The Modular Degree, Congruence Primes and Multiplicity One

Amod Agashe  Kenneth A. Ribet  William A. Stein

Abstract.

The modular degree and congruence number are two fundamental invariants of an elliptic curve over the rational field. Frey and Müller have asked whether these invariants coincide. Although this question has a negative answer, we prove a theorem about the relation between the two invariants: one divides the other, and the ratio is divisible only by primes whose squares divide the conductor of the elliptic curve. We discuss the ratio even in the case where the square of a prime does divide the conductor, and we study analogues of the two invariants for modular abelian varieties of arbitrary dimension.

1 Introduction

Let E be an elliptic curve over Q. By [BCDT01], we may view E as an abelian variety quotient over Q of the modular Jacobian J_{0}(N), where N is the conductor of E. After possibly replacing E by an isogenous curve, we may assume that the kernel of the map J_{0}(N) → E is connected, i.e., that E is an optimal quotient of J_{0}(N).

Let f_{E} = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) be the newform attached to E. The congruence number r_{E} of E is the largest integer such that there is an element g = ∑ b_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) with integer Fourier coefficients b_{n} that is orthogonal to f_{E} with respect to the Peterson innner product, and congruent to f_{E} modulo r_{E} (i.e., a_{n} ≡ b_{n} (mod r_{E}) for all n). The modular degree m_{E} is the degree of the composite map X_{0}(N) → J_{0}(N) → E, where we map X_{0}(N) to J_{0}(N) by sending P ∈ X_{0}(N) to [P] − [∞] ∈ J_{0}(N).

Section 2 is about relations between r_{E} and m_{E}. For example, m_{E} | r_{E}. In [FM99, Q. 4.4], Frey and Müller asked whether r_{E} = m_{E}. We give examples in which r_{E} ≠ m_{E}, then conjecture that for any prime p, ord_{p}(r_{E}/m_{E}) ≤ \frac{1}{2} ord_{p}(N). We prove this conjecture when ord_{p}(N) ≤ 1.

In Section 3, we consider analogues of congruence primes and the modular degree for optimal quotients that are not necessarily elliptic curves; these are quotients of J_{0}(N) and J_{1}(N) of any dimension associated to ideals of the relevant Hecke algebras. In Section 4 we prove the main theorem of this paper, and in Section 5 we give some new examples of failure of multiplicity one motivated by the arguments in Section 4.

Acknowledgment. The authors are grateful to A. Abbes, R. Coleman, B. Conrad, J. Cremona, H. Lenstra, E. de Shalit, B. Edixhoven, L. Merel, and R. Taylor for several discussions and advice regarding this paper.

2 Congruence Primes and the Modular Degree

Let N be a positive integer and let X_{0}(N) be the modular curve over Q that classifies isomorphism classes of elliptic curves with a cyclic subgroup of order N. The Hecke algebra T of level N is the subring of the ring of endomorphisms of J_{0}(N) = Jac(X_{0}(N)) generated by the Hecke operators T_{n} for all n ≥ 1. Let f be a newform of weight 2 for Γ_{0}(N) with integer Fourier coefficients, and let I_{f} be kernel of the homomorphism T → Z[…, a_{n}(f), …] that sends T_{n} to a_{n}. Then the quotient E = J_{0}(N)/I_{f}J_{0}(N) is an elliptic curve over Q. We call E the optimal quotient associated to f. Composing the embedding X_{0}(N) ↪ J_{0}(N) that sends ∞ to 0 with the quotient map J_{0}(N) → E, we obtain a surjective morphism of curves ϕ_{E} : X_{0}(N) → E.

Definition 2.1. The modular degree m_{E} of E is the degree of ϕ_{E}.

Congruence primes have been studied by Doi, Hida, Ribet, Mazur and others (see, e.g., [Rib83, §1]), and played an important role in Wiles’s work [Wil95] on Fermat’s last theorem. Frey and Mai-Murty have observed that an appropriate asymptotic bound on the modular degree is equivalent to the abc-conjecture (see [Fre97, p.544] and [Mur99, p.180]). Thus, results that relate congruence primes and the modular degree are of great interest.

Theorem 2.2. Let E be an elliptic curve over Q of conductor N, with modular degree m_{E} and congruence number r_{E}. Then m_{E} | r_{E} and if ord_{p}(N) ≤ 1 then ord_{p}(r_{E}) = ord_{p}(m_{E}).

We will prove a generalization of Theorem 2.2 in Section 4 below.

The divisibility m_{E} | r_{E} was first discussed in [Zag85, Th. 3], where it is attributed to the second author (Ribet); however in [Zag85] the divisibility was mistakenly written in the opposite direction. For some other expositions of the proof, see [AU96, Lem 3.2] and [CK04]. We generalize this divisibility in Proposition 4.5. The second part of Theorem 2.2, i.e., that if ord_{p}(N) ≤ 1 then ord_{p}(r_{E}) = ord_{p}(m_{E}), follows from the more general Theorem 3.7 below. Note that [AU96, Prop. 3.3–3.4] implies the weaker statement that if p ∤ N then ord_{p}(r_{E}) = ord_{p}(m_{E}), since [AU96, Prop. 3.3] implies

ord_{p}(r_{E}) − ord_{p}(m_{E}) = ord_{p}(#C) − ord_{p}(c_{E}) − ord_{p}(#D),

Table 1: Differing Modular Degree and Congruence Number

Curve m_{E} r_{E}
54B1 2  6
64A1 2  4
72A1 4  8
80A1 4  8
88A1 8  16
92B1 6  12
96A1 4  8
96B1 4  8

Curve m_{E} r_{E}
99A1 4  12
108A1 6  18
112A1 8  16
112B1 4  8
112C1 8  16
120A1 8  16
124A1 6  12
126A1 8  24

Curve m_{E} r_{E}
128A1 4  32
128B1 8  32
128C1 4  32
128D1 8  32
135A1 12 36
144A1 4  8
144B1 8  16

and by [AU96, Prop. 3.4] ord_{p}(#C) = 0. (Here c_{E} is the Manin constant of E, which is an integer by results of Edixhoven and Katz-Mazur; see e.g., [ARS06] for more details.)

