The Manin Constant, Congruence Primes,

and the Modular Degree

Amod Agashe  Kenneth Ribet  William A. Stein

Abstract.

We obtain relations between the modular degree and congruence modulus of elliptic curves, and answer a question raised in a paper of Frey and Müller about whether or not the congruence number and modular degree of elliptic curves are equal; they are not, but we prove a theorem relating them and make a conjecture. We also prove results and make conjectures about Manin constants of quotients of J_{1}(N) of arbitrary dimension. For optimal elliptic curves E, we give a new condition under which the Manin constant of E is odd.

1 Introduction

Let E be an elliptic curve over Q. By [BCDT01], we may view E as a quotient 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).

The pullback of a minimal differential on E is a multiple c of some normalized new cuspidal eigenform f_{E} ∈ S_{2}(Γ_{0}(N)). The absolute value of c is the Manin constant c_{E} of E. Manin conjectured that c_{E} = 1. In Section 2.1, we summarize results about c_{E}, then extend techniques of Abbes and Ullmo [AU96] to show that 2 ∤ c_{E} under certain hypothesis.

The congruence number r_{E} of E is the largest integer such that there is a nonzero element of S_{2}(Γ_{0}(N)) that is orthogonal to f_{E} and congruent to f_{E} modulo r_{E}. The modular degree m_{E} is the degree of the composite map X_{0}(N) → J_{0}(N) → E. Section 2.2 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.

We generalize the Manin constant, congruence primes, and modular degree to optimal quotients of J_{0}(N) and J_{1}(N) of any dimension associated to ideals of the Hecke algebra. Section 3 is concerned with the congruence number and the modular degree and Section 4 with the Manin constant. We also conjecture that the Manin constant is 1 for newform quotients of J_{0}(N) and J_{1}(N). Acknowledgment. The authors are grateful to A. Abbes, R. Coleman, B. Conrad, E. de Shalit, B. Edixhoven, L. Merel, and R. Taylor for several discussions and advice regarding this paper. They would also like to thank J. Cremona for explaining his computations involving the Manin constant.

2 Optimal Elliptic Curve Quotients

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 (Modular Degree). The modular degree m_{E} of E is the degree of ϕ_{E}.

2.1 The Manin Constant

Let E_{Z} denote the Néron model of E over Z (see, e.g., [Sil92, App. C, §15],

[Sil94] and [BLR90]). Let ω be a generator for the rank one Z-module of invariant differential one forms on E_{Z}. The pullback of ω to X_{0}(N) is a differential ϕ∗_{E}ω on X_{0}(N). The newform f defines another differential 2πif(z)dz = f(q)dq/q on X_{0}(N). Because the action of Hecke operators is compatible with the map X_{0}(N) → E, [AL70] implies that ϕ∗_{E}ω = c·2πif(z)dz for some c ∈ Q^{∗} (see also [Man72, §5]).

Definition 2.2 (Manin Constant). The Manin constant c_{E} of E is the absolute value of c, where c is as above.

The Manin constant plays a role in the Birch and Swinnerton-Dyer conjecture (see Section 4.1), and its integrality is important to Cremona’s computations of elliptic curves (see [Cre97, pg. 45]).

The following conjecture is implicit in [Man72, §5].

Conjecture 2.3 (Manin). c_{E} = 1.

Significant progress has been made towards this conjecture. In the following list of theorems, p denotes a prime and N denotes the conductor of E.

The Manin Constant, Congruences, and Modular Degrees 3

Theorem 2.4 (Edixhoven [Edi91, Prop. 2]). c_{E} is an integer.

Edixhoven proved this using an integral q-expansion map, whose existence and properties follow from results in [KM85]. We generalize his argument to quotients of arbitrary dimension in Section 4.1.

Theorem 2.5 (Mazur, [Maz78, Cor. 4.1]). If p | c_{E}, then p^{2} | 4N.

Mazur proved this by applying theorems of Raynaud about exactness of sequences of differentials, then using the “q-expansion principle” in characteristic p and a property of the Atkin-Lehner involution. We generalize Mazur’s argument in Section 4.1.

The following two results refine the above results at p = 2.

Theorem 2.6 (Raynaud [AU96, Prop. 3.1]). If 4 | c_{E}, then 4 | N.

Theorem 2.7 (Abbes-Ullmo [AU96, Thm. A]). If p | c_{E}, then p | N.

We generalize Theorem 2.6 in Section 4.1. However, it is not clear if one can generalize Theorem 2.7 to dimension greater than 1 (see Remark 4.16). It would be fantastic if the theorem could be generalized, since it would imply that for newform quotients A_{f} of J_{0}(N), with N odd and square free, that the Manin constant is 1, which would be useful for computations regarding the Birch and Swinnerton-Dyer conjecture.

B. Edixhoven also has unpublished results (see [Edi89]) which assert that the only primes that can divide c_{E} are 2, 3, 5, and 7; he also gives bounds that are independent of E on the valuations of c_{E} at 2, 3, 5, and 7. His arguments rely on construction of certain stable integral models for X_{0}(p^{2}).

Cremona verified computationally that the Manin constant is 1 for every elliptic curve of conductor up to at least 10000. Cremona computes lattice invariants c_{4} and c_{6} from a rational newform f, and verifies in each case that c_{4} and c_{6} are the invariants of a minimal Weierstrass equation, to conclude that the Manin constant for the corresponding elliptic curve is 1.

Definition 2.8 (Congruence Number). The congruence number r_{E} of E is the largest integer r such that there exists a cusp form g ∈ S_{2}(Γ_{0}(N)) that has integer Fourier coefficients, is orthogonal to f with respect to the Petersson inner product, and satisfies g ≡ f (mod r). The congruence primes of E are the primes that divide r_{E}.

To the above list we add the following theorem. Our proof builds on the techniques of [AU96].

Theorem 2.9. If p | c_{E} then p^{2} | N or p | m_{E}.

This theorem is a special case of Theorem 4.13 below, which we prove in Section 4.4. In fact, Theorem 4.13 asserts that if p | c_{E} then p^{2} | N or p | r_{E}. However, Theorem 2.10 implies that when ord_{p}(N) = 1 then ord_{p}(r_{E}) = ord_{p}(m_{E}). In view of Theorem 2.5, our new contribution is that if m_{E} is odd and ord_{2}(N) = 1, then c_{E} is odd. This hypothesis is very stringent—of the 125357 optimal elliptic curve quotients of conductor ≤ 30000, only 31 of them satisfy the hypothesis. In the notation of [Cre97], they are 14A, 46A, 142C, 206A, 302B, 398A, 974C, 1006B, 1454A, 1646A, 1934A, 2606A, 2638B, 3118B, 3214B, 3758D, 4078A, 7054A, 7246C, 11182B, 12398B, 12686C, 13646B, 13934B, 14702C, 16334B, 18254A, 21134A, 21326A, 22318A, 26126A.

It is unknown if there are infinitely many elliptic curves that satisfy our hypothesis. The third author conjectured in [SW04, Conj. 4.2] that there are infinitely many elliptic curves (of prime conductor) with odd modular degree.

2.2 Congruence Primes and the Modular Degree

Congruence primes have been studied by Doi, Hida, Ribet, Mazur and others (e.g., see [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 2.10. Let E be an elliptic curve over Q of conductor N, with modular degree m_{E} and congruence modulus r_{E}. Then m_{E} | r_{E} and if ord_{p}(N) ≤ 1 then ord_{p}(r_{E}) = ord_{p}(m_{E}).

The divisibility m_{E} | r_{E} was first discussed in [Zag85, Th. 3], where it is attributed to 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 3.11. The second part of Theorem 2.10, i.e., that if ord_{p}(N) = 1 then ord_{p}(r_{E}) = ord_{p}(m_{E}), follows from the more general Theorem 3.5 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 Prop. 3.3 implies

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

and by Prop. 3.4 ord_{p}(#C) = 0.

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, we quickly found that the answer is no. The first 16 countexamples occur at levels

54, 64, 72, 80, 88, 92, 96, 99, 108, 112, 120, 124, 126, 128, 135, 144.

For example, the elliptic curve 54B1 of [Cre97], with 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}) is congruent to f modulo 3. To prove this congruence, we checked it for 18 Fourier coefficients, where the precision 18 was determined using [Stu87]. In accord with Theorem 2.10, since ord_{3}(r_{E}) ≠ ord_{3}(c_{E}), we have ord_{3}(54) ≥ 2.

