⟶^{κ′} ∏_{g∈N′} K.

The pushforward of ρ_{Q}^{univ} is equivalent to ∏ ρ_{g} ⊗ ξ_{g} and therefore ∏ ρ_{g} is equivalent to the pushforward of

ρ_{Q}^{univ} ⊗ ξ_{Q}^{−1}

(where ξ_{Q} was defined in section 2.8). For primes p dividing N_{∅}, we recover κ(T_{p}) as the image of Frob_{p} on the I_{p}-coinvariants, hence κ(T_{p}) is in κ′(𝕋_{Q}). For q ∈ Q and g ∈ N′, the pushforward of ξ_{Q}|_{G_{q}}^{−1} ·ξ_{q,Q} to the g-component is an unramified summand of ρ_{g}|_{G_{q}} whose reduction sends Frob_{q} to α_{q}. It follows that this character maps Frob_{q} to κ(T_{q}) and we conclude that κ(T_{q}) ∈ κ′(𝕋_{Q}). Finally in the case that ℓ does not divide N_{∅}, we appeal to lemma 4.1 to conclude that κ(T_{ℓ}) ∈ κ′(𝕋_{Q}). □

Auxiliary primes: For Σ as in proposition 4.10, i.e. a set of primes satisfying (4.2.2), a somewhat simpler argument provides a similar a description of 𝕋_{∅} with Γ replaced by

Γ′ = Γ_{0}(N_{∅}) ∩ Γ_{H}(M)  (4.2.5)

where H is the ℓ-Sylow subgroup of (ℤ/Mℤ)^{×}. While we shall make no direct use of this, the group Γ′ will play a role in the proof of theorem 3.31 and we shall need to choose Σ so that Γ′ has no elliptic elements; i.e., non-trivial elements of finite order. This is the case for example if Σ contains a prime p ≢ 1 mod ℓ with p > 3. The group Γ′ is then contained in Γ_{1}(N) for some integer N > 3, and therefore has no elliptic elements. Indeed if (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ has finite order then the roots of X^{2} − (a + d)X + 1 are roots of unity and we deduce that this matrix is the identity.

In order to show that Σ can be so chosen, we appeal to the following lemma (cf. [DT2], lemma 3).

Lemma 4.11 Suppose that G is a finite group, χ : G → k̄^{×} is a character of order d and ρ̄: G → GL_{2}(k̄) is a representation. Suppose that for all g ∉ ker χ we have

(tr ρ̄(g))^{2}/ det ρ̄(g) = (1 + χ(g))^{2}/χ(g).  (4.2.6)

• If d > 3, then ρ̄ is reducible.

• If d = 3, then ρ̄ is reducible or has projective image isomorphic to A_{4}.

• If d = 2, then ρ̄|_{ker χ} is reducible.

Proof: Note that if ρ̄(g) is a scalar then χ(G) = 1. Hence χ induces induces a surjective homomorphism χ′ : G′ → C_{d} where G′ is the projective image of ρ̄ and C_{d} is cyclic of order d. Furthermore if d = 2, then every element of G′ − ker χ′ has order 2. The lemma then follows from theorem 2.47(b). □

4.3 Proof of theorem 3.31

We shall give a proof of theorem 3.31 which is based on the q-expansion principle rather than the method of de Shalit [dS] employed in [TW]. It will be more convenient to consider the action of the Hecke operators on the full space of modular forms M_{2}(Γ). The Riemann-Roch theorem shows that the dimension of M_{2}(Γ) is g + s − 1 where g is the genus of the modular curve X_{Γ} associated to Γ and s is the number of cusps on X_{Γ} (see for example [Shi2] thm. 2.23 or [DI] (12.1.5)).

Eisenstein maximal ideals: One can give an explicit description of a space of Eisenstein series G_{2}(Γ) so that

M_{2}(Γ) = S_{2}(Γ) ⊕ G_{2}(Γ).

(See for example [Hi3], lemma 5.2.)

There is a natural action on M_{2}(Γ) by the Hecke operators ⟨d⟩ for all d ∈ (ℤ/Nℤ)^{×} and T_{p} for all primes p. We shall only need to use the operators ⟨d⟩ and T_{p} for p not dividing N, and we let 𝕋̃_{ℤ} denote the subring of End (M_{2}(Γ)) these generate. The ring 𝕋̃_{ℤ} is commutative and is finitely generated and free as a ℤ-module. That 𝕋̃_{ℤ} is finitely generated is proved for example by showing that M_{2}(Γ, ℤ), the set of forms with integer Fourier coefficients at ∞, is stable under the Hecke operators and contains a basis for M_{2}(Γ) (see [DI], cor. 12.3.12 and prop. 12.4.1). Alternatively, one can show that 𝕋̃_{ℤ} acts faithfully on the cohomology of the non-compact modular curve Y_{Γ} (cf. section 1.3).

For any ring A, we write M_{2}(Γ, A) for M_{2}(Γ, ℤ) ⊗ A and regard this as a module for 𝕋̃_{A} := 𝕋̃_{ℤ} ⊗ A. If 𝔫 is a maximal ideal of 𝕋̃_{O} we write 𝕋̃_{𝔫} for the localization. If 𝔪 is a maximal ideal of 𝕋_{O} then we let 𝔪̃ denote its preimage in 𝕋̃_{O}.

We say that a maximal ideal 𝔫 of 𝕋̃_{O} is Eisenstein if

T_{p} ≡ p + 1 mod 𝔫 for all p ≡ 1 mod N.

One sees from the explicit description of the Eisenstein series that T_{p} = p+1 on G_{2}(Γ) if p ≡ 1 mod N. We shall be interested in the non-Eisenstein maximal ideals because of the following lemma (see [R5], thm. 5.2(c)).

Lemma 4.12 Suppose that ℓ is odd. The representation

ρ_{𝔪} : G_{ℚ} → GL_{2}(𝕋_{O}/𝔪)

is absolutely irreducible if and only if 𝔪̃ is not Eisenstein.

Proof: If ρ_{𝔪} is not absolutely irreducible then ρ_{𝔪} ∼ χ_{1} ⊕χ_{2} with N divisible by the product of the conductors of χ_{1} and χ_{2}. Furthermore if ℓ does not divide N, then one of the characters is unramified at ℓ while the other coincides with the cyclotomic character ϵ on I_{ℓ}. Therefore 𝔪̃ is Eisenstein. Conversely if 𝔪̃ is Eisenstein then proposition 2.6 implies that ρ_{𝔪} restricted to G_{ℚ(ζ_{Nℓ})} has trivial semisimplification from which it follows that ρ_{𝔪} is also reducible. □

Differentials: Suppose now that Σ is a finite set of primes satisfying (4.2.2) Moreover we assume that Σ contains a prime p > 3 with p ≢ 1 mod ℓ. Let Γ and Γ′ be defined as in (4.2.4) and (4.2.5); i.e.

\begin{aligned}Γ & = & Γ_{0}(N_{∅}) ∩ Γ_{1}(M) \\ Γ′ & = & Γ_{0}(N_{∅}) ∩ Γ_{H}(M) \end{aligned}

where H is the ℓ-Sylow subgroup of (ℤ/Mℤ)^{×}. Let X and X′ denote the modular curves associated to Γ and Γ′.

We first consider the case where ℓ ∤ N_{∅}. The curve X has a smooth proper model X over ℤ[1/N_{∅}M] such that the complement of the cusps parametrizes cyclic isogenies (E_{1}, i_{1}) → (E_{2}, i_{2}) of degree N_{∅} where E_{j} is an elliptic curve and i_{j} is an embedding μ_{M} ↪ E_{j}. (See [DR] or [Kat] for example.) The action of H on X extends to X and the quotient X′ = X /H is a smooth model over ℤ[1/N_{∅}M] for X′. The natural projection X → X′ is étale. (The fact that it is étale on the complement of the cusps follows from the natural moduli-theoretic description of X′ using the fact that Γ_{H}(M) ⊂ Γ_{1}(p) for some p > 3. One then need only verify that X → X′ is unramified at the cusps.)

For a ℤ[1/N_{∅}M]-algebra A, we define • Ω_{A} = H^{0}(X_{A}, Ω_{X_{A}/A}^{1}),

• Ω′_{A} = H^{0}(X′_{A}, Ω_{X′_{A}/A}^{1}),

• Ω̃_{A} = H^{0}(X_{A}, Ω_{X_{A}/A}^{1}(D)),

• Ω̃′_{A} = H^{0}(X′_{A}, Ω_{X′_{A}/A}^{1}(D′)),

where D (resp. D′) denotes the reduced divisor defined by the cusps of X (resp. X′). The q-expansion principle and standard base-change arguments allow us to identify these with S_{2}(Γ, A), S_{2}(Γ′, A), M_{2}(Γ, A) and M_{2}(Γ′, A) (see [Kat], sec. 1.6, 1.7). We may therefore regard the first two of these as modules for 𝕋_{A} and all of them as modules for 𝕋̃_{A}.

Lemma 4.13 Suppose that A = k or K.

(a) Ω̃_{O} ⊗_{O} A ≅ Ω̃_{A} and Ω̃′_{O} ⊗_{O} A ≅ Ω̃′_{A}.

(b) The natural map X → X′ induces an isomorphism

Ω̃′_{A} ⟶^{∼} Ω̃_{A}^{H}.

(c) If 𝔫 is a non-Eisenstein maximal ideal of 𝕋̃_{O}, then the localization at 𝔫 of Ω_{O} → Ω̃_{O} is an isomorphism.

(d) If 𝔪 is a maximal ideal of 𝕋_{A} then Ω_{A}[𝔪] is one-dimensional over 𝕋_{A}/𝔪.

Sketch of proof:

(a) In the case A = K this follows from the fact that K is flat over O, so suppose A = k. Identify Ω̃_{O}/λΩ̃_{O} with the direct image of Ω̃_{k} under X_{k} → X_{O} and use that H^{1}(X_{k}, Ω_{X_{k}/k}^{1}(D)) vanishes by Serre duality. The case of Ω̃′ is similar. (See [Maz1] sec. II.3.)

(b) Using the fact that X → X′ is étale one identifies the pull-back of Ω_{X′_{A}/A}^{1} with Ω_{X_{A}/A}^{1} and that of D′ with D.

(c) The map is injective and one proves that its cokernel is free and annihilated by the operators T_{p} − (p + 1) for all p ≡ 1 mod N_{∅}M. To prove the latter assertion, first observe that it holds with O replaced by ℂ, then by ℤ[1/N_{∅}M].

(d) This follows from

Ω_{A}[𝔪] ≅ S_{2}(Γ, A)[𝔪] ≅ Hom_{A}(𝕋_{A}, A)[𝔪] ≅ Hom_{A}(𝕋_{A}/𝔪, A),

where the middle isomorphism is that of proposition 1.34, but let us reformulate the argument in a way more easily generalized to the case of ℓ dividing N_{∅} discussed below. Note that Ω_{A}[𝔪] is non-zero since 𝕋_{𝔪} acts faithfully on Ω_{𝔪}. To prove that the dimension is at most one we can enlarge the field A and assume 𝕋_{A}/𝔪 ≅ A. One then shows that an eigenform f in Ω_{A} (for all the Hecke operators T_{p} and ⟨d⟩) is determined by its eigenvalues and the first coefficient of its q-expansion. This follows from the fact that f is determined by its q-expansion and for all n, a_{n}(f) = a_{1}(T_{n}f) and T_{n} can be expressed in terms of the T_{p} and ⟨d⟩. (See [Maz1], sec. II.9.) □

We now explain how the situation changes if ℓ divides N_{∅}. In that case X has a regular model X over ℤ[ℓ/N_{∅}M] with the same moduli-theoretic description as above. This model is smooth over ℤ[1/N_{∅}M], but X_{𝔽_{ℓ}} has two smooth irreducible components crossing at ordinary double points as in [DR] sec. V.1. The quotient X′ = X /H is a regular model for X′ over ℤ[ℓ/N_{∅}M] and X → X′ is étale. For a ℤ[ℓ/N_{∅}M]-algebra A we define Ω_{A}, Ω′_{A}, Ω̃_{A} and Ω̃′_{A} as before, except that Ω^{1} is replaced by the sheaf of regular differentials (see [DR] sec. I.2, [Maz1] sec. II.6 or [MRi] sec. 7). Formation of these modules again commutes with change of the base A, but we can no longer identify them with S_{2}(Γ, A), S_{2}(Γ′, A), M_{2}(Γ, A) and M_{2}(Γ′, A) if ℓ is not invertible in A. For every A, there is a natural action of 𝕋_{A} (resp. 𝕋̃_{A}) on Ω_{A} and Ω′_{A} (resp. Ω̃_{A} and Ω̃′_{A}). In the case of 𝕋_{A} this is proved by identifying Ω_{A} (resp. Ω′_{A}) with the cotangent space at the origin for the Néron model over A for the Jacobian of X (resp. X′). In the case of 𝕋̃_{A} the action is defined using Grothendieck-Serre duality (as in [Maz1] sec. II.6 or [MRi] sec. 7).