Frey and Müller [FM99, Ques. 4.4] asked whether r_{E} = m_{E} in general. After implementing an algorithm to compute r_{E} in Magma [BCP97], we quickly found that the answer is no. The counterexamples at conductor N ≤ 144 are given in Table 1, where the curve is given using the notation of [Cre97]:

For example, the elliptic curve 54B1, given by the equation y^{2} + xy + y = x^{3} − x^{2} + x − 1, has r_{E} = 6 and m_{E} = 2. To see explicitly that 3 | r_{E}, observe that the newform corresponding to E is f = q + q^{2} + q^{4} − 3q^{5} − q^{7} + ⋯ and the newform corresponding to X_{0}(27) if g = q − 2q^{4} − q^{7} + ⋯, so g(q) + g(q^{2}) appears to be congruent to f modulo 3. To prove this congruence, we checked it for 18 Fourier coefficients, where the sufficiency of precision to degree 18 was determined using [Stu87].

In our computations, there appears to be no absolute bound on the p that occur. For example, for the curve 242B1 of conductor N = 2 · 11^{2} we have^{1}

m_{E} = 2^{4} ≠ r_{E} = 2^{4} · 11.

We propose the following replacement for Question 4.4 of [FM99]:

Conjecture 2.3. Let E be an optimal elliptic curve of conductor N and p be any prime. Then

ord_{p} (\frac{r_{E}}{m_{E}} ) ≤ \frac{1}{2} ord_{p}(N).

We verified Conjecture 2.3 using Magma for every optimal elliptic curve quotient of J_{0}(N), with N ≤ 539.

If p ≥ 5 then ord_{p}(N) ≤ 2, so a special case of the conjecture is

ord_{p} (\frac{r_{E}}{m_{E}} ) ≤ 1 for any p ≥ 5.

1The curve 242a1 in “modern notation.”

Remark 2.4. It is often productive to parametrize elliptic curves by X_{1}(N) instead of X_{0}(N) (see, e.g., [Ste89] and [Vat05]). Suppose E is an optimal quotient of X_{1}(N), let m′_{E} be the degree of the modular parametrization, and let r′_{E} be the Γ_{1}(N)-congruence number, which is defined as above but with S_{2}(Γ_{0}(N)) replaced by S_{2}(Γ_{1}(N)). For the optimal quotient of X_{1}(N) isogenous to 54B1, we find using Magma that m′_{E} = 18 and r′_{E} = 6. Thus the equality m′_{E} = r′_{E} fails, and the analogous divisibility m′_{E} | r′_{E} no longer holds. Also, for a curve of conductor 38 we have m′_{E} = 18 and r′_{E} = 6, so equality need not hold even if the level is square free. We hope to investigate this in a future paper.

3 Modular abelian varieties of arbitrary dimension

For N ≥ 4, let Γ be a fixed choice of either Γ_{0}(N) or Γ_{1}(N), let X be the modular curve over Q associated to Γ, and let J be the Jacobian of X. Let I be a saturated ideal of the corresponding Hecke algebra T ⊂ End(J), so T/I is torsion free. Then A = A_{I} = J/IJ is an optimal quotient of J since IJ is an abelian subvariety.

Definition 3.1. If f = ∑ a_{n}(f)q^{n} ∈ S_{2}(Γ) and I_{f} = ker(T → Z[…, a_{n}(f), …]), then A = A_{f} = J/I_{f}J is the newform quotient associated to f. It is an abelian variety over Q of dimension equal to the degree of the field Q(…, a_{n}(f), …).

In this section, we generalize the notions of the congruence number and the modular degree to quotients A = A_{I}, and state a theorem relating the two numbers, which we prove in Sections 4.1–4.2.

Let ϕ_{2} denote the quotient map J → A. By Poincare reducibility over Q there is a unique abelian subvariety A^{∨} of J that projects isogenously to the quotient A (equivalently, which has finite intersection with ker(ϕ_{2})), and so by Hecke equivariance of J → A it follows that A^{∨} is T-stable. Let ϕ be the composite isogeny

ϕ : A^{∨} ⟶^{ϕ_{1}} J ⟶^{ϕ_{2}} A.

Remark 3.2. Note that A^{∨} is the dual abelian variety of A. More generally, if C is any abelian variety, let C^{∨} denote the dual of C. There is a canonical principal polarization J ≅ J^{∨}, and dualizing ϕ_{2}, we obtain a map ϕ_{2}^{∨} : A^{∨} → J^{∨}, which we compose with θ^{−1} : J^{∨} ≅ J to obtain a map ϕ_{1} : A^{∨} → J. Note also that φ is a polarization (induced by pullback of the theta divisor).

The exponent of a finite group G is the smallest positive integer n such that every element of G has order dividing n.

Definition 3.3. The modular exponent of A is the exponent of the kernel of the isogeny ϕ, and the modular number of A is the degree of ϕ.

We denote the modular exponent of A by ñ_{A} and the modular number by n_{A}. When A is an elliptic curve, the modular exponent is equal to the modular degree of A, and the modular number is the square of the modular degree (see, e.g., [AU96, p. 278]).

If R is a subring of C, let S_{2}(R) = S_{2}(Γ; R) denote the subgroup of S_{2}(Γ) consisting of cups forms whose Fourier expansions at the cusp ∞ have coefficients in R. (Note that Γ is fixed for this whole section.) Let S_{2}(Γ; Z)[I]^{⊥} denote the orthogonal complement of S_{2}(Γ; Z)[I] in S_{2}(Γ; Z) with respect to the Petersson inner product.

The following is well known, but we had difficulty finding a good reference.

Proposition 3.4. The group S_{2}(Γ; Z) is of finite rank as a Z-module.

Proof. Using the standard pairing between T and S_{2}(Γ, Z) (see also [Rib83, Theorem 2.2]) we see that S_{2}(Γ, Z) ≅ Hom(T, Z). Thus S_{2}(Γ, Z) is finitely generated over Z if and only if T is finitely generated over Z. But the action of T on H_{1}(J, Z) is a faithful representation that embeds T into Mat_{2d}(Z) ≅ Z^{(2d)^{2}}. But Z is Noetherian, so T is finitely generated over Z.