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

m_{E} = 2^{4} ≠ r_{E} = 2^{4} · 11.

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

Conjecture 2.11. 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).

In particular, for p ≥ 5, the conjecture simply asserts that

ord_{p} (\frac{r_{E}}{m_{E}} ) ≤ 1,

because ord_{p}(N) ≤ 2 for any p ≥ 5. As evidence, we verified Conjecture 2.11 for every optimal elliptic curve quotient of J_{0}(N), with N ≤ 539.

3 Quotients of arbitrary dimension: generalization of the congruence number and the modular degree

Let Γ be either Γ_{0}(N) or Γ_{1}(N), for N ≥ 4, 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, 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 (Newform quotient). If f ∈ 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 to the degree of the field Q(…, a_{n}(f), …).

In Section 3.1, we generalize the notions of the congruence number and the modular degree to quotients A = A_{I}, and state a theorem relating the two, which we prove in Sections 3.2–3.3.

3.1 The congruence number and the modular degree

If C is an abelian variety, let C^{∨} denote the dual of C. Let ϕ_{2} denote the quotient map J → A. There is a canonical principal polarization θ : J ≅ J^{∨} arising from the theta divisor 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.

Since ϕ_{2} is a surjection, by [Lan83, §VI.3, Prop 3], ker(ϕ_{2}^{∨}) is finite. Since ker(ϕ_{2}) is connected, ker(ϕ_{2}^{∨}) is trivial, so ϕ_{2}^{∨} and ϕ_{1} are injections. Let ϕ be the composition

ϕ : A^{∨} ⟶^{ϕ_{1}} J ⟶^{ϕ_{2}} A.

Proposition 3.2. The map ϕ is a polarization.

Proof. Let i be the injection ϕ_{2}^{∨} : A^{∨} → J^{∨}, and let Θ denote the theta divisor. From the definition of the polarization attached to an ample divisor, we see that the map ϕ is induced by the pullback i^{∗}(Θ) of the theta divisor. The theta divisor is effective, and hence so is i^{∗}(Θ). By [Mum70, §6, Application 1, p. 60], ker ϕ is finite. Since the dimensions of A and A^{∨} are the same, ϕ is an isogeny. Moreover, since Θ is ample, some power of it is very ample. Then the pullback of this very ample power by i is again very ample, and hence a power of i^{∗}(Θ) is very ample, so i^{∗}(Θ) is ample (by [Har77, II.7.6]).

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}(Γ; R) denote the subgroup of S_{2}(Γ) consisting of cups forms whose Fourier expansions at the cusp ∞ have coefficients in R. Let W (I) = 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.

Definition 3.4. The exponent of the quotient group

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

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

Our definition of r_{A} generalizes the definition in Section 2.2 when A is an elliptic curve (see [AU96, p. 276]), and the following generalizes Theorem 2.10:

Theorem 3.5. 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 3.6. When A_{f} is an elliptic curve, Theorem 3.5 implies that the modular degree divides the congruence number, 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^{10} · 6947)^{2}.

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

The following Magma session illustrates how to verify the above assertion about n_{A_{f}} and r_{A_{f}}. The commands were implemented by the second author, and are parts of Magma V2.11 or greater.

   > A := ModularSymbols("431F");
   > Factorization(ModularDegree(A));
   [ <2, 11>, <6947, 1> ]
   > Factorization(CongruenceModulus(A));
   [ <2, 10>, <6947, 1> ]

3.2 Proof of Theorem 3.5 (a)

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, Remark 10.2.2]). However, on J^{new}, the action of the Hecke operators commutes with that of W_{N}. 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 for such quotients we may identify A^{∨} with ϕ_{1}(A^{∨}) as modules over T.

Recall that f is a newform, I_{f} = Ann_{T}(f), 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 3.7. Hom(A^{∨}, B) = 0.

Proof. If there were a nonzero element of Hom(A^{∨}, B), then for all ℓ, the Tate module Tate_{ℓ}(A^{∨}) = Q ⊗ \underleftarrow{lim}_{n} A^{∨}[ℓ^{n}] would be a factor of Tate_{ℓ}(B). One could then extract almost all prime-indexed coefficients of the corresponding eigenforms from the Tate modules, which would violate multiplicity one (see [Li75, Cor. 3, pg. 300]).

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 3.7 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 the order of e_{2} is a divisor of the order of e_{1}, which is the crucial ingredient in the proof of Proposition 3.11 below.

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 3.8. 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^{∨}}.

Lemma 3.9. The group (T_{1} ⊕ T_{2})/T is isomorphic to the quotient (1) in Definition 3.4, 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)

The term Ext^{1}(T_{1} ⊕ T_{2}, Z) is 0 is because Ext^{1}(M, Z) = 0 for any finitely generated free abelian group. Also, Hom((T_{1} ⊕ T_{2})/T, Z) = 0 since (T_{1} ⊕ T_{2})/T is torsion. 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}] ⊕ W (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}]^{⊥} = W (I_{f}). Finally note that if G is any finite abelian group, then Ext^{1}(G, Z) ≈ G as groups, to get the desired result.

Lemma 3.10. The element e_{1} ∈ (T_{1} ⊕ T_{2})/T has order r̃_{A}.

Proof. By Lemma 3.9, 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 an 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 3.11. 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 3.8 and 3.10.

This finishes the proof of the first statement in Theorem 3.5.

3.3 Proof of the Theorem 3.5 (b)

Write N = pM with p prime and p ∤ M. (Note: The argument below also works if p = 1, which addresses the case when no prime exactly divides N.) Let T = Z[…, T_{n}, …] be the subring of End(J_{0}(N)) generated by the Hecke operators T_{n} for all n ≥ 1. Let T′ be the saturation of T in End(J_{0}(N)), so

T′ = (T ⊗ Q) ∩ End(J_{0}(N)),

where the intersection is taken inside End(J_{0}(N)) ⊗ Q. The quotient T′/T is a finitely generated abelian group because both T and End(J_{0}(N)) are finitely generated over Z.

Suppose for the moment that N = 1, so p = pM. In [Maz77], Mazur proves that T = T′. He combines this result with the equality

T ⊗ Q = End(J_{0}(p)) ⊗ Q

of [Rib75] or [Rib81], to deduce that T = End(J_{0}(p)).

3.3.1 Multiplicity One

Mazur’s argument (see [Maz77, pg. 95]) is quite general; it relies on a multiplicity 1 statement for spaces of differentials in positive characteristic (see [Maz77, Prop. 9.3, pg. 94]). His method shows in the general case (where M is no longer constrained to be 1) that Supp_{T}(T′/T) contains no maximal ideal 𝔪 of T for which his space H^{0}(X_{0}(pM)_{F_{ℓ}}, Ω)[𝔪] has dimension ≤ 1. (Here ℓ is the residue characteristic of 𝔪.) In other words, multiplicity one for H^{0}(X_{0}(pM)_{F_{ℓ}}, Ω)[𝔪] implies that T and T′ agree at 𝔪. We record this fact as a lemma.

Lemma 3.12. Suppose 𝔪 is a maximal ideal of T of residue characteristic ℓ and that

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

Then 𝔪 is not in the support of T/T′.

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

Lemma 3.13. If ℓ ∤ pM, then ℓ ∤ #(T/T′).

Proof. The standard q-expansion argument of [Maz77] proves that

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

for all 𝔪 | ℓ. Now apply Lemma 3.12

In the context of Mazur’s paper, where p = pM, we see from Lemma 3.13 that T and T′ agree away from p. At p, we can still use the q-expansion principle because of the arguments in [Maz77, Ch.II §4]. Thus in this case T = T′, as we asserted above.

The question of multiplicity 1 at p for H^{0}(X_{0}(pM)_{F_{ℓ}}, 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 is p-old if it arises from S_{2}(Γ_{0}(M)).)

Lemma 3.14 (Wiles). If 𝔪 is an ordinary prime of T of characteristic ℓ and ord_{ℓ}(pM) = 1, then 𝔪 is not in the support of T′/T.