Lemma 4.14 If ℓ divides N_{∅} then lemma 4.13 carries over with the above notation and the additional hypothesis T_{ℓ} ∉ 𝔪 for part (d).

The proof is essentially the same except that part (d) is more delicate in the case A = k. For the proof in that case we refer the reader to [W3], lemma 2.2 (from which the result stated here is immediate). We remark however that we shall only use lemma 4.14 if ρ_{𝔪} is not good. In that case one can also deduce the statement here from the argument used in the proof of [MRi] prop. 20, which shows instead that dim_{k} Ω_{k}[𝔪′] = 1 where 𝔪′ is a certain maximal ideal corresponding to 𝔪, but contained in an algebra defined using the operators ⟨d⟩, T_{p} for p ≠ ℓ and w_{ℓ}.

Remark 4.15 The approach in [TW] to proving theorem 3.31 is based on the result that under certain hypotheses (T_{ℓ}(J) ⊗_{ℤ_{ℓ}} O)_{𝔪} is free of rank two over 𝕋_{𝔪} (see [W3], thm. 2.1 and its corollaries). This result generalizes work of Mazur Ribet, and Edixhoven (see secs. 14 and 15 of ch. II of [Maz1], thm. 5.2 of [R5], [MRi] and sec. 9 of [Edi]), and the key to its proof is lemmas 4.13(d) and 4.14. We shall instead give a proof of theorem 3.31 which is based directly on lemma 4.13, (d).

The fact that (T_{ℓ}(J) ⊗_{ℤ_{ℓ}} O)_{𝔪} is free of rank two over 𝕋_{𝔪} actually underlies Wiles’ approach to many intermediate results along the way to proving the Shimura-Taniyama conjecture for semistable elliptic curves. We shall appeal to a special case in the course of proving theorem 3.36 below.

Proof of the theorem: Suppose we are given a representation ρ̄ as in section 3.3 and a set of primes Q as in the statement of theorem 3.31. We apply lemma 4.11 with G = G_{ℚ} and χ = ϵ to choose an auxiliary prime p not dividing 6N_{∅}ℓ ∏_{q∈Q} q such that

• p ≢ 1 mod ℓ

• tr (ρ̄(Frob_{p}))^{2} ≠ (p + 1)^{2}.

To prove theorem 3.31 we may replace K by a larger field and assume that k contains the eigenvalues of ρ̄(Frob_{p}).

Let Σ = Q ∪ {p}. Thus Σ is a set of primes as in proposition 4.10 and Σ contains a prime p > 3 such that p ≢ 1 mod ℓ. We let Γ be as in (4.2.4); thus

Γ = Γ_{0}(N_{∅}) ∩ Γ_{1}(M)

where M = p^{2} ∏_{q∈Q} q, and we choose the maximal ideal 𝔪 of 𝕋_{O} as in proposition 4.10. Let H denote the ℓ-Sylow subgroup of (ℤ/Mℤ)^{×} and regard 𝕋_{O} and hence 𝕋_{𝔪} as an O[H]-algebra via d ↦ ⟨d⟩. In view of proposition 4.10, theorem 3.31 is equivalent to the following:

Theorem 4.16 𝕋_{𝔪} is free over O[H].

Proof: Consider the 𝕋̃_{O}-module

L̃ = Hom_{O}(Ω̃_{O}, O)

and the 𝕋_{O}-module

L = Hom_{O}(Ω_{O}, O).

Since X → X′ is unramified, the Riemann-Hurwitz formula implies that

dim M_{2}(Γ) = g + s − 1 = #H(g′ + s′ − 1) = #H dim M_{2}(Γ′),

where g (resp. g′) is the genus and s (resp. s′) is the number of cusps of X (resp. X′). It follows from lemma 4.13 (a) (and lemma 4.14) that

dim_{K}(L̃ ⊗_{O} K) = #H dim_{k} Ω̃′_{k}  (4.3.1)

On the other hand, by (b) we have that

L̃/(𝔞, λ)L̃ ≅ Hom_{k}(Ω̃_{k}^{H}, k) ≅ Hom_{k}(Ω̃′_{k}, k)  (4.3.2)

where 𝔞 = ker (O[H] → O) is the augmentation ideal. By Nakayama’s lemma and (4.3.2), L̃ is generated as an O[H]-module by d elements where d = dim_{k} Ω̃′_{k}. By (4.3.1) any surjective homomorphism

O[H]^{d} → L̃

is in fact an isomorphism. Hence L̃ is free over O[H], as is

L_{𝔫} = Hom_{O}(Ω_{𝔫}, O) ≅ Hom_{O}(Ω̃_{𝔫}, O)

for each non-Eisenstein maximal ideal 𝔫 of 𝕋̃_{O} (by part (c) of lemmas 4.13 and 4.14).

Since ρ_{𝔪} ∼ ρ̄ is absolutely irreducible, 𝔪̃ is not Eisenstein (by lemma 4.12), and it follows that L_{𝔪} is free over O[H].

Since L ⊗_{O} K is free of rank one over 𝕋_{K} (lemma 1.34), we have that L_{𝔪} ⊗_{O} K is free of rank one over 𝕋_{𝔪} ⊗_{O} K. If ℓ ∤ N_{∅} then lemma 4.13 implies that

L/𝔪L ≅ Hom_{k}(Ω_{k}[𝔪], k)

is one-dimensional over k. The same assertion holds if ℓ|N_{∅} by lemma 4.14 (from the definition of 𝔪 in this case, we see that it does not contain T_{ℓ}). It follows from Nakayama’s lemma that L_{𝔪} is free of rank one over 𝕋_{𝔪}. Therefore 𝕋_{𝔪} is free over O[H]. □

4.4 Proof of theorem 3.36

Our proof of theorem 3.36 is based on Wiles’ arguments in ch. 2 of [W3], which in turn are based on a method of Ribet [R4]. We shall reformulate the proof somewhat to underscore the relationship observed by Doi and Hida [Hi1] between the size of O/η and the value of an L-function, but after doing so we shall also sketch the more direct argument used by Wiles. Two important ingredients appear in both versions of the argument, and we shall discuss their proofs in section 4.5. These ingredients are • the generalization of a result of Ihara [Ih] used in Ribet’s argument;

• the generalization of a result of Mazur [Maz1] on the structure of the Tate module T_{ℓ}(J_{0}(N)) as a module for 𝕋_{ℤ_{ℓ}}.

Assume now that we are in the setting of theorem 3.36. In particular, ρ̄ is a representation as in section 3.3, Σ is a finite set of primes contained in Σρ̄ and π is an O-algebra homomorphism 𝕋_{Σ} → O arising from a newform f in N_{Σ} with coefficients in O.

The symmetric square L-function: For each prime p let α_{p}(f) and β_{p}(f) denote the roots of the polynomial X^{2} − a_{p}(f)X + δ_{p}p = 0, where δ_{p} = 0 or 1 according to whether p divides N_{f}. Thus we have

L_{p}(f, s) = (1 − α_{p}(f)p^{−s})^{−1}(1 − β_{p}(f)p^{−s})^{−1}.

If p = ℓ divides N/N_{f}, then we require that α_{ℓ}(f) be the root which is a unit in O, i.e. a_{ℓ}(g). We define the symmetric square L-function associated to f as

L(Symm^{2}f, s) = ∏_{p} L_{p}(Symm^{2}f, s),

where

L_{p}(Symm^{2}f, s) = (1 − α_{p}^{2}(f)p^{−s})^{−1}(1 − α_{p}(f)β_{p}(f)p^{−s})^{−1}(1 − β_{p}^{2}(f)p^{−s})^{−1}.

(We caution the reader we have defined the Euler factors rather naively at primes p such that p^{2} divides N_{f}.) The product converges absolutely for real part of s > 2 and can be analytically continued to an entire function.

The calculation of η_{Σ}: We write ℘ for the kernel of π and I for the annihilator of ℘ in 𝕋_{Σ}. Since 𝕋_{Σ} is reduced, we have ℘ ∩ I = 0 and ℘ ⊕ I has finite index in 𝕋_{Σ}. Note that

O/η_{Σ} ≅ 𝕋_{Σ}/(℘ ⊕ I).

Suppose that we are given a 𝕋_{Σ}-module L which is finitely generated and free over O, and such that L ⊗ K is free of rank d over 𝕋_{Σ} ⊗_{O} K. Suppose also that L is endowed with a perfect O-bilinear pairing

\begin{matrix}L × L & → & O \\ (x, y) & ↦ & ⟨x, y⟩ \end{matrix}

such that ⟨T x, y⟩ = ⟨x, T y⟩ for all x, y ∈ L and T ∈ 𝕋_{Σ}. Note that to give such a pairing is equivalent to giving an isomorphism

\begin{matrix}L & ⟶^{∼} & Hom_{O}(L, O) \\ x & ↦ & ⟨x, ·⟩ \end{matrix}

of 𝕋_{Σ}-modules. We shall refer to the module L together with the pairing ⟨ , ⟩ as a self-dual 𝕋_{Σ}-module of rank d. We then can give a lower bound for the size of O/η_{Σ} in terms of a basis {x_{1}, x_{2}, …, x_{d}} for the free O-module L[℘]:

Lemma 4.17 We have

η_{Σ}^{d} ⊂ O det(⟨x_{i}, x_{j}⟩)_{i,j},

and equality holds if L is free over 𝕋_{Σ}.

Proof: The modules L[℘] and L/L[I] are free of rank d over O and the pairing ⟨ , ⟩ induces an isomorphism

L/L[I] ⟶^{∼} Hom_{O}(L[℘], O).  (4.4.1)

The O-module M = L/(L[℘] ⊕ L[I]) is annihilated by η_{Σ} and is generated by d elements. Furthermore M is isomorphic to (O/η_{Σ})^{d} if L is free over 𝕋_{Σ}. It follows that

#(O/η_{Σ})^{d} ≥ #M,

with equality if L is free. The cardinality of M is that of the cokernel of the map

L[℘] ↪ Hom_{O}(L[℘], O)

arising from the pairing ⟨ , ⟩, and this is precisely

O/ det(⟨x_{i}, x_{j}⟩)_{i,j}.
□

Recall that proposition 4.7 establishes an isomorphism between 𝕋_{Σ} and 𝕋_{𝔪}, the localization at a certain maximal ideal 𝔪 of 𝕋_{O} = 𝕋_{ℤ} ⊗ O, where

𝕋_{ℤ} ⊂ End (S_{2}(Γ_{0}(N))),

and where N = N_{Σ} is defined in (4.2.1). Write X for X_{0}(N) and J for J_{0}(N). Recall that H_{1}(X, ℤ) is endowed with the structure of a 𝕋_{ℤ}-module and

T_{ℓ}(J) ⊗_{ℤ_{ℓ}} O ≅ H_{1}(X, ℤ) ⊗ O

with that of a 𝕋_{O}-module. We also regard

H^{1}(X, O) ≅ Hom(H_{1}(X, ℤ), O)

as a 𝕋_{O}-module; as such it is naturally isomorphic to

Hom_{O}(T_{ℓ}(J) ⊗_{ℤ_{ℓ}} O, O).

The Weil pairing defines a perfect pairing

T_{ℓ}(J) × T_{ℓ}(J) → ℤ_{ℓ}

(with (e^{2πi/ℓ^{n}})_{n} as the chosen generator for lim_{←} μ_{ℓ^{n}}(ℂ)). The composite

T_{ℓ}(J) →^{w_{∗}} T_{ℓ}(J) → Hom_{ℤ_{ℓ}}(T_{ℓ}(J), ℤ_{ℓ})  (4.4.2)

is an isomorphism of 𝕋_{ℤ_{ℓ}}-modules where w_{∗} is induced by the involution w = w_{N} of X (see section 1.4). Tensoring with O and localizing at 𝔪, we regard

L_{T} = (T_{ℓ}(J) ⊗_{ℤ_{ℓ}} O)_{𝔪}

as a self-dual 𝕋_{Σ}-module of rank 2 via the isomorphism 𝕋_{Σ} ≅ 𝕋_{𝔪} of proposition 4.7. We shall write ⟨, ⟩_{T} for the pairing obtained from (4.4.2). Similarly, using w^{∗}, the cup product and Poincaré duality, we regard

L_{H} = H^{1}(X, O)_{𝔪}

as a self-dual 𝕋_{Σ}-module of rank 2, with ⟨x, y⟩_{H} defined by the image of x ∪ w^{∗}y under the canonical isomorphism of H^{2}(X, O) with O. Note that L_{H} is canonically isomorphic as a 𝕋_{Σ}-module to Hom_{O}(L_{T} , O). In the next section we shall discuss the following generalization of a result of Mazur [Maz1].