Definition 3.5. The exponent of the quotient group

\frac{S_{2}(Γ; Z)}{S_{2}(Γ; Z)[I] + S_{2}(Γ; Z)[I]^{⊥}}  (1)

is the congruence exponent r̃_{A} of A and its order is the congruence number r_{A}.

Remark 3.6. Note that S_{2}(Γ, Z) ⊗_{Z} R = S_{2}(Γ, R); see, e.g., the discussion in [DI95, §12]. Thus the analogue of Definition 3.5 with Z replaced by an algebraic integer ring (or even \overline{Z}) gives a torsion module whose annihilator ideal meets Z in the ideal generated by the congruence exponent.

Our definition of r_{A} generalizes the definition in Section 2 when A is an elliptic curve (see [AU96, p. 276]), and the following generalizes Theorem 2.2:

Theorem 3.7. If f ∈ S_{2}(C) is a newform, then

(a) We have ñ_{A_{f}} | r̃_{A_{f}}, and

(b) If p^{2} ∤ N, then ord_{p}(r̃_{A_{f}}) = ord_{p}(ñ_{A_{f}}).

Remark 3.8. When A_{f} is an elliptic curve, Theorem 3.7 implies that the modular degree divides the congruence number (since for an elliptic curve the modular degree and modular exponent are the same), i.e., \sqrt{n_{A_{f}}} | r_{A_{f}}. In general, the divisibility n_{A_{f}} | r_{A_{f}}^{2} need not hold. For example, there is a newform of degree 24 in S_{2}(Γ_{0}(431)) such that

n_{A_{f}} = (2^{11} · 6947)^{2} ∤ r_{A_{f}}^{2} = (2^{10} · 6947)^{2}.

Note that 431 is prime and mod 2 multiplicity one fails for J_{0}(431) (see [Kil02]).

4 Proof of the Main Theorem

In this section we prove Theorem 3.7. We continue using the notation introduced so far.

4.1 Proof of Theorem 3.7 (a)

We begin with a remark about compatibilities. In general, the polarization of J induced by the theta divisor need not be Hecke equivariant, because if T is a Hecke operator on J, then on J^{∨} it acts as W_{N}T W_{N}, where W_{N} is the Atkin-Lehner involution (see e.g., [DI95, Rem. 10.2.2]). However, on J^{new} the action of the Hecke operators commutes with that of W_{N}, so if the quotient map J → A factors through J^{new}, then the Hecke action on A^{∨} induced by the embedding A^{∨} → J^{∨} and the action on A^{∨} induced by ϕ_{1} : A^{∨} → J are the same. Hence A^{∨} is isomorphic to ϕ_{1}(A^{∨}) as a T-module.

Recall that f is a newform, I_{f} = Ann_{T}(f), and J = J_{0}(N). Let B = I_{f}J, so that A^{∨} + B = J, and J/B ≅ A. The following lemma is well known, but we prove it here for the convenience of the reader.

Lemma 4.1. Hom_{Q}(A^{∨}, B) = 0.

Proof. Pick a prime ℓ. Then \overline{Q}_{ℓ}⊗V_{ℓ}(J)^{ss} as a \overline{Q}_{ℓ}[G_{Q}]-module is a direct sum of copies of the representations ρ_{g} as g ranges through all normalized eigenforms of weight 2 and level N with coefficients in \overline{Q}; by a well-known result of the second author, these representations are absolutely irreducible. Now since f is a newform and A^{∨} → A is an isogeny, \overline{Q}_{ℓ} ⊗ V_{ℓ}(A^{∨})^{ss} is a direct sum of copies of ρ_{σ(f)} as σ ranges over all embeddings of K_{f} into \overline{Q}. Thus, by the analytic theory of multiplicity one (see [Li75, Cor. 3, pg. 300]), the Galois modules V_{ℓ}(A^{∨}) and V_{ℓ}(B) = V_{ℓ}(J)/V_{ℓ}(A^{∨}) share no common Jordan-Hölder factors even when coefficients are extended to \overline{Q}_{ℓ}, so Hom_{Q}(A′, B) = 0.

Let T_{1} be the image of T in End(A^{∨}), and let T_{2} be the image of T in End(B). We have the following commutative diagram with exact rows:

\xymatrix{0 \ar[r] & T \ar[r] \ar[d] & T_{1} ⊕ T_{2} \ar[r] \ar[d] & \frac{T_{1} ⊕ T_{2}}{T} \ar@{.>}[d] \ar[r] & 0 \\ 0 \ar[r] & End(J) \ar[r] & End(A^{∨}) ⊕ End(B) \ar[r] & \frac{End(A^{∨}) ⊕ End(B)}{End(J)} \ar[r] & 0.}

(2)

Let

e = (1, 0) ∈ T_{1} ⊕ T_{2},

and let e_{1} and e_{2} denote the images of e in the groups (T_{1} ⊕ T_{2})/T and (End(A^{∨}) ⊕ End(B))/End(J), respectively. It follows from Lemma 4.1 that the two quotient groups on the right hand side of (2) are finite, so e_{1} and e_{2} have finite order. Note that because e_{2} is the image of e_{1}, the order of e_{2} is a divisor of the order of e_{1}.

The denominator of any φ ∈ End(J) ⊗ Q is the smallest positive integer n such that nφ ∈ End(J).

Let π_{A^{∨}}, π_{B} ∈ End(J) ⊗ Q be projection onto A^{∨} and B, respectively. Note that the denominator of π_{A^{∨}} equals the denominator of π_{B}, since π_{A^{∨}} +π_{B} = 1_{J}, so that π_{B} = 1_{J} − π_{A^{∨}}.

Lemma 4.2. The element e_{2} ∈ (End(A^{∨}) ⊕ End(B))/End(J) defined above has order ñ_{A}.