Proof. This follows from [Wil95, Lem. 2.2, pg. 485], which proves, under a suitable hypothesis, that H^{0}(X_{0}(pM)_{F_{p}}, Ω)[𝔪] is 1-dimensional if 𝔪 is a maximal ideal of T that divides p. The “suitable hypothesis” is that 𝔪 is ordinary, in the sense that T_{p} ∉ 𝔪. (Note that T_{p} is often denoted U_{p} in this context.) It follows from Wiles’s lemma that T′ = T locally at 𝔪 whenever 𝔪 is an ordinary prime whose residue characteristic exactly divides the level (which is pM here). We make a few further comments about the proof of this lemma.

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 the lemma; 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 forms of level N.) Here is an example with p = 2 and N = 11: 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.

We now summarize the conclusions we can make from the lemmas so far. Wiles’s lemma and the standard q-expansion argument (Lemma 3.13 and Lemma 3.14) imply that T and T′ agree locally at each rational prime that is prime to the level pM, and also at each maximal ideal 𝔪 dividing p that is ordinary, in the sense that T_{p} ∉ 𝔪. A more palatable description of the situation involves considering the Hecke algebra T and its saturation T′ at some level N ≥ 1. Then T = T′ 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. In Mazur’s original context, the level N is prime. Moreover, 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.

3.3.2 Degrees and Congruences

Let e ∈ T ⊗ Q be as in Section 3.2. Let A ⊂ J_{0}(pM) be the image of e (note that we denoted this image by A^{∨} in Section 3.2). 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 3.15 (Congruence Ideal). 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. Also, we may view T as embedded in T_{A} × T_{B}, via the map j.

Lemma 3.16. 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 3.17. We have I ⊂ J.

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

Lemma 3.18. We have J = T_{A} ∩ End(J_{0}(pM)) = T_{A} ∩ T′.

Proof. This is elementary; it is an analogue of Lemma 3.16.

Proposition 3.19. 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 3.16 and Lemma 3.18, 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 3.20. 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 Lemma 3.14 and Proposition 3.19 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 Lemma 3.14. By Lemma 3.13, 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 degree”. In this case, the inclusion I ⊂ J corresponds to the divisibility ñ_{A_{f}} | r̃_{A_{f}} , 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 3.5.

Remark 3.21. 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.

4 Quotients of arbitrary dimension: generalization of the Manin Constant

Let the notation be as in the beginning of Section 3. Let J_{old} denote the abelian subvariety of J generated by the images of the degeneracy maps from levels that properly divide N (e.g., see [Maz78, §2(b)]) and let J^{new} denote the quotient of J by J_{old}.

In Section 4.1, we generalize the notion of the Manin constant to quotients A as above, and conjecture that this constant is 1 for newform quotients of J_{0}(N) and J_{1}(N). In Section 4.2, we show that the (generalized) Manin constant is an integer. In the next two sections, we give generalizations of some of the results from Section 2 to quotients A that factor through J_{0}(N)^{new}. In Section 4.3, we show that if the level N is squarefree and A is a factor of J_{0}(N)^{new}, then the Manin constant is a power of 2, whose exponent is bounded above by the dimension of A if A is a newform quotient. In Section 4.4, we prove that if A is a newform quotient of J_{0}(N) and the level is squarefree, then the Manin constant is coprime to the congruence number.

4.1 The definition of the generalized Manin constant and a conjecture

As in Section 3.1, if R is a subring of C, let S_{2}(R) = S_{2}(Γ; R) denote the T-submodule of S_{2}(Γ; C) consisting of modular cuspforms whose Fourier expansions at ∞ have coefficients in R. Note that S_{2}(R) ≅ S_{2}(Z) ⊗ R.

If A is an abelian variety over Q and n is a positive integer, let A_{Z[1/n]} denote the Néron model of A over Z[1/n]. On a Néron model, the global differentials are the same as the group of invariant differentials, so the group H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{1}) is free of rank d, where d = dim(A) and Ω_{A_{Z}/Z}^{1} is the sheaf of differentials on

the Néron model A_{Z} of A. Let D be a generator of ⋀^{d} H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{1}).

The real volume Ω_{A} of A is the volume of A(R) with respect to the measure given by D. This quantity is of interest because it appears in the Birch and Swinnerton-Dyer conjecture, which expresses the ratio L(A, 1)/Ω_{A} in terms of certain arithmetic invariants of A (see [Lan91, Chap. III, §5] and [AS05]). Let g_{1}, …, g_{d} be a Z-basis of S_{2}(Z)[I], and for j = 1, …, d, let

ω′_{j} = 2πig_{j}(z)dz ∈ H^{0}(X, Ω_{X/Q}) = H^{0}(J, Ω_{J/Q})

(where we use the standard map X → J that sends the cusp ∞ to 0). As before, let ϕ_{2} denote the quotient map J → A. Then ϕ_{2}^{∗} induces an isomorphism H^{0}(A, Ω_{A/Q}) → ⨁_{j} Qw′_{j}. For j = 1, …, d, let ω_{j} = (ϕ_{2}^{∗})^{−1}ω′_{j}.

In calculations (see [AS05]), or while proving formulas regarding the ratio mentioned above (see [Aga99, §2]), instead of working with Ω_{A}, it is easier to work with the volume Ω′_{A} of A(R) with respect to the measure given by ∧_{j}ω_{j}. There exists c ∈ Q^{∗} such that D = c · ∧_{j}ω_{j}. The absolute value of c depends only on I, and is independent of other choices made above.

Definition 4.1. Let A be an optimal quotient of J attached to an ideal I of the Hecke algebra, as above. The Manin constant c_{A} of the optimal quotient A is the absolute value of the constant c defined above.

If A has dimension one, then c_{A} is as in Definition 2.2. The constant c as defined above was considered by Gross [Gro82, (2.5) on p. 222] and Lang [Lan91, III.5, p.95], although they did not explicitly state its relation to the usual Manin constant (for elliptic curves). The constant c_{A} was defined for a particular quotient A in [Aga99], where it was called the generalized Manin constant. In [CES03] it is called the Manin index.

If one works with the easier-to-compute volume Ω′_{A} instead of Ω_{A}, it is necessary to obtain information about c_{A} in order to make conclusions regarding the Birch and Swinnerton-Dyer conjecture, since Ω_{A} = c_{A} · Ω_{A′}. This is our motivation for studying the Manin constant. Cremona’s method for proving that c_{A} = 1 for a specific elliptic curve, i.e., computing c_{4} and c_{6} and checking that they are invariants of a minimal Weierstrass model, is of little use when A has dimension greater than one, since there is no simple analogue of the minimal Weierstrass model for general A.

Note that the Manin constants c_{A} might not equal 1, especially if A is not a quotient of J^{new} (see Remark 4.10). At the same time, if A is a newform quotient and the level N is squarefree, then Theorems 4.11, 4.12, and 4.13 suggest that the Manin constant is 1 for such quotients.

In the case when the level is not square free, computations of [FpS^{+}01] involving Jacobians of genus 2 curves that are quotients of J_{0}(N)^{new} show that c_{A} = 1 in 28 case of 2-dimensional newform quotients. These include quotients having the following non-square-free levels:

3^{2} · 7, 3^{2} · 13, 5^{3}, 3^{3} · 5, 3 · 7^{2}, 5^{2} · 7, 2^{2} · 47, 3^{3} · 7.

The above observations are evidence for the following conjecture, which generalizes Conjecture 2.3 of Manin:

Conjecture 4.2. If f is a newform on Γ_{0}(N) or Γ_{1}(N), then c_{A_{f}} = 1.

Remark 4.3. The above conjecture does not hold if all we know is that J_{0}(N) → A factors through J_{0}(N)^{new}. Adam Joyce [Joy03] found an optimal quotient of J_{0}(431)^{new} whose Manin constant is 2 (this example is motivated by [Kil02]); note that this optimal quotient is not attached to a single newform.

4.2 Integrality of the Manin constant

We continue to use the notation introduced so far in Section 4. In this section we prove Theorem 4.7, which asserts that the (generalized) Manin constant is an integer, which generalizes Thm. 2.4 of Edixhoven (see also [CES03, §6.1.2] for a similar argument). The proof is itself a generalization of that of [Edi91, Prop. 2]. The main idea is to construct an injective map on H^{0}(A, Ω_{A/Q}^{1}) using “q-expansions”, then show that the image of H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{1}) under this map is contained in the image of ⊕_{j}Zω_{j}. We assume that N > 4, which is harmless, since J_{1}(N) has dimension 0 for N ≤ 10.