Theorem 4.18 The 𝕋_{Σ}-modules L_{T} and L_{H} are free.

Corollary 4.19 If {x, y} is a basis for L_{T} [℘] (resp. L_{H}[℘]), then η_{Σ} is generated by ⟨x, y⟩_{T} (resp. ⟨x, y⟩_{H}).

This follows from lemma 4.17, theorem 4.18 and skew-symmetry of the pairings.

Hida’s formula: We now explain how the value of ⟨x, y⟩_{H} in corollary 4.19 is related to L(Symm^{2}f, 2) by a formula of Hida (see [Hi1] sec. 5, [W3] sec. 4.1).

Recall that the homomorphism

𝕋_{O} → 𝕋_{𝔪} ≅ 𝕋_{Σ} →^{π} O

arises as T_{n} ↦ a_{n}(g) for a normalized eigenform g in S_{2}(Γ_{0}(N), K) whose associated newform is f. We shall write P for the kernel of this homomorphism; thus P is the preimage in 𝕋_{O} of the ideal ℘ of 𝕋_{Σ}. Note that P_{𝔪} corresponds to ℘ under the isomorphism 𝕋_{𝔪} ≅ 𝕋_{Σ}.

Choose a number field K_{0} containing the Fourier coefficients of g and let O_{0} denote the valuation ring O ∩ K_{0}. We choose the basis {x, y} for L_{H}[℘] in the image of

H^{1}(X, O_{0})[P_{0}] ⊂ H^{1}(X, O)[P] ⟶^{∼} H^{1}(X, O)_{𝔪}[P] = L_{H}[℘],

where P_{0} = P ∩ 𝕋_{O_{0}}. Let P_{ℂ} denote the kernel of the homomorphism

θ_{g} : 𝕋_{ℂ} → ℂ

associated to the eigenform g. The two-dimensional complex vector space

(H^{1}(X, O_{0})[P_{0}]) ⊗_{O_{0}} ℂ = H^{1}(X, ℂ)[P_{ℂ}]

has a canonical basis

{ω_{g}, ω̄_{g^{c}}},

where we view these differential forms as cohomology classes and the superscript c indicates complex conjugation applied to the Fourier coefficients. (Recall that ω_{g} is the holomorphic differential on X defined by ∑ a_{n}(g)q^{n−1}dq, so ω̄_{g^{c}} is the antiholomorphic differential defined by ∑ a_{n}(g)q̄^{n−1}dq̄ where q = e^{2πiτ}.) Let A denote the matrix in GL_{2}(ℂ) such that

(ω_{g}, ω̄_{g^{c}}) = (x, y)A.

Theorem 4.20 With the above notation we have (up to sign)

⟨x, y⟩ = (iπ det A)^{−1}N_{f}L(Symm^{2}f, 2) ∏_{p|N/N_{f}} c_{p}(f)),

where

c_{p}(f) = \begin{cases}p^{δ_{p}}L_{p}(Symm^{2}f, 2)^{−1} & if p ≠ ℓ \\ (α_{ℓ}(f) − β_{ℓ}(f))(1 − α_{ℓ}^{−2}(f)) & if p = ℓ. \end{cases}

Proof: We have

⟨x, y⟩_{H} det A = ∫_{X} ω_{g} ∧ ω̄_{wg^{c}} = 8π^{2}i⟨g, (wg)^{c}⟩

where the last ⟨, ⟩ denotes the Petersson inner product and we have used that wg^{c} = (wg)^{c} for forms on Γ_{0}(N). We then appeal to a formula of Shimura (see [Shi5] (2.5) and [Hi1] (5.13)) to obtain

⟨g, (wg)^{c}⟩ = (48π)^{−1}[SL_{2}(ℤ) : Γ_{0}(N)]res_{s=2}D(g, wg, s).

By [Shi5] lemma 1, the Dirichlet series D(g, h, s) is defined by ∑ a_{n}(g)a_{n}(h)n^{−s} and if g and h are normalized eigenforms then this has an Euler product expression in which the factors are

\begin{matrix}(1 − α_{p}(g)β_{p}(g)α_{p}(h)β_{p}(h)p^{−2s}) & (1 − α_{p}(g)α_{p}(h)p^{−s})^{−1}· \\ (1 − α_{p}(g)β_{p}(h)p^{−s})^{−1} & (1 − β_{p}(g)α_{p}(h)p^{−s})^{−1}(1 − β_{p}(g)β_{p}(h)p^{−s})^{−1}, \end{matrix}

(where the α_{p}’s and β_{p}’s are defined as they were for f). Using the recipe for obtaining g from f (see lemma 4.6) we find that if ℓ is not in Σ then

D(g, wg, s) = ±D_{Σ}(f, f, s) ∏_{p|N/N_{f}} p^{−1},

(where the subscript Σ indicates that the primes in Σ are removed from the defining Euler product). If ℓ is in Σ, then we must also multiply by the factor

• (1 − ℓ^{−s})^{−1} if ℓ divides N_{f};

• (1 − ℓ^{1−s})^{−1}(1 − α_{ℓ}^{2}(f)ℓ^{−s})^{−1}(β_{ℓ}(f) − α_{ℓ}^{2−s}(f)) otherwise.

To obtain the formula in this last case, consider the eigenform

f_{α} = f − β_{ℓ}(f)f(ℓτ)

on Γ_{0}(N_{f}ℓ). Since

w_{N_{f} ℓ}f_{α} = ±(ℓf(ℓτ) − α_{ℓ}^{−1}(f)f)

and

D(f_{α}, ℓf(ℓτ), s) = α_{ℓ}(f)ℓ^{1−s}D(f_{α}, f, s),

we get

D(f_{α}, w_{N_{f} ℓ}f_{α}, s) = ±ℓ^{−1}(β_{ℓ}(f) − α_{ℓ}(f)ℓ^{2−s})D(f_{α}, f, s)

and use the Euler product. The Euler product expression also gives

D_{Σ}(f, f, s) = \frac{ζ_{N}(s − 1)L_{Σ}(Symm^{2}f, s)}{ζ_{N}(2s − 2)},

where ζ_{N}(s) is the Riemann zeta function with the Euler factors removed at primes dividing N. Thus

res_{s=2}D(g, wg, s) = ±\frac{6}{π^{2}} \frac{NL_{Σ}(Symm^{2}f, 2) ∏_{p|N/N_{f}} p}{[SL_{2}(ℤ) : Γ_{0}(N)]},

with the extra factor of (1 − ℓ^{−2})^{−1} or

(β_{ℓ}(f) − α_{ℓ}(f))(1 − β_{ℓ}^{−2}(f))^{−1}(1 − ℓ^{−1})^{−1}

if ℓ is in Σ. The theorem follows. □

Corollary 4.21 With the above notation (in particular, Σ ⊂ Σρ̄), we have

\frac{ℓ^{m}N_{f}L_{Σ}(Symm^{2}f, 2)}{iπ det A} ∈ O_{0}

is a generator of η_{Σ}, where

m = \begin{cases}0, & if ℓ ∉ Σ, \\ 2, & if ℓ ∈ Σ and ℓ|N_{f}, \\ 3, & if ℓ ∈ Σ and ℓ ∤ N_{f}. \end{cases}

Remark 4.22 Without theorem 4.18 we obtain instead that η_{Σ} is contained in the ideal generated by the expression in the corollary.

Comparing η_{Σ}’s: Suppose that Σ′ ⊂ Σρ̄ is a finite set of primes containing Σ. In view of corollary 4.21, theorem 3.36 reduces to a comparison of det A and det A′ where A′ is defined using Σ′ instead of Σ. We shall make the desired comparison using Wiles’ generalization of a result of Ihara.

Applying proposition 4.7 to Σ′, we have an isomorphism between 𝕋_{Σ′} and 𝕋′_{𝔪′} where 𝔪′ is a certain maximal ideal of 𝕋′_{O} = 𝕋′_{ℤ} ⊗ O with 𝕋′_{ℤ} acting on S_{2}(Γ_{0}(N_{Σ′})). We write P′ for the preimage in 𝕋′_{O} of ℘′. We let N′ = N_{Σ′}, X′ = X_{0}(N′), J′ = J_{0}(N′) and define self-dual 𝕋′_{𝔪′}-modules L′_{T} and L′_{H} as above.

We now define a homomorphism

ϕ_{T} : L′_{T} → L_{T}

of 𝕋′_{𝔪′}-modules in the case that Σ′ = Σ∪{p} for some prime p ∈ Σρ̄−Σ. We first suppose that p ≠ ℓ, in which case N′ = Np^{2} and the surjective homomorphism of O-algebras 𝕋′_{𝔪′} → 𝕋_{𝔪} (corresponding to 𝕋_{Σ′} ↦ 𝕋_{Σ}) is defined by T_{r} ↦ T_{r} if r ≠ p and T_{p} ↦ 0. The maps τ ↦ τ, τ ↦ pτ and τ ↦ p^{2}τ on the upper half-plane induce maps X′ → X which we denote α, β and γ. These give rise by functoriality to a map

\begin{matrix}J′ & → & J × J × J \\ x & ↦ & (α_{∗}x, β_{∗}x, γ_{∗}x) \end{matrix}

which induces an O-linear homomorphism

T_{ℓ}(J′) ⊗_{ℤ_{ℓ}} O → (T_{ℓ}(J) ⊗_{ℤ_{ℓ}} O)^{3}.

We compose with the projection to

L_{T}^{3} = (T_{ℓ}(J) ⊗_{ℤ_{ℓ}} O)_{𝔪}^{3}

followed by the map L_{T}^{3} → L_{T} defined by the matrix (1, −p^{−1}T_{p}, p^{−1}). The reason for this choice of matrix is that

g′ = (α^{∗} − p^{−1}β^{∗}T_{p} + p^{−1}γ^{∗})g  (4.4.3)

is the eigenform corresponding to P′.

Lemma 4.23 The composite

T_{ℓ}(J′) ⊗_{ℤ_{ℓ}} O → L_{T}  (4.4.4)

is 𝕋′_{O}-linear where L_{T} is regarded as a 𝕋′_{O}-module via the surjection

𝕋′_{O} → 𝕋′_{𝔪′} → 𝕋_{𝔪}.

We leave the proof to the reader after pointing out that the map on Hecke algebras can be rewritten as

\begin{matrix}𝕋′_{O} & → & 𝕋_{O}[u]/(f(u)) & → & 𝕋_{𝔪}[u]/(f(u)) & → & 𝕋_{𝔪} \\ T_{p} & ↦ & u & & u & ↦ & 0, \end{matrix}

where f(u) = u^{3} − T_{p}u^{2} + pu.

It follows that (4.4.4) factors through L′_{T} and we thus obtain the desired homomorphism

ϕ_{T} : L′_{T} → L_{T}

of 𝕋′_{𝔪′}-modules. Note that ϕ_{T} is also the composite of the canonical splitting

L′_{T} ↪ T_{ℓ}(J′) ⊗_{ℤ_{ℓ}} O

with (4.4.4).

The construction of ϕ_{T} is similar for p = ℓ except that we have only two copies of J and the map L_{T}^{2} → L_{T} is given by the matrix (1, −α_{ℓ}^{−1}) where α_{ℓ} is the unit root of (3.3.2). For arbitrary Σ′ satisfying Σ ⊂ Σ′ ⊂ Σρ̄ we define ϕ_{T} as a composite of the maps defined above. It is independent of the choice of ordering of Σ′ − Σ. Recall that ϕ′_{T} is used to denote the dual map L_{T} → L′_{T}. We define ϕ_{H} : L′_{H} → L_{H} as the adjoint of the map ϕ′_{H} which renders the diagram

ϕ′_{H} : H^{1}(X, O)_{𝔪} → H^{1}(X′, O)_{𝔪′}
↓  ↓
Hom_{O}(L_{T} , O) → Hom_{O}(L′_{T} , O)

commutative (where the vertical maps are the natural isomorphisms and the bottom one is dual to ϕ_{T}).

The second crucial result whose discussion we postpone until the next section is the following generalization of a lemma of Ihara.

Lemma 4.24 ϕ_{T} and ϕ_{H} are surjective.