Proof. Let n be the order of e_{2}, so n is the denominator of π_{A^{∨}}, which, as mentioned above, is also the denominator of π_{B}. We want to show that n is equal to ñ_{A}, the exponent of A^{∨} ∩ B.

Let i_{A^{∨}} and i_{B} be the embeddings of A^{∨} and B into J, respectively. Then

φ = (nπ_{A^{∨}}, nπ_{B}) ∈ Hom(J, A^{∨} × B)

and φ ∘ (i_{A^{∨}} + i_{B}) = [n]_{A^{∨}×B}. We have an exact sequence

0 → A^{∨} ∩ B ⟶^{x↦(x,−x)} A^{∨} × B ⟶^{i_{A^{∨}} +i_{B}} J → 0.

Let Δ be the image of A^{∨} ∩ B. Then by exactness,

[n]Δ = (φ ∘ (i_{A^{∨}} + i_{B}))(Δ) = φ ∘ ((i_{A^{∨}} + i_{B})(Δ)) = φ({0}) = {0},

so n is a multiple of the exponent ñ_{A} of A^{∨} ∩ B.

To show the opposite divisibility, consider the commutative diagram

\xymatrix{0 \ar[r] & A^{∨} ∩ B \ar[r]^{x↦(x,−x)} \ar[d]^{[ñ_{A}]} & A^{∨} × B \ar[d]^{([ñ_{A}],0)} \ar[r] & J \ar[r] \ar@{.>}[d]^{ψ} & 0 \\ 0 \ar[r] & A^{∨} ∩ B \ar[r]^{x↦(x,−x)} & A^{∨} × B \ar[r] & J \ar[r] & 0,}

where the middle vertical map is (a, b) ↦ (ñ_{A}a, 0) and the map ψ exists because [ñ_{A}](A^{∨} ∩ B) = 0. But ψ = ñ_{A}π_{A^{∨}} in End(J) ⊗ Q. This shows that ñ_{A}π_{A^{∨}} ∈ End(J), i.e., that ñ_{A} is a multiple of the denominator n of π_{A^{∨}}.

Let Ext^{1} = Ext_{Z}^{1} denote the first Ext functor in the category of Z-modules.

Lemma 4.3. The group (T_{1} ⊕ T_{2})/T is isomorphic to the quotient (1) in Definition 3.5, so r_{A} = #((T_{1} ⊕ T_{2})/T) and r̃_{A} is the exponent of (T_{1} ⊕ T_{2})/T. More precisely, Ext^{1}((T_{1} ⊕ T_{2})/T, Z) is isomorphic as a T-module to the quotient (1).

Proof. Apply the Hom(−, Z) functor to the first row of (2) to obtain a threeterm exact sequence

0 → Hom(T_{1} ⊕ T_{2}, Z) → Hom(T, Z) → Ext^{1}((T_{1} ⊕ T_{2})/T, Z) → 0. (3)

There is a T-equivariant bilinear pairing T × S_{2}(Z) → Z given by (t, g) ↦ a_{1}(t(g)), which is perfect by [AU96, Lemma 2.1] (see also [Rib83, Theorem 2.2]). Using this pairing, we transform (3) into an exact sequence

0 → S_{2}(Z)[I_{f}] ⊕ S_{2}(Γ; Z)[I_{f}]^{⊥} → S_{2}(Z) → Ext^{1}((T_{1} ⊕ T_{2})/T, Z) → 0

of T-modules. Here we use that Hom(T_{2}, Z) is the unique saturated Heckestable complement of S_{2}(Z)[I_{f}] in S_{2}(Z), hence must equal S_{2}(Z)[I_{f}]^{⊥}. Finally note that if G is any finite abelian group, then Ext^{1}(G, Z) ≈ G as groups, which gives the desired result.

Lemma 4.4. The element e_{1} ∈ (T_{1} ⊕ T_{2})/T has order r̃_{A}.

Proof. By Lemma 4.3, the lemma is equivalent to the assertion that the order r of e_{1} equals the exponent of M = (T_{1} ⊕ T_{2})/T. Since e_{1} is an element of M, the exponent of M is divisible by r.

To obtain the reverse divisibility, consider any element x of M. Let (a, b) ∈ T_{1} ⊕ T_{2} be such that its image in M is x. By definition of e_{1} and r, we have (r, 0) ∈ T, and since 1 = (1, 1) ∈ T, we also have (0, r) ∈ T. Thus (Tr, 0) and (0, Tr) are both subsets of T (i.e., in the image of T under the map T → T_{1} ⊕ T_{2}), so r(a, b) = (ra, rb) = (ra, 0) + (0, rb) ∈ T. This implies that the order of x divides r. Since this is true for every x ∈ M, we conclude that the exponent of M divides r.

Proposition 4.5. If f ∈ S_{2}(C) is a newform, then ñ_{A_{f}} | r̃_{A_{f}}.

Proof. Since e_{2} is the image of e_{1} under the right-most vertical homomorphism in (2), the order of e_{2} divides that of e_{1}. Now apply Lemmas 4.2 and 4.4.

This finishes the proof of the first statement in Theorem 3.7.

4.2 Proof of Theorem 3.7 (b)

Let T′ be the saturation of T = Z[…, T_{n}, …] in End(J_{0}(N)), i.e., the set of elements of End(J_{0}(N))⊗Q some positive multiple of which lie in T. The quotient T′/T is a finitely generated abelian group because both T and End(J_{0}(N)) are finitely generated over Z. Since T′/T is also a torsion group, it is finite.

In Section 4.2.1, we will give some conditions under which T and T′ agree locally at maximal ideal of T. In Section 4.2.2, we will explain how the ratio of the congruence number to the modular degree is closely related to the order of T′/T, and finally deduce that this ratio is 1 (for quotients associated to newforms) locally at a prime p such that p^{2} ∤ N.