Using the standard immersion X ↪ J sending ∞ to 0, we have maps

X ↪ J → A.  (4)

If X = X_{1}(N), then we obtain a map X_{1}(N) → A. If X = X_{0}(N), then composing with the standard map X_{1}(N) → X_{0}(N) we get a map X_{1}(N) → A. In either case, denote the resulting map X_{1}(N) → A by ϕ_{A}.

Consider the model X_{μ}(N) over Z for X_{1}(N) whose affine points parametrize isomorphism classes of pairs (E, i), where E is an elliptic curve and i : μ_{N} ↪ E^{reg} is an immersion, as in [Kat76] (see also [DI95, §9.3.6, p. 80]). Since X_{μ}(N) is smooth over Z (by [Kat76, §II.2.5]), the Néron mapping property implies that there is a map

X_{μ}(N) → A_{Z},

which we again denote by ϕ_{A}.

The Tate curve E_{q} over Z[[q]] with the canonical immersion of μ_{N} gives a map (see, e.g., [DI95, p. 112])

τ : Spec Z[[q]] → X_{μ}(N).  (5)

Pulling back differentials, gives a map

H^{0} (X_{μ}(N), Ω_{X_{μ}(N)/Z}^{1}) ⟶ H^{0} (Spec Z[[q]], Ω_{Z[[q]]/Z}^{1}) .

Now H^{0}(Spec Z[[q]], Ω_{Z[[q]]/Z}^{1}) is free of rank one over Z[[q]] with generator dq, so we get a map

H^{0} (X_{μ}(N), Ω_{X_{μ}(N)/Z}^{1}) ⟶ Z[[q]].

Let q-exp denote the composite

H^{0} (X_{μ}(N), Ω_{X_{μ}(N)/Z}^{1}) ⟶ Z[[q]] ⟶^{q·} Z[[q]],

where the second map is multiplication by q.

Next, we relate q-exp to the usual Fourier-expansion over C. Since X_{μ}(N)⊗ C ≅ X_{1}(N)_{C}, the Tate curve over C (see [DR73, VII.4.2]) gives a map

τ_{C} : Spec C[[q]] → X_{1}(N)_{C},

which is the base extension of (5) and which identifies q with the local parameter e^{2πiz} on X_{1}(N)_{C} at the cusp ∞. As above, pulling back differentials, gives a map

H^{0} (X_{1}(N)_{C}, Ω_{X_{1}(N)/C}^{1}) ⟶ C[[q]].

Let F -exp denote the composite

H^{0} (X_{1}(N)_{C}, Ω_{X_{1}(N)/C}^{1}) ⟶ C[[q]] ⟶^{q·} C[[q]],

where the second map is multiplication by q.

We obtain a commutative diagram

\xymatrix{H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{1}) \ar[r]^{ϕ_{A}^{∗}} \ar[d] & H^{0}(X_{μ}(N), Ω_{X_{μ}(N)/Z}^{1}) \ar[r]^{^{q-exp}} \ar[d] & Z[[q]] \ar[d] \\ H^{0}(A_{C}, Ω_{A_{C}/C}^{1}) \ar[r]^{ϕ_{A}^{∗}⊗C} & H^{0}(X_{1}(N)_{C}, Ω_{X_{1}(N)/C}^{1}) \ar[r]^{_{F -exp}} & C[[q]]}  (6)

in which the first and last vertical maps are injections.

The relation of F -exp to the Fourier expansion of cusp forms is given by the following lemma. Let ψ be the isomorphism

ψ : S_{2}(Γ_{1}(N), C) →^{≅} H^{0}(X_{1}(N)_{C}, Ω_{X_{1}(N)/C}^{1})

given by f(z) ↦ 2πif(z)dz.

Lemma 4.4. Let f ∈ S_{2}(Γ_{1}(N), C), and let {a_{n}} be the coefficients of the Fourier expansion of f. Then F -exp(ψ(f)) = ∑_{n} a_{n}q^{n}.

Proof. If f ∈ S_{2}(Γ_{1}(N), C), and its Fourier series is ∑_{n} a_{n}e^{2πizn}, then ψ(f) = 2πi ∑_{n} a_{n}e^{2πizn}dz. Since τ_{C} identifies q with the local parameter e^{2πiz}, we see that the pullback of ψ(f) via τ_{C} to H^{0}(Spec C[[q]], Ω_{C[[q]]/C}^{1}) is ∑_{n} a_{n}q^{n−1}dq. So F -exp(ψ(f)) = ∑_{n} a_{n}q^{n}.

Lemma 4.5. The group q-exp (ϕ∗_{A} (H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{1}))) is a subgroup of F -exp(ψ(S_{2}(Γ_{1}(N), Z)[I])).

Proof. If x ∈ H^{0}(X_{μ}(N), Ω_{X_{μ}(N)/Z}^{1}) maps to y ∈ H^{0}(X_{1}(N)_{C}, Ω_{X_{1}(N)/C}^{1}), then by the commutativity of the right half of the commutative diagram above and by Lemma 4.4, the Fourier expansion of ψ^{−1}(y) ∈ S_{2}(Γ_{1}(N), C) is the same as q-exp(x), i.e., has integral Fourier coefficients; hence ψ^{−1}(y) ∈ S_{2}(Γ_{1}(N), Z). This gives an injection

q-exp (H^{0}(X_{μ}(N), Ω_{X_{μ}(N)/Z}^{1})) ↪ F -exp(ψ(S_{2}(Γ_{1}(N), Z))).

Now the lemma follows from the fact that ϕ∗_{A}(H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{1}))[I] = 0.

Proposition 4.6. We have H^{0}(A_{Z}, Ω_{A_{Z}}^{1}) ⊆ ⊕_{j}Zω_{j}, considered as subgroups of H^{0}(A, Ω_{A/Q}^{1}).

Proof. Let ϕ denote the composite

H^{0}(A_{C}, Ω_{A_{C}/C}^{1}) ⟶^{ϕ_{A}^{∗}⊗C} H^{0}(X_{1}(N)_{C}, Ω_{X_{1}(N)/C}^{1}) ⟶^{F -exp} C[[q]].

Now ϕ∗_{A} is injective: if X = X_{1}(N), this follows by considering pullbacks along the sequence of maps in (4); if X = X_{0}(N), then a similar argument works, noting that the pullback of differentials along X_{1}(N) → X_{0}(N) is injective. Also, F -exp is injective since the Fourier expansion map is injective. Thus ϕ is injective.

By Lemma 4.5 and diagram (6), we have ϕ(H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{1})) ⊆ ϕ(⊕_{j}Zω_{j}). As ϕ is injective, H^{0}(A_{Z}, Ω_{A_{Z}}^{1}) ⊆ ⨁_{j} Zω_{j}.

We obtain the following theorem as a corollary of Proposition 4.6:

Theorem 4.7. The Manin constant c_{A} is an integer.

We finish this section with a few remarks.

Remark 4.8. The quotient

\frac{⊕_{j}Zω_{j}}{H^{0}(A_{Z}, Ω_{A_{Z}}^{1})} ≅ \frac{ψ(S_{2}(Γ_{1}(N), Z))}{ϕ∗_{A}(H^{0}(A_{Z}, Ω_{A_{Z}}^{1}))} ≅ \frac{F -exp(ψ(S_{2}(Γ_{1}(N), Z)[I]))}{q-exp (ϕ∗_{A} (H^{0}(A_{Z}, Ω_{A_{Z}}^{1})))}

is in fact a module over T, and hence one may in general be interested in its module structure, as opposed to just the Manin constant, which is its order.

Remark 4.9. The reason we used the model X_{μ}(N) was that we needed a smooth model over Z so that we can use the Néron mapping property to define a q-expansion map over Z that agreed with the usual one over C. When A is a quotient of J_{0}(N), (i.e., when J = J_{0}(N)), we could use a model for X_{0}(N) in the proof above, as we describe now.

By [KM85, 6.6.1], the moduli problem [Γ_{0}(N)] ([KM85, 3.4]) is relatively representable and finite. The moduli problem [Γ_{0}(N)] is also regular (by [KM85, 6.6.1] again), and hence normal, and so the associated coarse moduli scheme M([Γ_{0}(N)]) is normal (by [KM85, 8.1.2]). So one can use [KM85, §8.6] to compactify it; call the resulting compactification M_{0}(N). Let M_{0}(N)^{0} be the open part of M_{0}(N) where the projection to Spec Z is smooth. For the case where J = J_{0}(N), we could have used M_{0}(N)^{0} instead of X_{μ}(N) for proving integrality of the Manin constant. This is what was done in the proof of Prop. 2 in [Edi91], but some of the details were skipped, which we mention two paragraphs below.