Suppose again we are in the case Σ′ = Σ ∪ {p} with p ≠ ℓ. One need only unravel the definition of ϕ′_{H} to see that the diagram

H^{1}(X, O_{0})[P_{0}] → H^{1}(X′, O_{0})[P′_{0}]
↓  ↓
L_{H}  →^{ϕ′_{H}}  L′_{H},

commutes, where the top arrow is defined as

α^{∗} − p^{−1}π(T_{p})β^{∗} + p^{−1}γ^{∗}.

Extending scalars to ℂ, this map sends ω_{g} to ω_{g′} and ω̄_{g^{c}} to ω̄_{g^{′c}} where g′ is defined by (4.4.3). In the case Σ′ = Σ ∪ {ℓ}, the same assertion holds if O_{0} is chosen so that it contains π(α_{ℓ}) = a_{ℓ}(g′) and the top arrow is defined as

α^{∗} − π(α_{ℓ})^{−1}β^{∗}.

We conclude that if {x, y} is a basis for H^{1}(X, O_{0})[P_{0}] and the matrices A and A′ are defined using the bases {x, y} for H^{1}(X, O_{0})[P_{0}] and {ϕ′_{H}x, ϕ′_{H}y} for H^{1}(X′, O_{0})[P′_{0}], then A′ = A. Theorem 3.36 is now immediate from corollary 4.21. In fact, we have proved that

η_{Σ′} = π( ∏_{p∈Σ′−Σ} c_{p})η_{Σ}.

□

Remark 4.25 Note that to obtain the inclusion stated in theorem 3.36 it suffices to apply theorem 4.18 for Σ, rather than both Σ and Σ′ (cf. remark 4.22).

Wiles’ argument: The method of [W3] sec. 2.2, like that of Ribet in [R4], is more direct than the one given above but does not explicitly illustrate the relation with values of L-functions. The approach is simply to compute the composite ϕ_{T} ϕ′_{T}. It suffices to consider the case Σ′ = Σ ∪ {p} and we suppose first that p ≠ ℓ. Let Δ denote the endomorphism of J^{3} defined by the matrix

Δ = (\begin{matrix}α_{∗} \\ β_{∗} \\ γ_{∗} \end{matrix} ) w′_{∗}(α^{∗}, β^{∗}, γ^{∗})w_{∗}.

Using the relations αw′ = wγ and βw′ = wβ, we find that

Δ = (\begin{matrix}α_{∗}γ^{∗} & α_{∗}β^{∗} & α_{∗}α^{∗} \\ β_{∗}γ^{∗} & β_{∗}β^{∗} & β_{∗}α^{∗} \\ γ_{∗}γ^{∗} & γ_{∗}β^{∗} & γ_{∗}α^{∗} \end{matrix} ) ,

which can be computed for example by considering its effect on the cotangent space S_{2}(Γ_{0}(N))^{3}. The result is that

Δ = (\begin{matrix}T_{p}^{2} − (p + 1) & pT_{p} & p(p + 1) \\ pT_{p} & p(p + 1) & pT_{p} \\ p(p + 1) & pT_{p} & T_{p}^{2} − (p + 1) \end{matrix}) ,

which we note commutes with the action of 𝕋_{ℤ} on J^{3}. One then checks that

\begin{matrix}ϕ_{T} ϕ′_{T} = & (1, −p^{−1}T_{p}, p^{−1}) (T_{ℓ}(Δ) ⊗_{ℤ_{ℓ}} O)_{𝔪} (\begin{matrix}1 \\ −p^{−1}T_{p} \\ p^{−1} \end{matrix} ) \\ = & −p^{−2}(p − 1)(T_{p}^{2} − (p + 1)^{2}). \end{matrix}

The case of p = ℓ is similar but simpler. One uses

Δ = (\begin{matrix}α_{∗} \\ β_{∗} \end{matrix} ) w′_{∗}(α^{∗}, β^{∗})w_{∗} = (\begin{matrix}T_{ℓ} & ℓ + 1 \\ ℓ + 1 & T_{ℓ} \end{matrix})

and gets

ϕ_{T} ϕ′_{T} = −(α_{ℓ} − ℓα_{ℓ}^{−1})(1 − α_{ℓ}^{−2}).

Thus in either case we find that ϕ_{T} ϕ′_{T} acts on L′_{T} [P] by an element of O = 𝕋_{𝔪}/P which is a unit times π(c_{p}). Theorem 3.36 then follows from corollary 4.19 and lemma 4.24.

4.5 Homological results

In this section we sketch the proofs of theorem 4.18 and lemma 4.24, but shall often refer the reader to ch. 2 of [W3] for more details.

Multiplicities: We first consider theorem 4.18, generalizing a result proved by Mazur in sections II.14 and II.15 of [Maz1]. Recall that L_{T} (resp. L_{H}) is defined as (T_{ℓ}(J_{0}(N))⊗_{O})_{𝔪} (resp. H^{1}(X_{0}(N), O)_{𝔪}) where 𝔪 is a maximal of the ideal of the Hecke algebra 𝕋_{O} generated by the Hecke operators on S_{2}(Γ_{0}(N)). We are assuming moreover that ρ_{𝔪} is irreducible and that one of the following holds:

• ℓ does not divide N;

• ℓ^{2} does not divide N and T_{ℓ} ∉ 𝔪.

We wish to prove that L_{T} and L_{H} are free over 𝕋_{𝔪}.

Since L_{H} and L_{T} are isomorphic as 𝕋_{𝔪}-modules, it suffices to prove that L_{T} is free. Furthermore, to prove that L_{T} is free, it suffices to prove that the localization of T_{ℓ}(J_{0}(N)) at 𝔪 ∩ 𝕋_{ℤ_{ℓ}} is free; i.e., we may replace O by ℤ_{ℓ} and 𝔪 by its intersection with 𝕋_{ℤ_{ℓ}} and consider

T_{ℓ}(J_{0}(N))_{𝔪}.

Since T_{ℓ}(J_{0}(N)) ⊗_{ℤ_{ℓ}} ℚ_{ℓ} is free is free of rank two over 𝕋_{ℚ_{ℓ}}, it suffices to prove that

dim_{𝕋/𝔪}(L_{T} /𝔪L_{T}) = 2.

Since

T_{ℓ}(J_{0}(N))/ℓT_{ℓ}(J_{0}(N)) ≅ J_{0}(N)[ℓ] ≅ Hom_{𝔽_{ℓ}}(J_{0}(N)[ℓ], 𝔽_{ℓ})

as 𝕋_{𝔽_{ℓ}}-modules, it suffices to prove the following cases of [W3] thm. 2.1.

Theorem 4.26 Let 𝔪 be a maximal ideal of 𝕋_{ℤ_{ℓ}}. Suppose that ρ_{𝔪} is irreducible and that either

(a) ℓ does not divide N, or

(b) ℓ^{2} ∤ N and T_{ℓ} ∉ 𝔪.

Then

dim_{𝕋_{ℤ_{ℓ}}/𝔪} J_{0}(N)[𝔪] = dim_{𝕋_{ℤ_{ℓ}}/𝔪}(J_{0}(N)[ℓ]/𝔪) = 2.

The proof of theorem 4.26 in case (b) requires more of the theory of group schemes and Néron models than we wish to delve into here. We shall therefore only explain the proof in the case (a) (which is [R5] thm. 5.2(b), see Ribet’s paper for more details) and refer to [W3] sec. 2.1 for the general case. We remark however that the proof of theorem 3.42 appeals to theorem 3.36 only in the case Σ = ∅ (cf. remarks 4.22 and remark 4.25). Recall that ℓ divides N_{∅} only if ρ̄ is not good. So for the purposes of proving the Shimura-Taniyama conjecture for semistable elliptic curves, (b) can be replaced by the stronger hypothesis

(b′) ℓ^{2} ∤ N and ρ_{𝔪}|_{G_{ℓ}} is not good.

The result in this case is due to Mazur and Ribet (the main result of [MRi]) and the proof is slightly easier than in the case of (b).

Returning to case (a), we appeal to a general property of the functor 𝔻 of theorem 2.31 which follows from cor. 5.11 of [Oda].

Theorem 4.27 Suppose that A is an abelian variety over ℤ_{ℓ} (i.e., the Néron model of an abelian variety over ℚ_{ℓ} with good reduction). There is a canonical and functorial isomorphism of vector spaces over 𝔽_{ℓ}

𝔻(A(ℚ̄_{ℓ})[ℓ]^{∗})^{0} ≅ Cot_{0}(A/𝔽_{ℓ}),

where Cot_{0} denotes the cotangent space at the origin.

In the case that A is the Jacobian of a smooth proper curve X over ℤ_{ℓ}, we have also a canonical isomorphism

Cot_{0}(A/𝔽_{ℓ}) ≅ H^{0}(X, Ω_{X/𝔽_{ℓ}}^{1})

and in the case X = X_{0}(N) with ℓ ∤ N, this provides an isomorphism of 𝕋_{𝔽_{ℓ}}-modules

𝔻(J_{0}(N)(ℚ̄_{ℓ})[ℓ]^{∗})^{0} ≅ S_{2}(Γ_{0}(N), 𝔽_{ℓ}).  (4.5.1)

Consider now the action of G_{ℚ} on the points of

V = J_{0}(N)[ℓ]^{∗}[𝔪] ≅ (J_{0}(N)[ℓ]/𝔪)^{∗}.

By the argument in [Maz1] prop. II.14.2 or by the main result of [BLR], we know that every Jordan-Hölder constituent of the representation of G_{ℚ} on this 𝕋/𝔪-vector space is isomorphic to ρ_{𝔪}^{∗} ≅ ρ_{𝔪}. It follows that

dim_{𝕋/𝔪} 𝔻(V) = 2 dim_{𝕋/𝔪} 𝔻(V)^{0}

where we now regard V as a good G_{ℓ}-module. On the other hand (4.5.1) implies that

𝔻(V)^{0} ≅ S_{2}(Γ_{0}(N), 𝔽_{ℓ})[𝔪],

which is one-dimensional over 𝕋/𝔪 by the q-expansion principle (lemma 1.34 for example). We have now shown that 𝔻(V) is two-dimensional over 𝕋/𝔪, and it follows that so is V.

Ihara’s lemma: We now sketch the proof of lemma 4.24, which proceeds by analyzing the behavior of the homology of modular curves under certain degeneracy maps. Note that it suffices to prove the lemma for ϕ_{T}.

If N and M are positive integers, then we let Γ_{1}(N, M) = Γ_{1}(N) ∩ Γ_{0}(M) and write Y_{1}(N, M) (resp. X_{1}(N, M)) for the associated non-compactified (resp. compactified) modular curve. The key intermediate result is the following:

Lemma 4.28 Suppose that N is a positive integer and p is a prime not dividing N.

(a) The map

\begin{matrix}H_{1}(X_{1}(N, p), ℤ) & → & H_{1}(X_{1}(N), ℤ)^{2} \\ x & ↦ & (α_{∗}x, β_{∗}x) \end{matrix}

is surjective, where α is defined by τ ↦ τ and β by τ ↦ pτ.

(b) If N > 3 and ℓ is an odd prime different from p, then the sequence

\begin{matrix}H_{1}(Y_{1}(Np, p^{2}), ℤ_{ℓ}) & → & H_{1}(Y_{1}(Np), ℤ_{ℓ})^{2} & → & H_{1}(Y_{1}(N), ℤ_{ℓ}) \\ x & ↦ & (α_{1∗}x, β_{1∗}x); (y, z) & ↦ & β_{2∗}y − α_{2∗}z \end{matrix}

is exact, where the maps α_{1} and α_{2} are defined by τ ↦ τ and β_{1} and β_{2}

by τ ↦ pτ.

To prove the lemma, one first translates the statement into one about the homology (or cohomology) of the congruence subgroups involved. Part (a) of the lemma is due to Ihara [Ih], but see also the proof of thm. 4.1 of [R4]. Part (b) due to Wiles is more elementary and is established in the course of proving lemma 2.5 of [W3] (see the sequence (2.13)). The result stated there is in terms of cohomology rather than homology, but the one given here is immediate from it.

Remark 4.29 A method of Khare [Kh], sec. 2 using modular symbols yields an alternate proof of part (b) (after localization at a non-Eisenstein maximal ideal).