4.2.1 Multiplicity One

Fixt an integer N and a prime p | N. Suppose for a moment that N is prime, so p = N. In [Maz77], Mazur proves that T = T′; he combines this result with the equality

T ⊗ Q = End(J_{0}(p)) ⊗ Q,

to deduce that T = End(J_{0}(p)). This result, combined with Ribet’s result [Rib75] or [Rib81] to the effect that T ⊗ Q = (End\overline{Q}J_{0}(N)) ⊗ Q, shows that T is the full ring of endomorphisms of J_{0}(N) over \overline{Q}. When N is no longer necessarily prime, the method of [Maz77] shows that T and T′ agree locally at a maximal ideal 𝔪 of T that satisfies a simple condition involving differentials form mod ℓ, where ℓ is the residue characteristic of 𝔪.

For the sake of completeness, we state and prove a lemma that can be easily extracted from [Maz77]. Let m be the largest square dividing N and let R = Z[\frac{1}{m}]. Let X_{0}(N)_{R} denote the minimal regular model of X_{0}(N) over R. Let Ω = Ω_{X_{0}(N)/R} denote the sheaf of regular differentials on X_{0}(N)_{R}, as in [Maz78, §2(e)]. If ℓ is a prime such that ℓ^{2} ∤ N, then X_{0}(N)_{F_{ℓ}} denotes the special fiber of X_{0}(N)_{R} at the prime ℓ.

Lemma 4.6 (Mazur). Let 𝔪 be a maximal ideal of T of residue characteristic ℓ such that ℓ^{2} ∤ N. Suppose that

dim_{T/𝔪} H^{0}(X_{0}(N)_{F_{ℓ}}, Ω)[𝔪] ≤ 1.

Then T and T′ agree locally at 𝔪.

Proof. Let M denote the group H^{1}(X_{0}(N)_{R}, O_{X_{0}(N)}), where O_{X_{0}(N)} is the structure sheaf of X_{0}(N). As explained in [Maz77, p. 95], we have an action of End_{Q}J_{0}(N) on M, and the action of T on M via the inclusion T ⊆ End_{Q}J_{0}(N) is faithful, so likewise for the action by T′. Hence we have an injection ϕ : T′ ↪ End_{T}M. Suppose 𝔪 is a maximal ideal of T that satisfies the hypotheses of the lemma. To prove that T_{𝔪} = T′_{𝔪} it suffices to prove the following claim:

Claim: The map ϕ|_{T} is surjective locally at 𝔪.

Proof. By Nakayama’s lemma, to show that M is generated as a single element over T locally at 𝔪, it suffices to check that the dimension of the T/𝔪 -vector space M/𝔪M is at most one. Since ℓ^{2} ∤ N, M/𝔪M is dual to H^{0}(X_{0}(N)_{F_{ℓ}}, Ω)[𝔪] (see, e.g., [Maz78, §2]). Since we are assuming that dim_{T/𝔪}H^{0}(X_{0}(N)_{F_{ℓ}}, Ω)[𝔪] ≤ 1, we have dim_{T/𝔪}(M/𝔪M) ≤ 1, which proves the claim.

If 𝔪 is a maximal ideal of the Hecke algebra T of residue characteristic ℓ, we say that 𝔪 satisfies multiplicity one for differentials if

dim(H^{0}(X_{0}(N)_{F_{ℓ}}, Ω)[𝔪]) ≤ 1.

By Lemma 4.6, multiplicity one for H^{0}(X_{0}(N)_{F_{ℓ}}, Ω)[𝔪] implies that T and T′ agree at 𝔪.

There is quite a bit of literature on the question of multiplicity 1 for H^{0}(X_{0}(N)_{F_{ℓ}}, Ω)[𝔪]. The easiest case is that ℓ is prime to the level N:

Lemma 4.7 (Mazur). If 𝔪 is a maximal ideal of T of residue characteristic ℓ such that ℓ ∤ N, then

dim_{T/𝔪} H^{0}(X_{0}(N)_{F_{ℓ}}, Ω)[𝔪] ≤ 1.

Proof. Mazur deduces this lemma from injectivity of the q-expansion map. The reader may find the following alternative approach to part of the argument easier to follow than the one on p. 95 of [Maz77]. We have an F_{ℓ}-vector space that embeds in F_{ℓ}[[q]], for example a space V of differentials that is killed by a maximal ideal 𝔪. This space is a T/𝔪-vector space, and we want to see that its dimension over T/𝔪 is at most 1. Mazur invokes tensor products and eigenvectors; alternatively, we note that V embeds in Hom_{F_{ℓ}}(T/𝔪, F_{ℓ}) via the standard duality that sends v ∈ V to the linear form whose value on a Hecke operator T is the qth coefficient of v|T. The group Hom_{F_{ℓ}}(T/𝔪, F_{ℓ}) has the same size as T/𝔪, which completes the argument because Hom_{F_{ℓ}}(T/𝔪, F_{ℓ}) has dimension 1 as a T/𝔪-vector space.

In the context of Mazur’s paper, where the level N is prime, we see from Lemma 4.7 that T and T′ agree away from N. Locally at N, Mazur proved that T = T′ by an analogue of the arguments that he used away from N; see Chapter II of [Maz77] (and especially Prop. 9.4 and 9.5 of that chapter) as well as [MR91], where these arguments are taken up in a context where the level is no longer necessarily prime (and where one works locally at a prime whose square does not divide the level). Thus in the prime level case, T = T′, as we asserted above.

Now let p be a prime such that p ∥ N, and let M = N/p. The question of multiplicity 1 at p for H^{0}(X_{0}(pM)_{F_{p}}, Ω)[𝔪] is discussed in [MR91], where the authors establish multiplicity 1 for maximal ideals 𝔪 | p for which the associated mod p Galois representation is irreducible and not p-old. (A representation of level pM is p-old if it arises from S_{2}(Γ_{0}(M)).)

If 𝔪 is a maximal ideal of T of residue characteristic ℓ, then we say that 𝔪 is ordinary if T_{ℓ} ∉ 𝔪 (note that T_{ℓ} is often denoted U_{ℓ} if ℓ | N). For our purposes, the following lemma is convenient:

Lemma 4.8 (Wiles). If 𝔪 is an ordinary maximal ideal of T of characteristic p, then

dim_{T/𝔪} H^{0}(X_{0}(pM)_{F_{p}}, Ω)[𝔪] ≤ 1.