Note that q-expansion maps over Z or Z[1/m] (where m is the largest square that divides N) on differentials on certain models of X_{0}(N) have been constructed in several places in the literature (e.g., [Maz78, p.141], [AU96, p.271]), and the usual reference given is [DR73]. However, this seems inadequate, since in [DR73], one has to invert N to get a moduli-theoretic interpretation at the cusps. And in [KM85], while the models are over Z, they do not give a moduli interpretation at the cusps. We now indicate how the construction of a q-expansion map over Z for differentials on M_{0}(N)^{0} can be justified (this is probably well-known to experts).

One method, communicated to us by B. Edixhoven, is as follows: Consider the Tate curve Tate(q) over Z((q)) as in [KM85, p.258] along with its canonical subgroup μ_{N}. This gives us an element of M_{0}(N)(Z((q))) as in [KM85, §8.11]. One then verifies that this element extends uniquely to an element of M_{0}(N)(Z[[q]]). Thus we get a map τ : Spec Z[[q]] → M_{0}(N) and composing with the map Spec Z → Spec Z[[q]] (given by q ↦ 0)), we get a point in M_{0}(N)(Z), called the cusp ∞. The structure along ∞ of M_{0}(N) is described in [Edi90, §1.2]; in particular, the completion along ∞ is given by Z[[q]], and so ∞ is a smooth point. Thus the map τ factors through M_{0}(N)^{0}, and so we can define a q-expansion map on H^{0}(M_{0}(N)^{0}, Ω_{M_{0}(N)^{0}/Z}) as we did (for X_{μ}(N)) above. The usual q-expansion map over C is just given by extending scalars from Z to C in the description just above, and hence our q-expansion map is compatible with the usual one over C.

Another method, which is more moduli-theoretic, was communicated to us by B. Conrad, and is as follows: it is shown in [Con03] that one can merge the “affine” moduli-theoretic Z-theory in [KM85] with the “proper” modulitheoretic Z[1/N]-theory in [DR73]. Using this, one can show that the proper schemes over Z in [KM85] are in fact moduli schemes for generalized elliptic curves with “Drinfeld structure”. Then, by the moduli interpretation, the Tate curve with its canonical subgroup gives a map τ and the cusp ∞ as in the previous paragraph. Next, one can use a deformation theoretic argument to show that the cusp ∞ is a smooth point, i.e., that τ factors through M_{0}(N)^{0}. As in the previous paragraph, one can now pullback via τ to get the q-expansion map over Z, which by the moduli interpretation agrees with the usual q-expansions over C.

Remark 4.10. The Manin constants c_{A} might not equal 1. For example, let Γ = Γ_{0}(N), and suppose A = J_{0}(N) is the quotient by the trivial ideal. Let us work in the setting of Remark 4.9, using the model M_{0}(N)^{0} over Z of X_{0}(N). Then, since A = J_{0}(N), the map ϕ∗_{A} is just

H^{0}(J_{0}(N)_{Z}, Ω_{J_{0}(N)/Z}^{1}) → H^{0}(M_{0}(N)^{0}, Ω_{M_{0}(N)^{0}/Z}^{1}),

which is an isomorphism. Let us identify S_{2}(Z) with its image in Z[[q]]. Then using the argument in the proof of Proposition 4.6 we see that c_{A} is the order of the cokernel of the map

H^{0}(M_{0}(N)^{0}, Ω_{M_{0}(N)^{0}/Z}^{1}) ⟶^{q-exp} S_{2}(Z),  (7)

where q-exp is the q-expansion map discussed in Remark 4.9. The map (7) need not be surjective, and the order of its cokernel can be calculated by using methods in [DR73, VII.3.17] (see [Edi03]). For example, B. Edixhoven observed that for N = 33 the cokernel has order 3, so c_{J_{0}(33)} = 3. B. Edixhoven also informed us that if N is square free, then the map (7) is surjective if and only if there are no old spaces in S_{2}(Γ_{0}(N), C) (cf. [Edi03]). See also Remark 4.3 for an example of a quotient of J_{0}(N), with N prime, and with Manin constant 2.

Note that H^{0}(M_{0}(N)_{Z}^{0}, Ω_{M_{0}(N)^{0}/Z}^{1}) is precisely the subgroup of S_{2}(Q) = H^{0}(X_{0}(N), Ω_{X_{0}(N)/Q}^{1}) of elements that have integral Fourier expansion at all the cusps (this follows from the interpretation in [Edi03] of the integrality condition in terms of a differential having no pole along along any irreducible component of M_{0}(N)^{0}). Whereas S_{2}(Z) consists of differentials that are required only to have integral Fourier expansion at the cusp ∞.

If one assumes the BSD conjecture, then a comparison of formulas for the ratio L(J_{e}, 1)/Ω_{J_{e}}, where J_{e} is the winding quotient of prime level, and the corresponding formulas for winding quotients of level a product of two distinct primes (see [Aga00, Thm. 3.2.2 and Thm. 4.2.1]) suggests that the Manin constant of such winding quotients is not 1 when there are old forms involved (see [Aga00, §4.2.1] for details).

4.3 Generalizations of theorems of Mazur and Raynaud

In this section, we prove the following two theorems:

Theorem 4.11. Let A be a quotient of J = J_{0}(N) by an ideal of the Hecke algebra such that the quotient map factors through J_{0}(N)^{new}. If p is a prime such that p | c_{A}, then p^{2} | 4N.

Theorem 4.12. Let f be a newform on Γ_{0}(N), and let A_{f} be the associated newform quotient. If 4 ∤ N, then ord_{2}(c_{A_{f}}) ≤ dim A_{f}.

Theorem 4.11 generalizes Mazur’s Theorem 2.5, while Theorem 4.12 generalizes Raynaud’s Theorem 2.6.

The proofs of the theorems are similar. Suppose p || N. The reduction X_{0}(N)_{F_{p}} is a union of two copies of X_{0}(N/p)_{F_{p}}, identified at the supersingular points. A differential on X_{0}(N)_{F_{p}} has q-expansion 0 if and only if it vanishes on the component X of X_{0}(N)_{F_{p}} that contains ∞. Since there can be differentials that vanish on X, but not on the other component, the q-expansion map on differentials on X_{0}(N)_{F_{p}} need not be injective. However, as Mazur observed in [Maz78], if a differential is an eigenvector for the Atkin-Lehner involution W_{p}, then it is 0 on one component if and only if it is 0 on both components, since W_{p} swaps the two components. That the q-expansion map is injective on each eigenspace for W_{p} is one of the key ideas behind the proofs of Theorems 4.11, 4.12, and 4.13.

Proof of Theorem 4.11. We want to prove that that c_{A} is a unit in Z[\frac{1}{2m}], where m is the largest square dividing N. We do this by generalizing the proof of [Maz78, Prop. 3.1].

Let R = Z[\frac{1}{2m}], and let J_{R} denote the Néron model of J_{0}(N) over R. Let X be the smooth locus of a minimal proper regular model for X_{0}(N) over R, and let Ω_{X} denote the sheaf of “regular differentials”, denoted Ω in [Maz78, §2(e)].

Let π denote the map J_{0}(N) → A. Consider the diagram

H^{0}(A_{R}, Ω_{A_{R}}) ⟶^{π^{∗}} H^{0}(J_{R}, Ω_{J_{R}}) ≅ H^{0}(X , Ω_{X}) ⟶^{q-exp} R[[q]],  (8)

where the map q-exp is as in [Maz78, §2(e)]. (Note that we defined a different q-expansion map in Section 4.2.)

The composite of the maps in (8) must be an inclusion because H^{0}(A_{R}, Ω_{A_{R}}) is torsion free and the composite is an inclusion after tensoring with C. To show that the generalized Manin constant is a unit in R, it suffices to check that the image of H^{0}(A_{R}, Ω_{A_{R}}) in R[[q]] is saturated, in the sense that the cokernel is torsion free. This is because the image of S_{2}(Γ_{0}(N); R)[I] is saturated in R[[q]] and S_{2}(Γ_{0}(N); R)[I] ⊗ Q = H^{0}(A_{R}, Ω_{A_{R}}) ⊗ Q.