To deduce lemma 4.24 from lemma 4.28 we also need the following result:

Lemma 4.30 Suppose that O is the ring of integers of a finite extension of ℚ_{ℓ}. Let M be a positive integer and H a subgroup of (ℤ/Mℤ)^{×}. Let 𝕋̃_{O} = 𝕋̃_{ℤ} ⊗ O where 𝕋̃_{ℤ} is the subring of endomorphisms of M_{2}(Γ_{1}(M)) generated by the operators ⟨d⟩ and T_{r} for primes r ∤ M. If 𝔫 is a non-Eisenstein maximal ideal then the localization at 𝔫 of the natural map

(a) H_{1}(Y_{1}(M), O) → H_{1}(X_{1}(M), O) is an isomorphism;

(b) H_{1}(X_{1}(M), O) → H_{1}(X_{H}(M), O) is surjective.

To prove part (a), one checks that

H_{1}(Y_{1}(M), ℤ) → H_{1}(X_{1}(M), ℤ)

is surjective and that T_{r} = r + 1 on the kernel for primes r ≡ 1 mod M (cf. proposition 4.13, part (c)).

Part (b) is most easily proved by showing that G_{ℚ} acts trivially on the cokernel of the natural map

T_{ℓ}(J_{1}(M)) → T_{ℓ}(J_{H}(M))

(using [LO], prop. 6 for example) and then applying lemma 4.12.

Finally we explain how to use lemmas 4.28 and 4.30 to prove lemma 4.24. It suffices to consider the case Σ′ = Σ∪{p}. We discuss only the more difficult case of p ≠ ℓ. We define N = N_{Σ}, N′ = Np^{2}, 𝔪 and 𝔪′ as in the preceding section. Thus 𝔪′ is a maximal ideal of 𝕋′_{O} where 𝕋′_{ℤ} acts on S_{2}(Γ_{0}(N′)). All the maps we consider in the argument below will respect the action of the Hecke operators in 𝕋̃′′_{ℤ} where 𝕋̃′′_{ℤ} is the subring of endomorphisms of M_{2}(Γ_{1}(Np, p^{2})) generated by the operators ⟨d⟩ and T_{r} for primes r not dividing Np. We let 𝔫 denote the preimage of 𝔪′ in 𝕋̃′′_{O} = 𝕋̃′′_{ℤ} ⊗ O. We first apply lemma 4.28(b) to show that

H_{1}(Y_{1}(Np, p^{2}), O)_{𝔫} → H_{1}(Y_{1}(Np), O)_{𝔫}^{2} → H_{1}(Y_{1}(N), O)_{𝔫}

is exact. Next we apply lemma 4.30(a) to obtain the exactness of

H_{1}(X_{1}(Np, p^{2}), O)_{𝔫} → H_{1}(X_{1}(Np), O)_{𝔫}^{2} → H_{1}(X_{1}(N), O)_{𝔫}.

Applying lemma 4.30(b) with M = Np, we find that

H_{1}(X_{1}(N, p^{2}), O)_{𝔫} → H_{1}(X_{1}(N, p), O)_{𝔫}^{2} → H_{1}(X_{1}(N), O)_{𝔫}

is exact. We then apply lemma 4.28(a) to conclude that the map

\begin{matrix}H_{1}(X_{1}(N, p^{2}), O)_{𝔫} & → & H_{1}(X_{1}(N), O)_{𝔫}^{3} \\ x & ↦ & (α_{∗}x, β_{∗}x, γ_{∗}x) \end{matrix}

is surjective where α, β and γ are defined respectively by τ ↦ τ, τ ↦ pτ and τ ↦ p^{2}τ. Finally, we use lemma 4.30(b) again (now with M = N) to get the surjectivity of

H_{1}(X_{0}(Np^{2}), O)_{𝔫} → H_{1}(X_{0}(N), O)_{𝔫}^{3},

hence that of (4.4.4) upon localizing at 𝔪′.

5 Commutative algebra

In this section we collect some basic facts of commutative algebra that are used in the proof. We recall that O is the ring of integers in a finite extension K of ℚ_{ℓ}, and that O has residue field k. Let C_{O} denote as in section 2.6 the category of complete noetherian local O-algebras with residue field k.

5.1 Wiles’ numerical criterion

In this section we state a numerical criterion discovered by Wiles for a map between two rings in C_{O} to be an isomorphism.

Complete intersections: We say that a ring A in C_{O} is finite flat if it is finitely generated and torsion free as an O-module. A key ingredient in Wiles’ isomorphism criterion is played by the concept of complete intersection, for which we give the following naive definition.

Definition 5.1 An object A in C_{O} which is finite flat is called a complete intersection if it can be expressed as a quotient

A ≃ O[[X_{1}, …, X_{n}]]/(f_{1}, …, f_{n}),

where there are as many relations as there are variables.

Remark 5.2 It is also true that if an object O[[Y_{1}, …, Y_{r}]]/J in C_{O} which is finite flat is a complete intersection, then necessarily J can be generated by r elements. See for example [Mat], thm. 21.2 (and lemma 5.11 below). We will not use this fact in this chapter, although a special case is proved in the course of establishing lemma 5.30.

The category C_{O}^{•}: The numerical criterion of Wiles is stated more naturally in terms of rings A in the category C_{O} which are endowed with some extra structure: namely, a surjective O-algebra homomorphism π_{A} : A ⟶ O. Let C_{O}^{•} be the category whose objects are pairs (A, π_{A}), where A is an object of C_{O} and π_{A} : A ⟶ O is a surjective O-algebra homomorphism, also called the augmentation map attached to A. Morphisms in C_{O}^{•} are local ring homomorphisms which are compatible in the obvious way with the augmentation maps. By abuse of notation one often omits mentioning the augmentation map π_{A} when talking of objects in C_{O}^{•}, and simply uses A to denote (A, π_{A}), when this causes no confusion. Objects of C_{O}^{•} will also be called augmented rings.

The invariants Φ_{A} and η_{A}: One associates to an augmented ring (A, π_{A}) two basic invariants:

Φ_{A} = (ker π_{A})/(ker π_{A})^{2};
η_{A} = π_{A}(Ann_{A}ker π_{A}).

Here Ann_{A}(I) denotes the annihilator ideal of the ideal I in A.

The invariant Φ_{A} can be thought of as a tangent space for the object A. (More precisely, it is the cotangent space of the scheme spec(A) at the point ker π_{A}.) It is a finitely generated O-module.

The invariant η_{A} seems less familiar at first sight. It is called the congruence ideal. (The reason for this terminology should become clearer shortly.) We are now ready to state Wiles’ numerical criterion:

Theorem 5.3 Let ϕ : R ⟶ T be a surjective morphism of augmented rings. Assume that T is finitely generated and torsion-free as an O-module, and that η_{T} ≠ (0). (And hence, in particular, #(O/η_{T}) < ∞.) Then the following are equivalent:

(a) The inequality #Φ_{R} ≤ #(O/η_{T}) is satisfied.

(b) The equality #Φ_{R} = #(O/η_{T}) is satisfied.

(c) The rings R and T are complete intersections, and the map ϕ : R ⟶ T is an isomorphism.

Remark 5.4 The above theorem is slightly different from the one that appears in [W3], where it is assumed from the outset that the ring T is Gorenstein. In the form in which we state it above, the theorem is due to H. Lenstra [Len]. Our presentation follows Lenstra’s very closely. For the original (and slightly different) point of view, the reader should consult the appendix of [W3].

Some examples: Before going further, it may be good to pause and consider some examples of objects of C_{O}^{•} and the invariants associated to them. While logically independent of the proof, the examples should help the reader develop some intuition. (For a systematic way to compute the tangent spaces Φ_{A}, see the paragraph at the end of section 5.2.)

Example 1:

\begin{aligned}A & = & \{(a, b) ∈ O × O, a ≡ b (mod λ^{n})\} \\ & ≃ & O[[T]]/(T (T − λ^{n})), with π_{A}(a, b) := a. \end{aligned}

Φ_{A} ≃ O/λ^{n}O, η_{A} = (λ^{n}).

Example 2:

\begin{aligned}A & = & \{(a, b, c) ∈ O × O × O, a ≡ b ≡ c (mod λ)\} \\ & ≃ & O[[X, Y]]/(X(X − λ), Y (Y − λ), XY), with π_{A}(a, b, c) := a. \end{aligned}

Φ_{A} ≃ O/λO × O/λO, η_{A} = (λ).

Example 3:

A = O[[X]]/(X^{2}), with π_{A}(f) := f(0).
Φ_{A} ≃ O, η_{A} = 0.

Example 4:

A = \{(a, b, c, d) ∈ O × ⋯ × O, \begin{matrix}a ≡ b ≡ c ≡ d (mod λ), \\ a + d ≡ b + c (mod λ^{2}) \end{matrix}\}
≃ O[[X, Y]]/(X(X − λ), Y (Y − λ)),  with π_{A}(a, b, c, d) := a.
Φ_{A} ≃ O/λO × O/λO, η_{A} = (λ^{2}).

Example 5:

A = O[[X_{1}, …, X_{n}]], with π_{A}(f) := f(0).
Φ_{A} ≃ O^{n}, η_{A} = (0).

Example 6: (λ = ℓ ≡ −1 (mod 4), O = ℤ_{ℓ}).

A = {(a, b + ci) ∈ ℤ_{ℓ} × ℤ_{ℓ}[i], a ≡ b (mod ℓ^{2}), c ≡ 0 (mod ℓ)}
≃ ℤ_{ℓ}[[X]]/(X(X^{2} + ℓ^{2})),  with π_{A}(a, b + ci) := a.
Φ_{A} ≃ ℤ/ℓ^{2}ℤ, η_{A} = (ℓ^{2}).

Example 7:

A = O[[T]]/(λT) ≃ O ⊕ kT ⊕ kT^{2} ⊕ ⋯, with π_{A}(f) = f(0).
Φ_{A} ≃ k, η_{A} = (λ).

5.2 Basic properties of Φ_{A} and η_{A}

In this section we collect some of the basic properties of the invariants Φ_{A} and η_{A}, and prove the equivalence of (a) and (b) in theorem 5.3.

Behaviour of Φ_{A} under morphisms: The assignment A ↦ Φ_{A} is a functor from the category C_{O}^{•} to the category of O-modules; a morphism A ⟶ B in C_{O}^{•} induces a homomorphism Φ_{A} ⟶ Φ_{B} of O-modules. Moreover, if A ⟶ B is surjective, then so is the induced map on the tangent spaces. Therefore, when A maps surjectively onto B we have

#Φ_{A} ≥ #Φ_{B}.  (5.2.1)

There is also a converse to this, which will be useful later:

Lemma 5.5 If the homomorphism Φ_{A} ⟶ Φ_{B} is surjective, then A ⟶ B is also surjective.

Proof: This follows from Nakayama’s lemma. (Cf. [Ha], ch. II, sec. 7.4 and [Mat], th. 8.4.) □

Behaviour of η_{A} under (surjective) morphisms: Unlike the assignment A ↦ Φ_{A}, the assignment A ⟶ η_{A} is not functorial, but it does have a nice behaviour under surjective morphisms: namely, if ϕ : A ⟶ B is surjective, then

η_{A} ⊂ η_{B}, i.e., #(O/η_{A}) ≥ #(O/η_{B}).  (5.2.2)

This is simply because in that case ϕ induces a map

Ann_{A}ker π_{A} ⟶ Ann_{B}ker π_{B}.

Relation between the invariants Φ_{A} and η_{A}: In general, we have the following inequality:

#Φ_{A} ≥ #(O/η_{A}).  (5.2.3)

The key behind proving this identity is to interpret #Φ_{A} in terms of Fitting ideals.

Digression on Fitting ideals: If R is a ring (in C_{O}, say) and M is a finitely generated R-module, we express M as a quotient of R^{n} for some n:

0 ⟶ M′ ⟶ R^{n} ⟶ M ⟶ 0.  (5.2.4)

The Fitting ideal of M, denoted Fitt_{R}(M), is the ideal of R generated by the determinants det(v_{1}, …, v_{n}), where the vectors v_{i} ∈ R^{n} range over all possible choices of elements of M′ ⊂ R^{n}. One checks that this ideal does not depend on the choice of exact sequence (5.2.4), and hence is an invariant of the R-module M. For example, if M is a finitely generated O-module, we may write

M = O^{r} ⊕ O/(λ^{n_{1}}) ⊕ O/(λ^{n_{2}}) ⊕ ⋯ ⊕ O/(λ^{n_{k}}),

with n_{1} ≥ n_{2} ≥ ⋯ ≥ n_{k}, and the sequence n_{i} is completely determined by M. The Fitting ideal Fitt_{O}(M) is then the ideal of O generated by λ^{n_{1}+···+n_{k}}, if r = 0, and is the 0-ideal if r > 0. Note in particular that, if M is a finite O-module, then

#M = #(O/Fitt_{O}(M)).  (5.2.5)

Furthermore, if M is any R-module, it follows directly from the definition that

Fitt_{R}(M) ⊂ Ann_{R}(M).  (5.2.6)

Fitting ideals behave well under tensor products: in particular, if M is a finitely generated A-module, where A is an object in C_{O}^{•}, then:

π_{A}(Fitt_{A}(M)) = Fitt_{O}(M ⊗_{A} O),  (5.2.7)

where the tensor product is taken with respect to the augmentation map π_{A}. For more details and references on the Fitting ideal, see [Len] for example.