This is essentially Lemma 2.2 in [Wil95, pg. 485]; we make a few comments about how it applies on our situation:

1. Wiles considers X_{1}(M, p) instead of X_{0}(pM), which means that he is using Γ_{1}(M)-structure instead of Γ_{0}(M)-structure. This surely has no relevance to the issue at hand.

2. Wiles assumes (on page 480) that p is an odd prime, but again this assumption is not relevant to our question.

3. The condition that 𝔪 is ordinary does not appear explicitly in the statement of Lemma 2.2 in [Wil95]; instead it is a reigning assumption in the context of his discussion.

4. We see by example that Wiles’s “ordinary” assumption is less stringent than the assumption in [MR91]; note that [MR91] rule out cases where 𝔪 is both old and new at p, whereas Wiles is happy to include such cases. (On the other hand, Wiles’s assumption is certainly nonempty, since it rules out maximal ideals 𝔪 that arise from non-ordinary (old) forms of level M. Here is an example with p = 2 and M = 11, so N = 22: There is a unique newform f = ∑ a_{n}q^{n} of level 11, and T = Z[T_{2}] ⊂ End(J_{0}(22)), where T_{2}^{2} − a_{2}T_{2} + 2 = 0. Since a_{2} = −2, we have T ≅ Z[\sqrt{−1}]. We can choose the square root of −1 to be T_{2} + 1. Then T_{2} is a generator of the unique maximal ideal 𝔪 of T with residue characteristic 2, and this maximal ideal is not ordinary.)

We now summarize the conclusions we can make from the lemmas so far:

Proposition 4.9. The modules T and T′ agree locally at each maximal ideal 𝔪 that is either prime to N or that satisfies the following supplemental hypothesis: the residue characteristic of 𝔪 divides N only to the first power and 𝔪 is ordinary.

Proof. This follows easily from Lemmas 4.6, 4.7, and 4.8.

In Mazur’s original context, where the level N is prime, we have T_{N}^{2} = 1 because there are no forms of level 1. Accordingly, each 𝔪 dividing N is ordinary, and we recover Mazur’s equality T = T′ in this special case.

4.2.2 Degrees and Congruences

Let e ∈ T ⊗ Q be as in Section 4.1, and let p, N, M be as before Lemma 4.8. The image of e in J_{0}(pM) is the T-stable abelian subvariety denoted A^{∨} in Section 4.1, but since we shall now exclusively work with this subvariety rather than the corresponding optimal quotient of J_{0}(pM) (which was denoted A earlier), we will now write A to denote the image of e (without risk of confusion). We also write B to denote the unique T-stable abelian subvariety of J_{0}(pM) complementary to A.

For t ∈ T, let t_{A} be the restriction of t to A, and let t_{B} be the image of t in End(B). Let T_{A} be the subgroup of End(A) consisting of the various t_{A}, and define T_{B} similarly. As before, we obtain an injection j : T ↪ T_{A} × T_{B} with finite cokernel. Because j is an injection, we refer to the maps π_{A} : T → T_{A} and π_{B} : T → T_{B}, given by t ↦ t_{A} and t ↦ t_{B}, respectively, as “projections”.

Definition 4.10. The congruence ideal associated with the projector e is I = π_{A}(ker(π_{B})) ⊂ T_{A}.

Viewing T_{A} as T_{A} × {0}, we may view T_{A} as a subgroup of T ⊗ Q ≅ (T_{A} × T_{B}) ⊗ Q. Also, we may view T as embedded in T_{A} × T_{B}, via the map j.

Lemma 4.11. We have I = T_{A} ∩ T.

A larger ideal of T_{A} is J = Ann_{T_{A}}(A ∩ B); it consists of restrictions to A of Hecke operators that vanish on A ∩ B.

Lemma 4.12. We have I ⊂ J.

Proof. The image in T_{A} of an operator that vanishes on B also vanishes on A ∩ B.

Lemma 4.13. We have J = T_{A} ∩ End(J_{0}(pM)) = T_{A} ∩ T′.

Proof. This is elementary; it is an analogue of Lemma 4.11.

Proposition 4.14. There is a natural inclusion J/I ↪ T′/T of T-modules.

Proof. Consider the map T → T ⊗ Q given by t ↦ te. This homomorphism factors through T_{A} and yields an injection ι_{A} : T_{A} ↪ T ⊗ Q. Symmetrically, we also obtain ι_{B} : T_{B} ↪ T ⊗ Q. The map (t_{A}, t_{B}) ↦ ι_{A}(t_{A}) + ι_{B}(t_{B}) is an injection T_{A} × T_{B} ↪ T ⊗ Q. The composite of this map with the inclusion j : T ↪ T_{A} × T_{B} defined above is the natural map T ↪ T ⊗ Q. We thus have a sequence of inclusions

T ↪ T_{A} × T_{B} ↪ T ⊗ Q ⊂ End(J_{0}(pM)) ⊗ Q.

By Lemma 4.11 and Lemma 4.13, we have I = T_{A} ∩ T and J = T_{A} ∩ T′. Thus I = J ∩ T, where the intersection is taken inside T′. Thus

J/I = J/(J ∩ T) ≅ (J + T)/T ↪ T′/T.

Corollary 4.15. If 𝔪 is a maximal ideal not in Supp_{T}(T′/T), then 𝔪 is not in the support of J/I, i.e., if T and T′ agree locally at 𝔪, then I and J also agree locally at 𝔪.

Note that the Hecke algebra T acts on J/I through its quotient T_{A}, since the action of T on I and on J factors through this quotient.

Now we specialize to the case where A is ordinary at p, in the sense that the image of T_{p} in T_{A}, which we denote T_{p,A}, is invertible modulo every maximal ideal of T_{A} that divides p. (This case occurs when A is a subvariety of the p-new subvariety of J_{0}(pM), since the square of T_{p,A} is the identity.)

