The generalized Manin constant, congruence primes, and the modular degree

Amod Agashe  William A. Stein

May 21, 2000

Department of Mathematics, 940 Evans Hall, University of California, Berkeley, CA 94720, U.S.A. Email: amod@math.berkeley.edu, was@math.berkeley.edu

Abstract. We extend the techniques of a paper of Abbes and Ullmo to show that if E is a strong modular elliptic curve and if p is a prime which divides the Manin constant of E, but whose square does not divide the conductor of E, then p divides the modular degree of E. We also show that except for primes whose squares divide the conductor, the congruence primes associated to the newform corresponding to E divide the modular degree of E. Finally we generalize the notion of the Manin constant to arbitrary quotients of J_{0}(N) under the action of ideals of the Hecke algebra and mention the corresponding results and a conjecture for it.

1 Introduction and results

We recall the definition of the Manin constant and what is known about it. Let N be a positive integer. Let E be a strong modular elliptic curve (see, e.g., [Edix]) of level N and ϕ : X_{0}(N)_{Q} → E be the modular parametrization, where X_{0}(N)_{Q} is the modular curve over Q classifying elliptic curves with a given cyclic subgroup of order N. The degree of the parametrization, deg ϕ, is called the modular degree of E. Let E_{Z} be the Néron model of E. Then the Z-module of invariant differentials on E_{Z} is free of rank one; let ω be one of its two generators. Pulling ω back to X_{0}(N)_{Q}, we get a differential form ϕ^{∗}ω on X_{0}(N)_{Q}. Now there is a newform f corresponding to E and it gives another differential f dq/q on X_{0}(N)_{Q}. By the multiplicity one theorem, we have ϕ^{∗}ω = c_{E}f dq/q for some c_{E} ∈ Q^{∗}. By replacing ω by −ω, if necessary, we may assume that c_{E} is positive; it is called the Manin constant of E.

Conjecture 1.1 (Manin). c_{E} = 1.

The following theorems bound c_{E}.

Theorem 1.2 (Edixhoven [Edix]). c_{E} ∈ Z.

Theorem 1.3 (Mazur [Maz-iso]). If p is a prime such that p | c_{E}, then p^{2} | N or p = 2.

Theorem 1.4 (Raynaud [AbbUll]). If 2^{2} ∤ N, then v_{2}(c_{E}) ≤ 1.

Theorem 1.5 (Abbes-Ullmo [AbbUll]). If p is a prime such that p | c_{E}, then p | N.

Also B. Edixhoven has unpublished results which say that the only primes that can possibly divide c_{E} are 2, 3, 5, 7 and that universally bound the valuation of c_{E} with respect to these primes.

To this, we add the following theorem, whose proof uses an extension of the techniques of [AbbUll].

Theorem 1.6. If p is a prime such that p | c_{E}, then p^{2} | N or p | deg ϕ.

Considering what was known before, the only new information is that if 2 | N, but 2^{2} ∤ N and 2 ∤ deg ϕ, then 2 ∤ c_{E}. For example, for the elliptic curve 46A1 of [Cre-alg, Table 1], earlier one could conclude that c_{E} is 1 or 2. But one finds [Cre-alg, Table 5] that the modular degree of E is 5, and so c_{E} = 1.

Let r be the largest integer such that there exists a modular form g having integral Fourier coefficients, orthogonal to f with respect to the Petersson inner product, and satisfying g ≡ f mod r. It is called the congruence number and primes dividing it are called the congruence primes.

Then we have

Theorem 1.7 (Ribet [Zag]). deg ϕ | r; moreover, if N is prime, then r = deg ϕ.

A natural question that has been raised [FrMu] is about what one can say in the other direction. We prove:

Theorem 1.8. Let p be a prime such that p^{2} ∤ N. If p | \frac{r}{deg ϕ} then p | deg ϕ; in particular if p | r, then p | deg ϕ also.

Thus, if N is squarefree and either deg ϕ or N are odd, then r = deg ϕ. K. Ribet has pointed out that results like this come up in the proofs of multiplicity one results (see Remark 3.1). As an application of this theorem, from [Cre-alg, Table 3] (and using [Sturm]), the eigenform corresponding to the elliptic curve 26A shares a 2-congruence with the form corresponding to 26B, and 2^{2} ∤ 26, so we conclude that 2 divides the modular degree; and indeed on looking up [Cre-alg, Table 5], we find that the modular degree is 2.