For the image of H^{0}(A_{R}, Ω_{A_{R}}) in R[[q]] to be saturated means that the quotient D is torsion free. Let ℓ be a prime not dividing 2m. Tensoring

0 → H^{0}(A_{R}, Ω_{A_{R}}) ⟶^{q-exp} R[[q]] → D → 0

with F_{ℓ}, we obtain

0 → D[ℓ] → H^{0}(A_{R}, Ω_{A_{R}}) ⊗ F_{ℓ} → F_{ℓ}[[q]] → D ⊗ F_{ℓ} → 0.

Here we have used either the snake lemma applied to multiplication-by-ℓ or that Tor^{1}(D, F_{ℓ}) is the ℓ-torsion in D, and that Tor^{1}(−, F_{ℓ}) vanishes on the torsion-free group R[[q]]. To show D[ℓ] = 0, it suffices to prove injectivity of

Φ : H^{0}(A_{R}, Ω_{A_{R}}) ⊗ F_{ℓ} ⟶ F_{ℓ}[[q]].

Since A is optimal, J has good or semistable reduction at ℓ, and ℓ ≠ 2, [Maz78, Cor 1.1] gives an exact sequence

0 → H^{0}(A_{Z_{ℓ}}, Ω_{A_{Z_{ℓ}}}) → H^{0}(J_{Z_{ℓ}}, Ω_{J_{Z_{ℓ}}}) → H^{0}(B_{Z_{ℓ}}, Ω_{B_{Z_{ℓ}}}) → 0

where B = ker(J → A). Since H^{0}(B_{Z_{ℓ}}, Ω_{B_{Z_{ℓ}}}) is torsion free, the map

H^{0}(A_{R}_{Z_{ℓ}}, Ω_{A_{R}_{Z_{ℓ}}}) ⊗ F_{ℓ} → H^{0}(J_{R}_{Z_{ℓ}}, Ω_{J_{R}_{Z_{ℓ}}}) ⊗ F_{ℓ} ≅ H^{0}(X_{F_{ℓ}}, Ω_{X_{F_{ℓ}}})

is injective. We also remark that

H^{0}(A_{R}, Ω_{A_{R}}) ⊗ F_{ℓ} ≅ H^{0}(A_{Z_{ℓ}}, Ω_{A_{Z_{ℓ}}}) ⊗ F_{ℓ},

because Z_{ℓ} is torsion free, hence flat over R. This proves injectivity of

H^{0}(A_{R}, Ω_{A_{R}}) ⊗ F_{ℓ} → H^{0}(X_{F_{ℓ}}, Ω_{X_{F_{ℓ}}}).

If ℓ ∤ N, then injectivity of Φ now follows from the q-expansion principle, which asserts that the q-expansion map H^{0}(X_{F_{ℓ}}, Ω_{X_{F_{ℓ}}}) → F_{ℓ}[[q]] is injective. (This part of the argument does not assume that A is new.)

Next suppose that ℓ | N; note that ℓ || N because ℓ ∤ m. As mentioned above, the reduction X_{F_{ℓ}} is a union of two copies of X_{0}(N/ℓ)_{F_{ℓ}} identified transversely at the supersingular points, and these two copies are swapped under the action of the Atkin-Lehner involution W_{ℓ}. If ω ∈ ker(Φ), then the q-expansion principle implies that ω vanishes on the irreducible component containing the cusp ∞. The action of W_{ℓ} on H^{0}(A_{R}, Ω_{A_{R}})⊗F_{ℓ} is diagonalizable since its minimal polynomial divides X^{2} − 1, and X^{2} − 1 has distinct roots since ℓ ≠ 2, and the eigenvalues ± 1 are in F_{ℓ}. Let ω ∈ ker(Φ) be in the +1 eigenspace for the action of W_{ℓ}. If ω is also nonzero on the component that does not contain ∞, then ω = W_{ℓ}(ω) is nonzero when restricted to the component that contains ∞, which is a contradiction. Therefore ω = 0. A similar argument shows that if ω ∈ ker(Φ) is in the −1 eigenspace for the action of W_{ℓ}, then ω = 0. Hence Φ is injective, as required.

Proof of Theorem 4.12. Recall that we want to prove that if A = A_{f} is a quotient of J = J_{0}(N) attached to a newform f, and 4 ∤ N, then ord_{2}(c_{A} ≤ dim(A). The proof closely follows the one in [AU96], except at the end we argue using indexes instead of multiples.

Let B denote the kernel of the quotient map J → A. Consider the exact sequence 0 → B → J → A → 0, and the corresponding complex B_{Z_{2}} → J_{Z_{2}} → A_{J_{Z_{2}}} of Néron models. Because J_{Z_{2}} has semiabelian reduction (since 4 ∤ N), Theorem A.1 of the appendix of [AU96, pg. 279–280], due to Raynaud, implies that there is an integer r and an exact sequence

0 → Tan(B_{Z_{2}}) → Tan(J_{Z_{2}}) → Tan(A_{Z_{2}}) → (Z/2Z)^{r} → 0.

Here Tan is the tangent space at the 0 section; it is a free abelian group of rank equal to the dimension. Note that Tan is Z_{2}-dual to the cotangent space, and the cotangent space is isomorphic to the global differential 1-forms. The theorem of Raynaud mentioned above is the generalization to e = p − 1 of [Maz78, Cor. 1.1], which we used above in the proof of Theorem 4.11.

Let C be the cokernel of Tan(B_{Z_{2}}) → Tan(J_{Z_{2}}). We have a diagram

\xymatrix{0 \ar[r] & Tan(B_{Z_{2}}) \ar[r] & Tan(J_{Z_{2}}) \ar[rr] \ar@{->>}[dr] &  & Tan(A_{Z_{2}}) \ar[r] & (Z/2Z)^{r} \ar[r] & 0. \\  &  &  & C \ar@{^{(}->}[ur]} (9)

Note that C ⊂ Tan(A_{Z_{2}}), so C is torsion free, hence C is a free Z_{2}-module of rank d = dim(A). Let C^{∗} = Hom_{Z_{2}}(C, Z_{2}) be the Z_{2}-linear dual of C. Applying the Hom_{Z_{2}}(−, Z_{2}) functor to the two short exact sequences in (9), we obtain exact sequences

0 → C^{∗} → H^{0}(J_{Z_{2}}, Ω_{J/Z_{2}}) → H^{0}(B_{Z_{2}}, Ω_{B/Z_{2}}) → 0,

and

0 → H^{0}(A_{Z_{2}}, Ω_{A/Z_{2}}) → C^{∗} → (Z/2Z)^{r} → 0.  (10)

Note that the (Z/2Z)^{r} on the right in (10) is really Ext_{Z_{2}}^{1}((Z/2Z)^{r}, Z_{2}), which is isomorphic to (Z/2Z)^{r}. Also, (10) implies that r ≤ d = dim(A).

Let X′ be the smooth locus a minimal proper regular model for X_{0}(N) over Z[1/m], where m is the largest square dividing N, and let Ω_{X′} denote the sheaf of “regular differentials” on X′ (denoted Ω in [Maz78, §2(e)]).

Arguing as in the last two paragraphs of the proof of Theorem 4.11 above (note that since A is attached to a single newform, the Atkin-Lehner involution W_{2} acts either as +1 or as −1), we see that the composition

C^{∗} ⊗ F_{2} → H^{0}(J_{Z_{2}}, Ω_{J/Z_{2}}) ⊗ F_{2} ≅ H^{0}(X′_{F_{2}}, Ω_{X′_{F_{2}}} ) ⟶^{q-exp} F_{2}[[q]]

is injective. Thus, just as in the proof of Theorem 4.11, we see that the image of C^{∗} in Z_{2}[[q]] is saturated. The Manin constant for A at 2 is the index of the image via q-expansion of H^{0}(A_{Z_{2}}, Ω) in Z_{2}[[q]] in its saturation. Since the image of C^{∗} in Z_{2}[[q]] is saturated, the image of C^{∗} is the saturation of the image of H^{0}(A_{Z_{2}}, Ω), so the Manin index at 2 is the index of H^{0}(A_{Z_{2}}, Ω) in C^{∗}, which is 2^{r} by (9), hence is at most 2^{d}.