Now we are ready to prove equation (5.2.3). For, noting that Φ_{A} = ker π_{A}⊗_{A}O, where the tensor product is taken with respect to π_{A}, and applying equation (5.2.7) with M = ker π_{A}, we have:

Fitt_{O}(Φ_{A}) = π_{A}(Fitt_{A}(ker π_{A})) ⊂ π_{A}(Ann_{A}ker π_{A}) = η_{A}, (5.2.8)

where the containment follows from equation (5.2.6). Now the inequality (5.2.3) follows by combining (5.2.5) and (5.2.8).

As a consequence, we have

Corollary 5.6 The statements (a) and (b) in theorem 5.3 are equivalent.

Proof: If R ⟶ T is a surjective map of augmented rings, then #Φ_{R} ≥ #Φ_{T}, by equation (5.2.1). But equation (5.2.3) gives the inequality #Φ_{T} ≥ #(O/η_{T}). Hence the inequality #Φ_{R} ≥ #(O/η_{T}) always holds, so that (a) implies (b) in theorem 5.3. The reverse implication is clear. □

Remark 5.7 (Computing the tangent spaces Φ_{A}): Any object (A, π_{A}) in C_{O}^{•} can be expressed as a quotient of the object U = O[[X_{1}, …, X_{n}]] of example 5 with augmentation map given by π_{U}(f) = f(0). Indeed, one can take a_{1}, …, a_{n} to be A-module generators of the finitely generated A-module ker π_{A}, and obtain the desired quotient map by sending X_{i} to a_{i}.

The tangent space Φ_{U} of U is a free O-module of rank n which can be written down canonically as

OX_{1} ⊕ OX_{2} ⊕ ⋯ ⊕ OX_{n},

the natural map from (ker π_{U}) being simply the map which sends a power series f ∈ U with no constant term to its degree 1 term, which we will denote by f̄.

If A is expressed as a quotient U/(f_{1}, …, f_{r}), then one has

Φ_{A} = Φ_{U}/(f̄_{1}, …, f̄_{r}).  (5.2.9)

5.3 Complete intersections and the Gorenstein condition

In this section we show that complete intersections satisfy a Gorenstein condition, and that (c) implies (a) and (b) in the statement of Wiles isomorphism criterion (theorem 5.3).

Definition 5.8 An object A in C_{O} which is finite flat is said to be Gorenstein if

Hom_{O}(A, O) ≃ A as A-modules.

Proposition 5.9 Suppose A in C_{O} is finite flat. If A is a complete intersection, then A is Gorenstein.

The remainder of this section will be devoted to proving proposition 5.9. Since the proof is a bit long and involved, and the concepts it uses are not used elsewhere, the reader is advised on a first reading to take it on faith and skip to the next section. A more direct proof of proposition 5.9 which bypasses the arguments of this section is explained in [Len].

We let A be a ring which is finite flat, and is a complete intersection. (Hence, A can be written as O[[X_{1}, …, X_{n}]]/(f_{1}, …, f_{n}).) We assume that the augmentation map for A is induced from the map on O[[X_{1}, …, X_{n}]] sending f to f(0). This implies that f_{i}(0) = 0, i.e., the f_{i} have no constant term.

We recall some definitions from commutative algebra that we will need. An ideal I of a local ring R is said to be primary if I ≠ R and every zero divisor in R/I is nilpotent. If (x_{1}, …, x_{n}) generates a primary ideal of R, and n = dim R, then (x_{1}, …, x_{n}) is called a system of parameters for R.

Lemma 5.10 The sequence (f_{1}, …, f_{n}, λ) is a system of parameters for U = O[[X_{1}, …, X_{n}]].

Proof: The quotient ring U/(f_{1}, …, f_{n}, λ) is local and is finitely generated as a k-vector space; therefore every element in its maximal ideal is nilpotent. □

A sequence (x_{1}, …, x_{n}) in a ring R is said to be a regular sequence if x_{i} is not a zero-divisor in R/(x_{1}, …, x_{i−1}) for i = 1, …, n.

Lemma 5.11 The sequence (f_{1}, …, f_{n}) is a regular sequence for U.

Proof: The ring U is Cohen Macaulay, since λ, X_{1}, …, X_{n} is a system of parameters of U which is also a regular sequence. Hence, by theorem 17.4 (iii) of [Mat], (f_{1}, …, f_{n}, λ) is a regular sequence in U. A fortiori, the sequence (f_{1}, …, f_{n}) is also a regular sequence. □

The proofs of lemma 5.10 and 5.11 use only the fact that A is finitely generated as an O-module, and not that A is flat. As a corollary of this proof, we therefore have:

Corollary 5.12 If R is an object of C_{O}^{•} which is finitely generated as an Omodule, and R ≃ O[[X_{1}, …, X_{n}]]/(f_{1}, …, f_{n}), then R is also flat, and hence is a complete intersection.

To go further, we will introduce the Koszul complex

K(\underline{x}, R) := ⊕_{p=0}^{n}K_{p}(\underline{x}, R)

associated to a local ring R and a sequence \underline{x} = (x_{1}, …, x_{n}) of elements in its maximal ideal. For more details on the Koszul complex and its relation to regular sequences, the reader can consult [Mat], especially §16, or [Bour] X, or [Se3]. This complex is defined to be the free graded differential algebra generated by symbols u_{1}, …, u_{n}:

K_{p}(\underline{x}, R) := ⊕_{i_{1}<i_{2}<···<i_{p}}R · u_{i_{1}} ∧ ⋯ ∧ u_{i_{p}},

with differential d : K_{p} ⟶ K_{p−1} defined by

d(u_{i_{1}} ∧ ⋯ ∧ u_{i_{p}}) = ∑_{t=1}^{p}(−1)^{t}x_{t} · u_{i_{1}} ∧ ⋯ ∧ u_{i_{t−1}} ∧ u_{i_{t+1}} ∧ ⋯ ∧ u_{i_{p}}.

We denote by H_{p}(\underline{x}; R) the homology groups of this complex. We record here the main properties of this complex that we will use.

Proposition 5.13 (a) H_{0}(\underline{x}; R) = R/(\underline{x}).

(b) There is a long exact homology sequence

⋯ ⟶ H_{p}(\underline{x}; R) ⟶ H_{p}(\underline{x}, x_{n+1}; R) ⟶ H_{p−1}(\underline{x}; R) ⟶^{±x_{n+1}}

H_{p−1}(\underline{x}; R) ⟶ H_{p−1}(\underline{x}, x_{n+1}; R) ⟶ H_{p−2}(\underline{x}; R) ⟶ ⋯

(c) H_{p}(\underline{x}; R) is annihilated by the ideal (\underline{x}), i.e., it has a natural R/(\underline{x})- module structure.

(d) If \underline{x} is a regular sequence, then H_{p}(\underline{x}; R) = 0 for all p > 0 (i.e., the complex K_{p}(\underline{x}; R) is a free resolution of R/(\underline{x}).)

Proof: The first assertion follows directly from the definition. For (b) and (c), see [Mat], th. 16.4. The assertion (d) can be proved by a direct induction argument on n, using the long exact homology sequence: For p > 1, this sequence becomes

0 ⟶ H_{p}(\underline{x}, x_{n+1}; R) ⟶ 0,

and for p = 1, it is

0 ⟶ H_{1}(\underline{x}, x_{n+1}; R) ⟶ H_{0}(\underline{x}; R) ⟶^{x_{n+1}} H_{0}(\underline{x}; R).

But the assumption that \underline{x}, x_{n+1} is a regular sequence means that multiplication by x_{n+1} is injective on H_{0}(\underline{x}; R) = R/(\underline{x}). Hence, the assertion (d) follows. □

Now, we turn to the proof of proposition 5.9, following a method of Tate which is explained in the appendix of [MRo]. For any ring R, write R[[\underline{X}]] := R[[X_{1}, …, X_{n}]]. Let a_{1}, …, a_{n} be the images in A of X_{1}, …, X_{n} by the natural map

α : O[[\underline{X}]] ⟶ A = O[[\underline{X}]]/(f_{1}, …, f_{n}),

and let

β : A[[\underline{X}]] ⟶ A

be the natural map which sends X_{i} to a_{i}. The sequence (g_{i}) = (X_{i} − a_{i}) generates the kernel of β. Since the f_{i}, viewed as polynomials in A[[\underline{X}]], also belong to ker β, we have:

(f_{1}, …, f_{n}) = (g_{1}, …, g_{n})M,  (5.3.1)

where M is an n × n matrix with coefficients in A[[\underline{X}]]. Let

D = det(M) ∈ A[[\underline{X}]].

Our goal is to construct an A-module isomorphism

Hom_{O}(A, O) ⟶ A.

We begin by constructing a surjective (O-linear) map

Hom_{O[[\underline{X}]]}(A[[\underline{X}]], O[[\underline{X}]]) ⟶ A.

Lemma 5.14 (Tate): The function Φ(f) = α(f(D)) induces an isomorphism of O[[\underline{X}]]-modules

Hom_{O[[\underline{X}]]}(A[[\underline{X}]], O[[\underline{X}]])/(g_{1}, …, g_{n}) ⟶ A.

Proof: By lemma 5.11, the sequence (\underline{f}) = (f_{1}, …, f_{n}) is a regular O[[\underline{X}]]- sequence. One can see directly from the definition that the sequence (\underline{g}) = (g_{i}) = (X_{i}−a_{i}) is a regular A[[\underline{X}]]-sequence. Let K(\underline{f}) and K(\underline{g}) be the Koszul complexes associated to these two sequences. It follows from proposition 5.13 that the Koszul complex K(\underline{f}) is a resolution of A by free O[[\underline{X}]]-modules, and the Koszul complex K(\underline{g}) is a resolution of A by free A[[\underline{X}]]-modules, and hence a fortiori, by free O[[\underline{X}]]-modules. We define a map Φ : K(\underline{f}) ⟶ K(\underline{g}) of complexes by letting

Φ_{0} : K_{0}(\underline{f}) ⟶ K_{0}(\underline{g})

be the natural inclusion of O[[\underline{X}]] into A[[\underline{X}]], and letting

Φ_{1} : K_{1}(\underline{f}) ⟶ K_{1}(\underline{g})

be the map defined by

(Φ_{1}(u_{1}), …, Φ_{1}(u_{n})) = (v_{1}, …, v_{n})M,

and extending it by skew-linearity to a map of exterior algebras. One can check that the resulting map Φ is a morphism of complexes which induces the identity map A ⟶ A, and satisfies

Φ_{n}(u_{1} ∧ ⋯ ∧ u_{n}) = D · v_{1} ∧ ⋯ ∧ v_{n}.

Applying the functor Hom_{O[[\underline{X}]]}(−, O[[\underline{X}]]) to these two free resolutions, and taking the homology of the resulting complexes, we find that since Φ is a homotopy equivalence of complexes it induces an isomorphism on the cohomology, and in particular, on the nth cohomology:

Φ_{n} : Hom_{O[[\underline{X}]]}(A[[\underline{X}]], O[[\underline{X}]])/(g_{1}, …, g_{n}) ⟶^{≃}

Hom_{O[[\underline{X}]]}(O[[\underline{X}]], O[[\underline{X}]])/(f_{1}, …, f_{n}) ≃ A,

which is given explicitly by the formula:

Φ_{n}(f) = α(f(D)).
□

We finally come to the proof of proposition 5.9, which we can state in a more precise form.

Lemma 5.15 The map Ψ : Hom_{O}(A, O) ⟶ A defined by Ψ(f) = α(f̃(D)), where f̃: A[[\underline{X}]] ⟶ O[[\underline{X}]] is the base change of f, is an A-module isomorphism, and hence, A is Gorenstein.

Proof: The key point is to show that Ψ is A-linear. By definition, if

a = α(a′) ∈ A, with a′ ∈ O[[\underline{X}]],

then

Ψ(af) = α(f̃(aD)) = α(f̃((a − a′)D)) + α(f̃(a′D)).