In the next section, we generalize the notion of the Manin constant to quotients of J_{0}(N) of arbitrary dimension associated to ideals of the Hecke algebra and indicate which of the above results apply in the general situation. Finally, in the last two sections, we give the proofs of all the new theorems mentioned in the first two sections.

Acknowledgements: We thank A. Abbes for pointing out some mistakes and explaining some of the results used in the proofs, and K. Ribet and E. DeShalit for conversations regarding Remark 3.1.

2 A generalization of the Manin constant

As before, let N be any positive integer. Let J = J_{0}(N)_{Q} be the Jacobian of X_{0}(N) and let T denote the sub-ring of endomorphisms of J generated by the Hecke-operators T_{l} for primes l ∤ N and by U_{p} for p | N. It is called the Hecke algebra. Let I be an ideal of T and A be the quotient abelian variety J/IJ defined over Q. If R = Z[1/n] for some non-zero integer n, then for any abelian variety B over Q, let B_{R} denote its Néron model over R. If R ⊆ C is any ring, let S_{2}(R) denote the subgroup of S_{2}(Γ_{0}(N), C) consisting of modular forms whose q-expansion (at the cusp ∞) has coefficients in R.

2.1 Motivation and definition

In the Birch-Swinnerton-Dyer conjecture for A, one considers the ratio of the residue of the Lfunction L_{A}(s) at s = 1 to the real volume of A (e.g., as in [Lang, III, §5]). This real volume is calculated by using a set of generators for H^{0}(A_{Z}, Ω_{A/Z}^{1}), which is the group of global differentials on the Néron model of A. But in calculations (for example, see [Ag-inv], [AgSt], [Ag-pq]), it is easier to use a set of generators of S_{2}(Z)[I], which is the group of cusp forms with integral Fourier expansion (at the cusp ∞) annihilated by I.

Both H^{0}(A_{Z}, Ω_{A/Z}^{1}) and S_{2}(Z)[I] are lattices of full rank in H^{0}(A_{Q}, Ω_{A/Q}^{1}) under the obvious embeddings.

Definition 2.1. The generalized Manin constant c_{A} is the module index [S_{2}(Z)[I] : H^{0}(A_{Z}, Ω_{A/Z}^{1})], i.e., if d is the dimension of A, {w_{1}, ..., w_{d}} is a basis of H^{0}(A_{Z}, Ω_{A/Z}^{1}), and {ω_{1}, ..., ω_{d}} a basis of S_{2}(Z)[I], then c_{A} is the absolute value of the element c of Q satisfying ∧_{i}w_{i} = c ∧_{j} ω_{j}.

Thus, in calculations regarding the Birch-Swinnerton-Dyer conjecture, one finds the real volume with respect to S_{2}(Z)[I] and corrects it by c_{A}. Note that the generalized Manin constant in this form was considered by Gross [Gross] and Lang [Lang, III, §5]. If A is an elliptic curve, then c_{A} is the usual Manin constant.

2.2 A finer definition

We essentially follow the ideas in [Edix, Prop. 2]. Let M_{0}(N) denote the compactified coarse moduli scheme \overline{M}([Γ_{0}(N)]) ([KaMa, 8.6]) and let M_{0}(N)^{0} be the open part of M_{0}(N) where the projection to SpecZ is smooth. By the Néron property we get maps M_{0}(N)^{0} → J_{Z} → A_{Z}. By abuse of notation, call the composite map ϕ also. It follows from [KaMa, Thm 8.11.10] that the formal completion of M_{0}(N) along the (unramified) cusp ∞ is Spf(Z[[q]]). Thus we have a map H^{0}(M_{0}(N)^{0}, Ω_{M_{0}(N)^{0}/Z}^{1}) → Z[[q]] denoted by q-exp. We have the series of maps:

H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{1}) →^{ϕ^{∗}} H^{0}(M_{0}(N)^{0}, Ω_{M_{0}(N)^{0}/Z}^{1}) →^{q-exp} Z[[q]].

The image of H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{1}) under the composite lands in S_{2}(Z)[I] ⊆ Z[[q]]. Let

C_{A} = \frac{S_{2}(Z)[I]}{q-exp(ϕ^{∗}(H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{1})))}

be the finite quotient group. It is in fact a module over T and c_{A} is just the order of C_{A}. In particular, the above argument shows

Theorem 2.2 (generalization of Thm 1.2 of Edixhoven). c_{A} ∈ Z.