4.4 The Manin constant and congruence primes

In this section, we prove the following theorem, whose proof builds on techniques of [AU96]:

Theorem 4.13. Let A = A_{f} be a quotient of J = J_{0}(N) attached to a newform f. If p | c_{A_{f}} is a prime, then p^{2} | N or p | r̃_{A_{f}}.

The key idea is to project the “Manin index” to the differentials on the dual of A and to use a “conjugate isogeny” to bring it back to differentials on a model of X_{0}(N), and then use an argument similar to the one in the last two paragraphs of the proof of Theorem 4.11. Note that the techniques of the proof of this theorem can be used to prove that if the quotient map J_{0}(N) → A factors through J_{0}(N)^{new}, and if p | c_{A}, then p^{2} | N or p = 2 or p | r̃_{A} (see Remark 4.17). However, this does not add anything new, in light of Theorem 4.11.

We will use notation consistent with [AU96] since we will follow their techniques closely. If G is a finite group, we denote its order by # G.

Suppose A_{1} and A_{2} are abelian varieties such that there is an isogeny f : A_{1} → A_{2}. If n is a positive integer which annihilates kerf, then the multiplication by n map on A_{1} factors through A_{1}/kerf ≅ A_{2}, thus giving an isogeny f′ : A_{2} → A_{1} such that f′ ∘ f is the multiplication by n map on A_{1}. Also one sees that f ∘ f′ is the multiplication by n map on A_{2}.

We apply this to our situation as follows. Recall that ϕ_{2} denotes the quotient map J → A, and ϕ_{1} denotes the composition of the dual map A^{∨} → J^{∨} with the canonical polarization J^{∨} ≅ J. By Proposition 3.2, the composite

A^{∨} ⟶^{ϕ_{1}} J ⟶^{ϕ_{2}} A  (11)

is an isogeny. As in Definition 3.3, we denote the exponent of the kernel of this isogeny by ñ_{A}. There is an isogeny ϕ′ : A → A^{∨} such that the composite

A^{∨} ⟶^{ϕ_{1}} J ⟶^{ϕ_{2}} A ⟶^{ϕ′} A^{∨}  (12)

is the multiplication by ñ_{A} map on A^{∨}, and the composite

A ⟶^{ϕ′} A^{∨} ⟶^{ϕ_{1}} J ⟶^{ϕ_{2}} A  (13)

is the multiplication by ñ_{A} map on A.

Pulling back differentials along ϕ_{2} then ϕ_{1} in (11), we obtain maps:

H^{0}(A_{C}, Ω_{A/C}^{1}) ⟶^{ϕ_{2}^{∗}} H^{0}(J_{C}, Ω_{J/C}^{1}) ⟶^{ϕ_{1}^{∗}} H^{0}(A_{C}^{∨}, Ω_{A^{∨}/C}^{1}).

Let m denote the largest square that divides the level N and let S = Spec Z[1/m]. Let M_{0}(N) be as in Remark 4.9. Then M_{0}(N)_{S} is semistable over S. Let Ω be the relative dualizing sheaf of M_{0}(N)_{S} over S. Consider the map

q-exp : H^{0}(M_{0}(N)_{S}, Ω) ↪ Z[1/m][[q]]

in [AU96, §2.1] (cf. Remark 4.9). Note that we are abusing notation slightly since we had defined a different q-expansion map in Section 4.2.

As mentioned in [AU96, §2.1] we have an inclusion

q-exp : H^{0}(M_{0}(N)_{S}, Ω) ↪ S_{2}(Z[1/m])

(this really follows from the discussion in Section 4.2). This map is not an isomorphism in general, but it induces an isomorphism

q-exp : H^{0}(M_{0}(N)_{F_{p}}, Ω) ⟶^{≅} S_{2}(F_{p})  (14)

for each prime p that does not divide N (see [AU96, §2.1], and the arguments in the proof of Theorem 4.11).

We have

H^{0}(M_{0}(N)_{S}, Ω) ↪ S_{2}(Z[1/m]) ↪ S_{2}(C) ≅ H^{0}(J_{C}, Ω_{J/C}^{1}).

Applying ϕ_{1}^{∗} to the first two groups, we get an injection

ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ↪ ϕ_{1}^{∗}(S_{2}(Z[1/m])),

where the source and target are viewed as sitting in H^{0}(A_{C}^{∨}, Ω_{A^{∨}/C}). Denote the cokernel of the above map by C. It is a finite group and, by (14), the only primes that can divide its order are the primes that divide N. An easy generalization of [AU96, Prop. 3.2] gives

ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) = H^{0}(A_{S}^{∨}, Ω_{A^{∨}/S}^{1}),

so we have an exact sequence

0 → H^{0}(A_{S}^{∨}, Ω_{A^{∨}/S}^{1}) → ϕ_{1}^{∗}(S_{2}(Z[1/m])) → C → 0.

On considering the quotient of the middle group above by the pullback of H^{0}(A_{S}, Ω_{A/S}^{1}) under ϕ_{2} ∘ ϕ_{1}, we obtain

0 → \frac{H^{0}(A_{S}^{∨}, Ω_{A^{∨}/S}^{1})}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1})} → \frac{ϕ_{1}^{∗}(S_{2}(Z[1/m]))}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1})} → C → 0.  (15)

Now ϕ_{1}^{∗} is injective when restricted to ϕ_{2}^{∗}H^{0}(A_{C}, Ω_{A/C}^{1}), because the pullback of the composite of the maps in (13) is injective, since it is multiplication by ñ_{A} on a vector space over C. So, since

S_{2}(Z)[I] ⊆ ϕ_{2}^{∗}H^{0}(A_{C}, Ω_{A/C}^{1}),

we have a natural isomorphism

\frac{S_{2}(Z)[I] ⊗ Z[1/m]}{ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1})} ⟶^{≅} \frac{ϕ_{1}^{∗}(S_{2}(Z)[I] ⊗ Z[1/m])}{ϕ_{1}^{∗}(ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1}))}.

If n and m are positive integers, let n_{m} denote the largest divisor of n that is coprime to m.

By the discussion in Section 4.2,

(c_{A})_{m} = # (\frac{S_{2}(Z)[I] ⊗ Z[1/m]}{ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1})}) .

So

(c_{A})_{m} = # (\frac{ϕ_{1}^{∗}(S_{2}(Z)[I] ⊗ Z[1/m])}{ϕ_{1}^{∗}(ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1}))}) .

Hence

# (\frac{ϕ_{1}^{∗}(S_{2}(Z[1/m]))}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1})}) = (c_{A})_{m} · # (\frac{ϕ_{1}^{∗}(S_{2}(Z[1/m]))}{ϕ_{1}^{∗}(S_{2}(Z)[I] ⊗ Z[1/m])}) . (16)

As in the proof of [AU96, Prop. 3.3], we have isomorphisms

(\frac{S_{2}(Z)}{S_{2}(Z)[I] ⊕ W (I} ) ⊗ Z[1/m] ⟶^{≅} \frac{S_{2}(Z[1/m])}{(S_{2}(Z)[I] ⊗ Z[1/m]) ⊕ (W (I) ⊗ Z[1/m])}
⟶^{≅} \frac{ϕ_{1}^{∗}(S_{2}(Z[1/m])))}{ϕ_{1}^{∗}(S_{2}(Z)[I] ⊗ Z[\frac{1}{m}])}.

Thus

# (\frac{ϕ_{1}^{∗}(S_{2}(Z[1/m])))}{ϕ_{1}^{∗}(S_{2}(Z)[I] ⊗ Z[1/m])}) = (r_{A})_{m}.

Putting this in (16) and then using (15), we get

(c_{A})_{m} · (r_{A})_{m} = # (\frac{H^{0}(A_{S}^{∨}, Ω_{A^{∨}/S}^{1})}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1})}) · #C.  (17)

Since the composite of the maps in (12) is multiplication by ñ_{A}, we see that

multiplication by some power of ñ_{A} kills (\frac{H^{0}(A_{S}^{∨},Ω_{A^{∨}/S}^{1})}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S},Ω_{A/S}^{1})}). Thus we obtain

the following lemma:

Lemma 4.14. If p | # (\frac{H^{0}(A_{S}^{∨},Ω_{A^{∨}/S}^{1})}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S},Ω_{A/S}^{1})}) is a prime, then p | ñ_{A}.

We already remarked that a prime can divide #C only if it divides N. The main addition to the techniques of [AU96] is the following result, which further controls the primes that can divide #C:

Proposition 4.15. If A = A_{f} is a newform quotient of J_{0}(N) and p | #C is a prime, then p^{2} | N or p | r̃_{A}.