If 𝔪 | p is a maximal ideal of T that arises by pullback from a maximal ideal of T_{A}, then 𝔪 is ordinary in the sense used above. When A is ordinary at p, it follows from Proposition 4.9 and Corollary 4.15 that I = J locally at p. The reason is simple: regarding I and J as T_{A}-modules, we realize that we need to test that I = J at maximal ideals of T_{A} that divide p. These ideals correspond to maximal ideals 𝔪 | p of T that are automatically ordinary, so we have I = J locally at 𝔪 because of Proposition 4.9. By Proposition 4.9, we have T = T′ locally at primes away from the level pM. Thus we conclude that I = J locally at all primes ℓ ∤ pM and also at p, a prime that divides the level pM exactly once.

Suppose, finally, that A is the abelian variety associated to a newform f of level pM. The ideal I ⊂ T_{A} measures congruences between f and the space of forms in S_{2}(Γ_{0}(pM)) that are orthogonal to the space generated by f. Also, A ∩ B is the kernel in A of the map “multiplication by the modular element e”. In this case, the inclusion I ⊂ J corresponds to the divisibility ñ_{A} | r̃_{A}, and we have equality at primes at which I = J locally. We conclude that the congruence exponent and the modular exponent agree both at p and at primes not dividing pM, which completes our proof of Theorem 3.7(b).

Remark 4.16. The ring

R = End(J_{0}(pM)) ∩ (T_{A} × T_{B})

is often of interest, where the intersection is taken in End(J_{0}(pM)) ⊗ Q. We proved above that there is a natural inclusion J/I ↪ T′/T. This inclusion yields an isomorphism J/I ⟶^{∼} R/T. Indeed, if (t_{A}, u_{B}) is an endomorphism of J_{0}(pM), where t, u ∈ T, then (t_{A}, u_{B}) − u = (t_{A}, 0) is an element of J. The ideals I and J are equal to the extent that the rings T and R coincide. Even when T′ is bigger than T, its subring R may be not far from T.

5 Failure of Multiplicity One

In this section, we discuss examples of failure of multiplicity one (in two different but related senses). The notion of multiplicity one, originally due to Mazur [Maz77], has played an important role in several places (e.g., in Wiles’s proof of Fermat’s last theorem [Wil95]). This notion is closely related to Gorensteinness of certain Hecke algebras (e.g., see [Til97]). Kilford [Kil02] found examples of failure of Gorensteinness (and multiplicity one) at the prime 2 for certain prime levels. Motivated by the arguments in Section 4, in this section we give examples of failure of multiplicity one for primes (including odd primes) whose square divides the level.

5.1 Multiplicity One for Differentials

In connection with the arguments in Section 4, especially Lemmas 4.6 and 4.8, it is of interest to compute the index [T′ : T] for various N. We can compute this index in Magma, e.g., the following commands com-

pute the index for N = 54: “J := JZero(54); T := HeckeAlgebra(J);

Index(Saturation(T), T);” We obtain Table 2, where the first column contains N and the second column contains [T′ : T]:

Let 𝔪 be a maximal ideal of the Hecke algebra T ⊂ End(J_{0}(N)) of residue characteristic p. Recall that we say that 𝔪 satisfies multiplicity one for differentials if dim(H^{0}(X_{0}(N)_{F_{p}}, Ω)[𝔪]) ≤ 1.

In each case in which [T′ : T] ≠ 1, Lemma 4.6 implies that there is some maximal ideal 𝔪 of T such that dim(H^{0}(X_{0}(N)_{F_{p}}, Ω)[𝔪]) > 1, which is an example of failure of multiplicity one for differentials.

In Table 2, whenever p | [T′ : T], then p^{2} | 2N. This is a consequence of Proposition 4.9, which moreover asserts that when 2 exactly divides N and 2 | [T′ : T] then there is a non-ordinary (old) maximal ideal of characteristic 2 in the support of T′/T.

Moreover, notice that Theorem 3.7(b) (whose proof is in Section 4.2) follows formally from two key facts: that A_{f} is new and that multiplicity one for differentials holds for ordinary maximal ideals with residue characteristic p || N and for all maximal ideals with residue characteristic p ∤ N. The conclusion of Theorem 3.7(b) does not hold for the counterexamples in Section 2 (e.g., for 54B1), which are all new elliptic curves, so multiplicity one for differentials does not hold for certain maximal ideals that arise from the new quotient of the Hecke algebra. Note that in all examples we have p | (r/m) with p^{2} | N, which raises the question: are there non-ordinary counterexamples with p || N?

5.2 Multiplicity One for Jacobians

We say that a maximal ideal 𝔪 of T satisfies multiplicity one if J_{0}(N)[𝔪] is of dimension two over T/𝔪. We sometimes use the phrase “multiplicity one for J_{0}(N)” in order to distinguish this notion from the notion of multiplicity one for differentials.

Proposition 5.1. Suppose E is an optimal elliptic curve over Q of conductor N and p is a prime such that p | r_{E} but p ∤ m_{E}. Let 𝔪 be the annihilator in T of E[p]. Then multiplicity one fails for 𝔪, i.e., dim_{T/𝔪} J_{0}(N)[𝔪] > 2.

Proof. Using the principal polarization E ≅ E^{∨} we view E as an abelian subvariety of J = J_{0}(N) and consider the complementary T-stable abelian subvariety A of E (thus A is the kernel of the modular parametrization map J → E). In this setup, J = E + A, and the intersection of E and A is E[m_{E}]. Here we use that the composite map E ≃ E^{∨} → J^{∨} → J → E is a polarization, and hence is multiplication by a positive integer m_{E}. Because p ∤ m_{E}, we have E[p] ∩ A = 0. On the other hand, let 𝔪 be the annihilator of E[p] inside T.