Since a − a′ ∈ ker β, it can be written as an A[[\underline{X}]]-linear combination of the g_{i}. By multiplying the relation

(f_{1}, …, f_{n}) = (g_{1}, …, g_{n})M

by the matrix D · M^{−1} ∈ M_{n}(A[[\underline{X}]]), one sees that the Dg_{i} can be written as A[[\underline{X}]]-linear combinations of the f_{i}’s. Hence, so can the expression (a − a′)D. By the O[[\underline{X}]]-linearity of f̃, and the fact that each f_{i} belongs to ker α, it follows that

α(f̃((a − a′)D)) = 0.

Therefore,

Ψ(af) = α(a′f̃(D)) = aΨ(f).

This shows that Ψ is A-linear. To show that Ψ is surjective, observe that if f_{1}, …, f_{r} is a O-basis of Hom_{O}(A, O), then f̃_{1}, …, f̃_{r} is a O[[\underline{X}]]-basis for Hom_{O[[\underline{X}]]}(A[[\underline{X}]], O[[\underline{X}]]). Hence, for all a ∈ A, there exist p_{1}, …, p_{r} such that

Φ_{n}(p_{1}f̃_{1} + ⋯ + p_{r}f̃_{r}) = a.

But this means that

Ψ(α(p_{1})f_{1} + ⋯ + α(p_{r})f_{r}) = a,

so that Ψ is surjective. Finally, since Hom_{O}(A, O) and A are free O-modules of the same finite rank, and Ψ is surjective, it must also be injective. Hence Ψ is an isomorphism, as was to be shown. □

Example 5.16 Let A ⊂ O ×O be the ring of example 1 in section 5.1. Then, f = T^{2} − λ^{n}T, and g = T − (0, λ^{n}). Hence, f = (T − (λ^{n}, 0))g, so that D = T − (λ^{n}, 0). It follows that

Φ(h) = α(h̃(T − (λ^{n}, 0))) = α(T h(1, 1) − h(λ^{n}, 0)) = (0, λ^{n})h(1, 1) − h(λ^{n}, 0).

The reader can check directly that Φ is indeed an A-linear isomorphism from the A-module Hom_{O}(A, O) to A.

Example 5.17 Let A = O[ϵ]/(ϵ^{2}) be the ring of example 3 in section 5.1. Then, f = T^{2}, and g = T − ϵ. Hence, f = (T + ϵ)g, so that D = T + ϵ. It follows that

Φ(f) = α(f̃(T + ϵ)) = α(T f(1) + f(ϵ)) = ϵf(1) + f(ϵ).

Exercise 5.18 Write down explicitly an isomorphism A →^{∼} Hom_{O}(A, O) for the complete intersection A = O[[X, Y]]/(X(X − λ), Y (Y − λ)) of example 4 of section 5.1.

5.4 The Congruence ideal for complete intersections

Let A be an object of C_{O}^{•}, which is finite flat and is a complete intersection as in the previous section, so that A ≃ O[[X_{1}, …, X_{n}]]/(f_{1}, …, f_{n}).

Using the result of the previous section, we will give an explicit formula for computing η_{A} in this case, and prove that (c) implies (b) in theorem 5.3.

Let A^{∨} := Hom_{O}(A, O), and let π_{A}^{∨} : O^{∨} ⟶ A^{∨} be the dual map. From the Gorenstein property of A, we may identify A^{∨} with A (as A-modules). Fix any identification Ψ : A^{∨} ⟶^{≃} A. (Any two such differ by a unit in A.) One checks that

Ψπ_{A}^{∨}(O^{∨}) = Ann_{A}ker π_{A}.

Hence, η_{A} is the image of the map π_{A}Ψπ_{A}^{∨}. By using the explicit construction of Ψ given in lemma 5.15, we find:

π_{A}Ψπ_{A}^{∨}(O^{∨}) = παπ̃(D),

where D is the determinant defined in section 5.3. By a direct calculation using equation (5.3.1), one sees that the right hand side is equal to det(∂f_{i}/∂X_{j}(0)). Hence we have shown:

Proposition 5.19 Suppose that A is an object of C_{O}^{•} which is finite flat. If A = O[[X_{1}, …, X_{n}]]/(f_{1}, …, f_{n}) is a complete intersection, then

η_{A} = (det(∂f_{i}/∂X_{j}(0))).

This proposition implies:

Corollary 5.20 Let A in C_{O}^{•} be finite flat. If A is a complete intersection, then #Φ_{A} = #(O/η_{A}). Hence, the statement (c) implies statement (b) in theorem 5.3.

Proof: The equation (5.2.9) of remark 5.7 implies that

#Φ_{A} = #O/(det(∂f_{i}/∂X_{j}(0))).

The result follows. □

5.5 Isomorphism theorems

The usefulness of the notion of complete intersections comes from the following two (vaguely stated) principles:

1. Isomorphisms to complete intersections can often be recognized by looking at their effects on the tangent spaces.

2. Isomorphisms from complete intersections can often be recognized by looking at their effects on the invariants η.

These vague principles are made precise in theorems 5.21 and 5.24 respectively.

Theorem 5.21 Let ϕ : A ⟶ B be a surjective morphism of augmented rings, with B a (finite, flat) complete intersection. If ϕ induces an isomorphism from Φ_{A} to Φ_{B}, and these modules are finite, then ϕ is an isomorphism.

Remark 5.22 Let A = O[[X, Y]]/(X(X − λ), Y (Y − λ)) be the ring of example 4, let B = O[[X, Y]]/(X(X − λ), Y (Y − λ), XY) be the ring of example 2, and let ϕ : A ⟶ B be the natural projection. The map ϕ induces an isomorphism Φ_{A} ⟶ Φ_{B}, even though ϕ is not an isomorphism. The assumption that B is a complete intersection is crucial for concluding that ϕ is an isomorphism.

Remark 5.23 The natural map

O[[X]]/(X^{3}) ⟶ O[[X]]/(X^{2})

is a surjective morphism inducing an isomorphism on tangent spaces, and the target ring is a complete intersection. Yet this map is not an isomorphism. This shows that the assumption on the finiteness of the tangent spaces cannot be dispensed with.

Proof of theorem 5.21: Recall that U = O[[X_{1}, …, X_{n}]] is the augmented ring of example 5 of section 5.1. Let

ν_{B} : U ⟶ B

be a surjective morphism of augmented rings with ker ν_{B} = (f_{1}, …, f_{n}). Let b_{1}, …, b_{n} ∈ ker π_{B} denote the images of X_{1}, …, X_{n} by ν_{B}, and let a_{1}, …, a_{n} ∈ ker π_{A} denote inverse images of b_{1}, …, b_{n} by ϕ. Since ϕ is an isomorphism on tangent spaces, the elements a_{i} generate (ker π_{A})/(ker π_{A})^{2}. Hence the morphism

ν_{A} : O[[X_{1}, …, X_{n}]] ⟶ A

defined by ν_{A}(X_{i}) = a_{i} induces a surjection Φ_{U} ⟶ Φ_{A}, and so it is surjective, by lemma 5.5.

We claim that ker ν_{B} is contained in ker ν_{A} (and hence, ker ν_{B} = ker ν_{A}). For, let g_{1}, …, g_{n} be elements of ker ν_{A} whose linear terms ḡ_{1}, …, ḡ_{n} generate the kernel of

ν̄_{A} : Φ_{U} ⟶ Φ_{A}.

Since ker ν_{A} ⊂ ker ν_{B}, it follows that there exists an n × n matrix M ∈ M_{n}(U) with entries in U such that

(g_{1}, …, g_{n}) = (f_{1}, …, f_{n})M.

Let M̄ be the matrix of constant terms of the matrix M. Then we have

(ḡ_{1}, …, ḡ_{n}) = (f̄_{1}, …, f̄_{n})M̄.

Since (ḡ_{1}, …, ḡ_{n}) and (f̄_{1}, …, f̄_{n}) generate the same submodules of rank n and finite index in Φ_{U}, it follows that det M̄ is a unit in O. Hence, M is invertible, and therefore the f_{i} can be expressed as a U-linear combination of the g_{j}. This implies that ker ν_{B} ⊂ ker ν_{A}. Now we see that ν_{A}ν_{B}^{−1} gives a well-defined inverse to ϕ, so that ϕ is an isomorphism. □

Theorem 5.24 Let ϕ : A ⟶ B be a surjective morphism of augmented rings. Suppose that A and B are finite flat, and that A a complete intersection. If η_{A} = η_{B} ≠ 0, then ϕ is an isomorphism.

Remark 5.25 The torsion-freeness assumption on B is essential: if n is large enough, then B = A/(ker π_{A})^{n} satisfies η_{A} = η_{B}, although the natural map A ⟶ B is not injective when A ≠ O.

Proof: By proposition 5.9, A is Gorenstein, i.e., it satisfies

A^{∨} := Hom_{O}(A, O) ≃ A as A-modules.

Now, we observe that

ker π_{A} ∩ Ann_{A}ker π_{A} = 0,  (5.5.1)

and likewise for B. For, let x be a non-zero element of η_{A}, and let x′ ∈ Ann_{A}ker π_{A} satisfy π_{A}(x′) = x. For all a ∈ ker π_{A} ∩ Ann_{A}ker π_{A}, we have

0 = a(x − x′) = ax,

the first equality because a belongs to Ann_{A}ker π_{A} and (x − x′) belongs to ker π_{A}, the second equality because a belongs to ker π_{A} and x′ belongs to Ann_{A}ker π_{A}. Hence a belongs to the O-torsion submodule of A, and therefore is 0.

It follows from (5.5.1) that the homomorphism π_{A} (resp. π_{B}) induces an isomorphism from Ann_{A}ker π_{A} (resp. Ann_{B}ker π_{B}) to η_{A} (resp. η_{B}). Since η_{A} is isomorphic to η_{B}, it follows that ϕ induces an isomorphism from Ann_{A}ker π_{A} to Ann_{B}ker π_{B}, i.e.,

ϕAnn_{A}ker π_{A} = Ann_{B}ker π_{B}.

From (5.5.1) it also follows a fortiori that

ker ϕ ∩ Ann_{A}ker π_{A} = 0,

hence there is an exact sequence of A-modules:

0 ⟶ ker ϕ ⊕ Ann_{A}ker π_{A} ⟶ A.  (5.5.2)

The cokernel of the last map is

A/(ker ϕ ⊕ Ann_{A}ker π_{A}) ≃ B/(ϕAnn_{A}ker π_{A}) ≃ B/(Ann_{B}ker π_{B}),

which is torsion-free, since there is a natural injection

B/(Ann_{B}ker π_{B}) ↪ End_{O}(ker π_{B}).

Hence, the exact sequence (5.5.2) splits over O. Taking O duals in (5.5.2) and using the Gorenstein condition for A, we thus get an exact sequence of A-modules:

A ⟶ (ker ϕ)^{∨} ⊕ (Ann_{A}ker π_{A})^{∨} ⟶ 0.

Applying the functor − ⊗_{A} k (relative to the map A ⟶ k), we find

1 = dim_{k}(A ⊗_{A} k) ≥ dim_{k}((ker ϕ)^{∨} ⊗_{A} k) + dim_{k}((Ann_{A}ker π_{A})^{∨} ⊗_{A} k).

Since η_{A} ≠ 0, it follows that (Ann_{A}ker π_{A})^{∨} ⊗_{A} k ≠ 0, and hence we must have

(ker ϕ)^{∨} ⊗_{A} k = 0.

Therefore by Nakayama’s lemma and duality, ker ϕ = 0, which proves the theorem. □

5.6 A resolution lemma

It turns out that objects in C_{O}^{•} can be “resolved” (in a weak sense) by a complete intersection, namely,

Theorem 5.26 Let A be an augmented ring which is finite flat over O. Then there is a morphism Ã ⟶ A in the category C_{O}^{•} such that:

(a) the ring Ã is finite flat over O and is a complete intersection;

(b) the map Ã ⟶ A induces an isomorphism ΦÃ ⟶ Φ_{A}.

Proof: Write A as a quotient of U = O[[X_{1}, …, X_{n}]] (with π_{U} : U ⟶ O the map which sends each X_{i} to 0.) Let f_{1}, …, f_{n} be elements in the kernel of the natural map U ⟶ A, such that f̄_{1}, …, f̄_{n} generate the kernel of Φ_{U} ⟶ Φ_{A}. Now letting

Ã = U/(f_{1}, …, f_{n})

would give the desired ring Ã, provided Ã is finitely generated: indeed, the flatness of Ã would follow from corollary 5.12.