Before proving Proposition 4.15 we use it to prove Theorem 4.13.

Proof of Theorem 4.13. Suppose p^{2} ∤ N and p | c_{A}. Then p | (c_{A})_{m}, and so

by equation (17), p | # (\frac{H^{0}(A_{S}^{∨},Ω_{A^{∨}/S}^{1})}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S},Ω_{A/S}^{1})}) or p | #C. In the former case, by

Lemma 4.14, p | ñ_{A}, and hence by Proposition 3.11, p | r̃_{A} and in the latter case, by Proposition 4.15, p | r̃_{A}.

Remark 4.16. The obstruction to proving a generalization of Theorem 2.7 to dimension greater than 1 lies in equation (17). When A is an elliptic curve, Abbes-Ullmo [AU96] prove that the quotient of differentials on the right hand side of (17) divides (r_{A})_{m}. Thus (c_{A})_{m} | #C, which proves Theorem 2.7, since the prime divisors of #C divide N. When A has dimension bigger than 1, the relationship between the quotient of differentials and (r_{A})_{m} is unclear. For example, Remark 3.6 suggests that divisibility might sometimes fail when multiplicity one fails.

Proof of Proposition 4.15. We have the exact sequence

0 → ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) → ϕ_{1}^{∗}(S_{2}(Z[1/m])) → C → 0.  (18)

Suppose p is a prime such that p^{2} ∤ N and p ∤ r̃_{A}. We want to show that p ∤ #C. We already know that the only primes that can divide #C are those that divide N; so we may assume that p exactly divides N. Then considering the multiplication by p map applied to each term of the sequence (18) and using the snake lemma, we get:

0 → C[p] → ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω))⊗F_{p} ⟶^{q-exp} ϕ_{1}^{∗}(S_{2}(Z[1/m]))⊗F_{p} → C⊗F_{p} → 0

(note the similarity to the situation in the proof of Theorem 4.11). Then to show that p ∤ #C, i.e., that C[p] is trivial, all we have to show is that the map

ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p} ⟶^{q-exp} ϕ_{1}^{∗} (S_{2} (Z[1/m])) ⊗ F_{p}  (19)

is injective.

The key idea is to use the isogeny ϕ′ defined at the beginning of this section. We have maps

A^{∨} →^{ϕ_{1}} J →^{ϕ_{2}} A →^{ϕ′} A^{∨}  (20)

such that the composite is multiplication by ñ_{A}. Let ϕ′′ = ϕ′ ∘ϕ_{2}. Pulling back differentials, we get

H^{0}(A_{C}^{∨}, Ω_{A^{∨}/C}^{1}) ⟶^{ϕ′′^{∗}} H^{0}(J_{C}, Ω_{J/C}^{1}) ⟶^{ϕ_{1}^{∗}} H^{0}(A_{C}^{∨}, Ω_{A^{∨}/C}^{1}),  (21)

where the composite is again multiplication by ñ_{A}.

By the Néron mapping property, the maps (20) extend to the corresponding Néron models, and we see that

ϕ^{′′∗}(ϕ_{1}^{∗}(H^{0}(J_{S}, Ω_{J/S}))) ⊆ H^{0}(J_{S}, Ω_{J/S}).

By [AU96, p.271], the canonical morphism X_{0}(N) → J_{0}(N) induces a canonical isomorphism

H^{0}(J_{S}, Ω_{J/S}) →^{≅} H^{0}(M_{0}(N)_{S}^{0}, Ω) = H^{0}(M_{0}(N)_{S}, Ω).

Thus we see that the image of

ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) = ϕ_{1}^{∗}(H^{0}(J_{S}, Ω_{J/S}))

under ϕ′′^{∗} lands in H^{0}(M_{0}(N)_{S}, Ω) = H^{0}(J_{S}, Ω_{J/S}). Also, since p ∤ r̃_{A}, we have

S_{2}(Z[1/m]) ⊗ F_{p} = S_{2}(Z[1/m])[I] ⊗ F_{p} ⊕ (W (I) ∩ S_{2}(Z[1/m])[I]) ⊗ F_{p}.

Thus if f ∈ S_{2}(Z[1/m]) ⊗ F_{p}, then there exist unique f_{1} ∈ S_{2}(Z[1/m])[I] ⊗ F_{p} and f_{2} ∈ (W (I) ∪ S_{2}(Z[1/m])[I]) ⊗ F_{p} such that f = f_{1} + f_{2}. It then follows that ϕ_{1}^{∗}f = ϕ_{1}^{∗}f_{1}, and so ϕ′′^{∗}(ϕ_{1}^{∗}f) = ñ_{A}f_{1} ∈ S_{2}(Z[1/m]) ⊗ F_{p}. Thus the image of ϕ_{1}^{∗}(S_{2}(Z[1/m])) ⊗ F_{p} under ϕ′′^{∗} lands in S_{2}(Z[1/m]) ⊗ F_{p}.

Hence, applying the maps in (21) to the groups in (19), which are subgroups of H^{0}(A_{C}^{∨}, Ω_{A^{∨}/C}^{1}), we get the following commutative diagram:

\xymatrix{ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω))_{F_{p}} \ar[r]^{^{ϕ′′^{∗}}} \ar[d]^{_{q-exp}} & H^{0}(M_{0}(N)_{S}, Ω)_{F_{p}} \ar[r]^{ϕ_{1}^{∗}} \ar[d]^{_{q-exp}} & ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω))_{F_{p}} \ar[d]^{_{q-exp}} \\ ϕ_{1}^{∗}(S_{2}(Z[1/m]))_{F_{p}} \ar[r]^{_{ϕ′′_{∗}}} & S_{2}(Z[1/m])_{F_{p}} \ar[r]^{ϕ_{1}^{∗}} & ϕ_{1}^{∗}(S_{2}(Z[1/m]))_{F_{p}}.}

The Atkin-Lehner involution W_{p} acts on ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p} and since A is attached to a newform, W_{p} acts as either +1 or −1. Suppose x is an element of ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p} that is in the +1 eigenspace for the action of W_{p} and in the kernel of the map in (19), i.e., the left-most q-exp map above. Then its image y = (ϕ′′^{∗})(x) in H^{0}(M_{0}(N)_{S}, Ω) ⊗ F_{p} above maps to zero in S_{2}(Z[1/m]) ⊗ F_{p} under the middle q-exp map, by commutativity of the first square. But we have H^{0}(M_{0}(N)_{S}, Ω) ⊗ F_{p} ≅ H^{0}(M_{0}(N)_{F_{p}}, Ω). Since p^{2} ∤ N, M_{0}(N)_{F_{p}} is a union of two irreducible components. Now q-exp(y) = 0 means that y ∈ H^{0}(M_{0}(N)_{F_{p}}, Ω) is zero on the component that contains the cusp ∞. But x is an eigenvector for W_{p}, and hence so is y. But W_{p} is an involution that swaps the two components of M_{0}(N)_{F_{p}}. Hence y is zero on all of M_{0}(N)_{F_{p}}; i.e., y = 0. Note that this part of the argument is very similar to the one towards the end of the proof of Theorem 4.11.

Looking at the top line in the diagram above, we find that x maps to zero under the composite. But its image under this composite is ñ_{A}x, and so ñ_{A}x = 0. Since p ∤ r̃_{A}, Proposition 3.11 shows that p ∤ ñ_{A}, and so x = 0. A similar argument shows that if x is an element of ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p} in the −1 eigenspace for the action of W_{p} and in the kernel of the map in (19), then x = 0. This shows that the map (19) is injective, which is what was left to prove.

Remark 4.17. Note that the fact that A is associated to a single newform was used only in the last two paragraphs of the proof above. We could have used the fact that the action of W_{p} on ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p} is diagonalizable if p ≠ 2 (e.g., see the last paragraph of the proof of Theorem 4.11; the paragraph at the beginning of Section 3.2 is also relevant here), to prove that if A is a quotient of J_{0}(N) by an ideal of the Hecke algebra such that the quotient map factors through J_{0}(N)^{new}, and if p | #C, then p^{2} | N or p = 2 or p | r̃_{A}. Then one would have the statement that for such a quotient, if p | c_{A}, then p^{2} | N or p = 2 or p | r̃_{A}.

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