Table 2: The Index [T′ : T]

11 1 12 1 13 1 14 1 15 1 16 1 17 1 18 1 19 1 20 1 21 1 22 1 23 1 24 1 25 1 26 1 27 1 28 1 29 1 30 1 31 1 32 1 33 1 34 1 35 1 36 1 37 1 38 1 39 1 40 1 41 1 42 1 43 1 44 2 45 1 46 2 47 1 48 1 49 1 50 1  51 1 52 1 53 1 54 3 55 1 56 2 57 1 58 1 59 1 60 2 61 1 62 2 63 1 64 2 65 1 66 1 67 1 68 2 69 1 70 1 71 1 72 2 73 1 74 1 75 1 76 2 77 1 78 2 79 1 80 4 81 1 82 1 83 1 84 2 85 1 86 1 87 1 88 8 89 1 90 1  91 1 92 16 93 1 94 4 95 1 96 8 97 1 98 1 99 9 100 1 101 1 102 1 103 1 104 4 105 1 106 1 107 1 108 54 109 1 110 2 111 1 112 8 113 1 114 1 115 1 116 4 117 1 118 2 119 1 120 32 121 1 122 1 123 1 124 16 125 25 126 18 127 1 128 64 129 1 130 1  131 1 132 8 133 1 134 1 135 27 136 16 137 1 138 4 139 1 140 8 141 1 142 8 143 1 144 32 145 1 146 1 147 7 148 4 149 1 150 5 151 1 152 32 153 9 154 1 155 1 156 32 157 1 158 4 159 1 160 256 161 1 162 81 163 1 164 8 165 1 166 2 167 1 168 128 169 13 170 1  171 9 172 8 173 1 174 4 175 5 176 512 177 1 178 1 179 1 180 72 181 1 182 1 183 1 184 1024 185 1 186 4 187 1 188 256 189 243 190 8 191 1 192 4096 193 1 194 1 195 1 196 14 197 1 198 81 199 1 200 80 201 1 202 1 203 1 204 32 205 1 206 4 207 81 208 256 209 1 210 2

Then J[𝔪] contains E[p] and also A[𝔪], and because p is a congruence prime, the submodule A[𝔪] ⊂ J[𝔪] is nonzero. Thus the sum E[p] + A[𝔪] is a direct sum and is larger than E[p], which is of dimension 2 over T/𝔪 = Z/pZ. Hence the dimension of J[𝔪] over T/𝔪 is bigger than 2, as claimed.

Proposition 5.1 implies that any example in which simultaneously p ∤ m_{E} and ord_{p}(r_{E}) ≠ ord_{p}(m_{E}) produces an example in which multiplicity one for J_{0}(N) fails. For example, for the curve 54B1 and p = 3, we have ord_{3}(r_{E}) = 1 but ord_{3}(m_{E}) = 0, so multiplicity one at 3 fails for J_{0}(54).

References

[ARS06] A. Agashe, K. Ribet and W. Stein, The Manin Constant, QJPAM, Coates Volume (2006), to appear.

[AU96] A. Abbes and E. Ullmo, À propos de la conjecture de Manin pour les courbes elliptiques modulaires, Compositio Math. 103 (1996), no. 3, 269–286.

[BCP97] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3–4, 235– 265, Computational algebra and number theory (London, 1993).

[BCDT01] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939 (electronic).

[CK04] Alina Carmen Cojocaru and Ernst Kani, The modular degree and the congruence number of a weight 2 cusp form, Acta Arith. 114 (2004), no. 2, 159–167.

[Cre97] J. E. Cremona, Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997,

         http://www.maths.nott.ac.uk/personal/jec/book/.

[CSS97] G. Cornell, J. H. Silverman, and G. Stevens (eds.), Modular forms and Fermat’s last theorem (boston,ma, 1995), New York, Springer-Verlag, 1997, Papers from the Instructional Conference on Number Theory and Arithmetic Geometry held at Boston University, Boston, MA, August 9–18, 1995.

[DI95] F. Diamond and J. Im, Modular forms and modular curves, Seminar on Fermat’s Last Theorem, Providence, RI, 1995, pp. 39–133.

[Fre97] G. Frey, On ternary equations of Fermat type and relations with elliptic curves, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 527–548.

[FM99] G. Frey and M. Müller, Arithmetic of modular curves and applications, Algorithmic algebra and number theory (Heidelberg, 1997), Springer, Berlin, 1999, pp. 11–48.

[Har77] R. Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52.

[Kil02] L. J. P. Kilford, Some non-Gorenstein Hecke algebras attached to spaces of modular forms, J. Number Theory 97 (2002), no. 1, 157– 164.

[Lan83] S. Lang, Abelian varieties, Springer-Verlag, New York, 1983, Reprint of the 1959 original.

[Li75] W-C. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285–315.

[Maz77] B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33–186 (1978).

[Maz78] B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162.

[MR91] B. Mazur and K. A. Ribet, Two-dimensional representations in the arithmetic of modular curves, Astérisque (1991), no. 196–197, 6, 215–255 (1992), Courbes modulaires et courbes de Shimura (Orsay, 1987/1988).

[Mum70] D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay, 1970.

[Mur99] M. R. Murty, Bounds for congruence primes, Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), Amer. Math. Soc., Providence, RI, 1999, pp. 177–192.

[Rib75] K. A. Ribet, Endomorphisms of semi-stable abelian varieties over number fields, Ann. Math. (2) 101 (1975), 555–562.

[Rib81] K. A. Ribet, Endomorphism algebras of abelian varieties attached to newforms of weight 2, Seminar on Number Theory, Paris 1979–80, Progr. Math., vol. 12, Birkhäuser Boston, Mass., 1981, pp. 263–276.

[Rib83] K. A. Ribet, Mod p Hecke operators and congruences between modular forms, Invent. Math. 71 (1983), no. 1, 193–205.

[Ste89] G. Stevens, Stickelberger elements and modular parametrizations of elliptic curves, Invent. Math. 98 (1989), no. 1, 75–106.

[Stu87] J. Sturm, On the congruence of modular forms, Number theory (New York, 1984–1985), Springer, Berlin, 1987, pp. 275–280.

[Til97] J. Tilouine, Hecke algebras and the Gorenstein property, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 327–342.

[Vat05] V. Vatsal, Multiplicative subgroups of J_{0}(N) and applications to elliptic curves, J. Inst. Math. Jussieu 4 (2005), no. 2, 281–316.

[Wil95] A. J. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551.

[Zag85] D. Zagier, Modular parametrizations of elliptic curves, Canad. Math. Bull. 28 (1985), no. 3, 372–384.