Thus we need to show that the f_{i} can be chosen so that Ã is finitely generated. Let a_{1}, …, a_{n} be O-module generators of the finitely generated module ker π_{A}, and define a homomorphism ϕ from the polynomial ring

V = O[X_{1}, …, X_{n}]

to A by sending X_{j} to a_{j}. Clearly ϕ is surjective. Let f_{1}, …, f_{n} be elements of ker ϕ chosen as above, and let m denote their maximal degree. Since the elements a_{i}^{2} belong to ker π_{A}, we may write

a_{i}^{2} = h_{i}(a_{1}, …, a_{n}),

where h_{i}(X_{1}, …, X_{n}) is a linear polynomial. Now, replacing the relations f_{i} by the relations

f_{i} + X_{i}^{m}h_{i} − X_{i}^{m+2},

we find that the ring V/(f_{1}, …, f_{n}) is a finitely generated O-module: it can be generated by the images of the monomials of degree ≤ n(m + 1), since the relations allow us to rewrite any monomial of higher degree in terms of ones of lower degree. Completing at the ideal (λ, X_{1}, …, X_{n}), we find that the ring

Ã = U/(f_{1}, …, f_{n})

has the desired properties: the natural homomorphism from Ã to A induces an isomorphism on the tangent spaces, since the linear terms of the f_{i} generate the kernel of the induced map Φ_{U} ⟶ Φ_{A} on the tangent spaces, and Ã is a finitely generated O-module, since V/(f_{1}, …, f_{n}) is. □

5.7 A criterion for complete intersections

The results we have accumulated so far allow us to give an important criterion for an object A to be a complete intersection:

Theorem 5.27 Let A be an augmented ring which is a finitely generated torsion-free O-module. If #Φ_{A} ≤ #(O/η_{A}) < ∞, then A is a complete intersection.

Proof: Let ϕ : Ã ⟶ A be the surjective morphism given by the resolution theorem (theorem 5.26). Then we have

#(O/η_{A}) ≥ (#Φ_{A}) = (#ΦÃ) ≥ #(O/ηÃ),

where the first inequality is by assumption, the second by the choice of Ã, and the third is by the equation (5.2.3). On the other hand, by equation (5.2.2), we have

#(O/ηÃ) ≥ #(O/η_{A}).

It follows that

η_{A} = ηÃ,

so that ϕ is an isomorphism by theorem 5.24. It follows that A is a complete intersection. □

5.8 Proof of Wiles’ numerical criterion

Theorem 5.28 Let R and T be augmented rings such that T is a finitely generated torsion-free O-module, and let ϕ : R ⟶ T be a surjective morphism. If

#Φ_{R} ≤ #(O/η_{T}) < ∞,

then R and T are complete intersections, and ϕ is an isomorphism.

Proof: We have:

#(O/η_{T}) ≤ #Φ_{T} ≤ #Φ_{R} ≤ #(O/η_{T}),

where the first inequality is by equation (5.2.3), the second follows from the surjectivity of ϕ, and the third is by hypothesis. Therefore,

#Φ_{T} = #(O/η_{T}),

and hence T is a complete intersection by theorem 5.27. Since the orders of Φ_{R} and Φ_{T} are the same, ϕ induces an isomorphism between them. Hence ϕ is an isomorphism R ⟶ T, by theorem 5.21. This completes the proof. □

Theorem 5.28 shows that the statements (a) and (b) in theorem 5.3 imply the statement (c). Combining corollaries 5.6, 5.20, and theorem 5.28 completes the proof of theorem 5.3.

5.9 A reduction to characteristic ℓ

Let C_{k} be the category of complete local noetherian k-algebras with residue field k. Again all morphisms are assumed to be local. There is a natural functor A ↦ Ā from C_{O} to C_{k} which send A to Ā := A/λ.

We say that an object A of C_{k} which is finite dimensional as a k-vector space is a complete intersection if it is isomorphic to a quotient

A = k[[X_{1}, …, X_{r}]]/(f_{1}, …, f_{r}).

Note that if an object A of C_{O} is a complete intersection, then Ā is a complete intersection in C_{k}. As a partial converse, we have:

Lemma 5.29 Suppose that R ↦ T is a map in the category C_{O}, and that T is finitely generated and free as an O-module. Then R ⟶ T is an isomorphism of complete intersections, if and only if R̄ ⟶ T̄ is.

Proof: This is an exercise and is left to the reader. □ We now come to the proof of lemma 3.39 of section 3.4:

Lemma 5.30 Suppose that K ⊂ K′ are local fields with rings of integers O ⊂ O′ and that A is an object of C_{O} which is finitely generated and free as an O-module. Then A is a complete intersection if and only if A ⊗_{O} O′ is.

Proof: One implication is clear. Let k and k′ be the residue fields of O and O′ respectively. By lemma 5.29 it is enough to prove that, if R is an object of C_{k} which is finite dimensional as a k-vector space, then

R′ = R ⊗_{k} k′ is a complete intersection ⇒ R is a complete intersection.

Let 𝔪 and 𝔪′ denote the maximal ideals of R and R′ respectively.

By assumption, we have R′ = k′[[Y_{1}, …, Y_{r}]]/J, where the ideal J can be generated by r elements. We can assume without loss of generality (by adding extra variables and relations if necessary) that the images of Y_{1}, … Y_{r} generate 𝔪′ as a k′-vector space. Now, let

ϕ : k[[X_{1}, …, X_{r}]] ⟶ R

be a ring homomorphism, such that the images of X_{1}, …, X_{r} generate 𝔪 as a k-vector space, and let

ϕ′ : k′[[X_{1}, …, X_{r}]] ⟶ R′

be the extension of scalars. Let I and I′ = I ⊗_{k} k′ be the kernels of these two maps. We claim that I′ can be generated by r elements. (In fact, this is also true without the assumption that the images of X_{1}, …, X_{r} generate 𝔪′ as a k′ vector space, although we use this assumption in the proof below. Cf. remark 5.2.) To see this, choose an isomorphism of k′-vector spaces k′X_{1} ⊕ ⋯ ⊕ k′X_{r} ⟶ k′Y_{1} ⊕ ⋯ ⊕ k′Y_{r} such that

k′X_{1} ⊕ ⋯ ⊕ k′X_{r} ⟶ k′Y_{1} ⊕ ⋯ ⊕ k′Y_{r}
↓  ↓
𝔪′  =  𝔪′

commutes. Extending this map to an isomorphism of k′-algebras,

ν : k′[[X_{1}, …, X_{r}]] ⟶ k′[[Y_{1}, …, Y_{r}]],

one sees that I′ = ν^{−1}(J), and hence can be generated by r elements, as claimed.

In particular we have dim_{k′}(I′/𝔪′I′) ≤ r, and

dim_{k}(I/𝔪I) = dim_{k′}((I/𝔪I) ⊗_{k} k′) ≤ dim_{k′}(I′/𝔪′I′) ≤ r.

Nakayama’s lemma now implies that I can be generated by r elements, and hence R is a complete intersection. This proves the lemma. □

5.10 J-structures

We now turn to the proof of theorem 3.41. In view of the last section we will work in characteristic ℓ. Thus let π : R → → T be a surjective morphism in the category C_{k}, where R and T are finite dimensional as k-vector spaces. Let r be a non-negative integer. If J ⊲ k[[S_{1}, ..., S_{r}]] and J ⊂ (S_{1}, ..., S_{r}) then by a strong J-structure we shall mean a commutative diagram in C_{k}

k[[S_{1}, ..., S_{r}]]
↓
k[[X_{1}, ..., X_{r}]] → → R′ → → T′
↓  ↓
R  → → T,

such that

(a) T′/(S_{1}, ..., S_{r})T′ →^{∼} T and R′/(S_{1}, ..., S_{r})R′ → → R,

(b) for each ideal I ⊃ J, I = ker (k[[S_{1}, ..., S_{r}]] → T′/I),

(c) J = ker (k[[S_{1}, ..., S_{r}]] → T′) and R′ ↪ T′ ⊕ R.

We refer to these as strong J-structures because we have slightly altered the conditions from section 3.4 for technical convenience in the rest of this section.

We will prove the following result.

Theorem 5.31 Suppose there exist a sequence of ideals J_{n}⊲k[[S_{1}, ..., S_{r}]] such that J_{0} = (S_{1}, ..., S_{r}), J_{n} ⊃ J_{n+1}, ⋂_{n} J_{n} = (0) and for each n there exists a strong J_{n}-structure. Then R →^{∼} T and these rings are complete intersections.

Before proving theorem 5.31 we shall explain how to deduce theorem 3.41 from it. If J is an ideal of O[[S_{1}, ..., S_{r}]] we will use J̄ to denote its image in k[[S_{1}, ..., S_{r}]]. If S given by

O[[S_{1}, ..., S_{r}]]
↓
O[[X_{1}, ..., X_{r}]] → → R′ → → T′
↓  ↓
R  → → T,

is a J-structure for R → T, let 𝔞 = ker (R ⟶ T), and let R′′ denote the ring Im (R′ → (R/𝔪_{R}𝔞 ⊕ (T′/J)) ⊗_{O} k). Then S̄ given by

k[[S_{1}, ..., S_{r}]]
↓
k[[X_{1}, ..., X_{r}]] → → R′′ → → (T′/J) ⊗_{O} k
↓  ↓
(R/𝔪_{R}𝔞) ⊗_{O} k → → T ⊗_{O} k,

is a strong J̄-structure for (R/𝔪_{R}𝔞)⊗_{O}k → T ⊗_{O}k. If J_{n} is a sequence of ideals as in theorem 3.41 then J̄_{n} is a sequence of nested ideals with J̄_{0} = (S_{1}, ..., S_{r}) and ⋂_{n} J̄_{n} = (0). Moreover if for each n there is a J_{n}-structure S_{n} for R → T then for each n there is a strong J̄_{n}-structure S̄_{n} for

(R/𝔪_{R}𝔞) ⊗_{O} k → T ⊗_{O} k.

Then by theorem 5.31 we see that this map is an isomorphism of complete intersections. Lemma 5.29 shows that theorem 3.41 follows. □

Before returning to the proof of theorem 5.31, let us first make some remarks about strong J-structures.

• The set of strong J structures for all ideals J ⊂ (S_{1}, ..., S_{r}) ⊂ k[[S_{1}, ..., S_{r}]] forms a category, with the obvious notion of morphism.

• If S is a strong J-structure and if (S_{1}, ..., S_{r}) ⊃ J′ ⊃ J then there is a natural J′-structure S mod J′ obtained by replacing T′ by T′/J′ and R′ by the image of R′ → R ⊕ (T′/J′).

• If R is a finite dimensional k-vector space and if J has finite index in k[[S_{1}, ..., S_{r}]] then there are only finitely many isomorphism classes of strong J-structure. (This follows because we can bound the order of R′ in any Jstructure. Explicitly we must have #R′ ≤ (#R)(#k[[S_{1}, ..., S_{r}]]/J)^{dim_{k} T}.)

Lemma 5.32 Suppose that R is a finite dimensional k vector space. Suppose also that {J_{n}} is a nested (decreasing) sequence of ideals and that J = ⋂_{n} J_{n}. If for each n a strong J_{n} structure exists then a strong J structure exists.

Proof: We may suppose that each J_{n} has finite index in k[[S_{1}, ..., S_{r}]]. Let S_{n} denote a strong J_{n}-structure. Let S_{n,m} = S_{n} mod J_{m} if m ≤ n. Because there are only finitely many isomorphism classes of strong J_{m} structure, we may recursively choose integers n(m) such that

• S_{n(m),m} ≅ S_{n,m} for infinitely many n,

• if m > 1 then S_{n(m),m−1} ≅ S_{n(m−1),m−1}.

Let S′_{m} = S_{n(m),m}. Then S′_{m} is a strong J_{m} structure and if m ≥ m_{1} then S′_{m} mod J_{m_{1}} ≅ S′_{m_{1}}. One checks that S = lim_{←} S′_{m} is the desired strong Jstructure. □

Lemma 5.33 Suppose that a strong (0) structure exists. Then the map R ⟶ T is an isomorphism, and these rings are complete intersections.

Proof: Because k[[S_{1}, ..., S_{r}]] ↪ T′ (and T′ is a finitely generated k[[S_{1}, ..., S_{r}]]- module by Nakayama’s lemma, cf. [Mat], thm. 8.4) we see that the Krull dimension of T′ is at least r. On the other hand k[[X_{1}, ..., X_{r}]] → → T′ and so by Krull’s principal ideal theorem this map must be an isomorphism. Thus

k[[X_{1}, ..., X_{r}]] →^{∼} R′ →^{∼} T′.

Hence we have that

k[[X_{1}, ..., X_{r}]]/(S_{1}, ..., S_{r}) →^{∼} R′/(S_{1}, ..., S_{r}) →^{∼} T,

and the lemma follows. □ Theorem 5.31 follows at once from these two lemmas. □

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