2.3 Results and a conjecture

At first, one might think that the generalized Manin constant should also be 1. In the formula for the ratio of the special L-value to the real volume (calculated using cusp forms) for the winding quotient of level a product of two distinct primes [Ag-pq], primes dividing the level appeared in the numerator. The only explanation, if one assumes the Birch-Swinnerton-Dyer formula, was that they divided the generalized Manin constant. For example, for A = J_{0}(33) considered as a quotient of itself (by the trivial ideal), they indicated that c_{J_{0}(33)} should be 3. When A = J_{0}(N), c_{A} is just the order of the cokernel of the q-expansion map, and in general, the q-expansion map is not surjective. In fact, B. Edixhoven pointed out that for N = 33, the cokernel has order 3, showing that c_{J_{0}(33)} is indeed 3. Thus the generalized Manin constant is not one in general.

However the similar ratio of the L-value to the real volume for winding quotients of prime level [Ag-inv] showed no appearance of primes dividing the level (in general). Also similar calculations for quotients of J_{0}(N) associated to newforms [AgSt] showed no appearance of such primes. Moreover, it is the presence of oldforms that makes the q-expansion map not surjective (in general). This suggests that c_{A} = 1 for quotients that factor through J^{new}, where J^{new} = J/J_{old}, where, in turn, J_{old} is the subvariety of J = J_{0}(N) generated by the images of the degeneracy maps from J_{0}(M) for M | N.

We have the following results in that direction:

Theorem 2.3 (generalization of Thm 1.3 of Mazur). If A factors through J^{new} and if p is a prime such that p | c_{A}, then p^{2} | N or p = 2.

Also, it is easy to see that Raynaud’s result generalizes:

Theorem 2.4 (Raynaud [AbbUll]). If A factors through J^{new} and 2^{2} ∤ N, then v_{2}(c_{A}) ≤ dimA.

Let ϕ_{2} denote the quotient map J → A. Since A is an optimal quotient of J, dualizing it, we get an injection ϕ_{1} : Â → J where Â is the dual abelian variety of A. Then the composite Â →^{ϕ_{1}} J →^{ϕ_{2}} A is an isogeny; let n_{A} be the smallest integer such that multiplication by n_{A} kills the kernel of this isogeny.

Then we have the following theorem, whose proof is given in §4:

Theorem 2.5 (extension of Thm 1.6). If A factors through J^{new} and if p is a prime such that p | c_{A}, but p^{2} ∤ N, then p | n_{A}.

It is hoped that one can improve this to prove the direct generalization of the statement of Thm 1.5 of Abbes-Ullmo: if A factors through J^{new} and p is a prime that divides c_{A}, then p divides N.

In [AgSt], for quotients of J associated to newforms, there are no instances of primes dividing the level appearing in the formula for the ratio of the special L-value to the real volume. Thus if the conjectural formula of Birch-Swinnerton-Dyer is true, then one does not expect primes whose squares divides the level to intervene in c_{A} either. Also computations of [FLSSSW] involving Jacobians of genus 2 curves that were quotients of J^{new} found c_{A} = 1 in 28 cases, including quotients at 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.

This has led the authors to make the following conjecture.

Conjecture 2.6. If A factors through J^{new} then c_{A} = 1.

Note that strong modular elliptic curves do factor through J^{new} and so this conjecture is a generalization of the conjecture of Manin (Conjecture 1.1).

2.4 Generalizations of congruence number and modular degree

Let W (I) denote the orthogonal complement of S_{2}(Z)[I] in S_{2}(C) with respect to the Petersson inner product. Let r_{A} = #(\frac{S_{2}(Z)}{S_{2}(Z)[I]⊕(W (I)∩S_{2}(Z))}). Note that this coincides with the definition of the congruence number r in §1 when A is an elliptic curve. Also, for an elliptic curve A, n_{A} equals the modular degree. An easy generalization of Thm. 1.7 (e.g., following the proof of [AbbUll, Lem. 3.2]) shows that n_{A} | r_{A}.

In §4, we show:

Proposition 2.7. If A factors through J^{new} and p is a prime such that p | r_{A}, then p^{2} | N, or p | n_{A}.

3 Proof of Theorem 2.3

Generalizing a theorem of Mazur, we write down a proof that c_{A} is a unit in Z[\frac{1}{2m}], where m is the largest square dividing N. That Mazur’s proof easily generalizes is undoubtedly well known to the experts, but has not been written down anywhere before.

Let π denote the map J_{0}(N) → A; let A denote the Néron model of A over R := Z[\frac{1}{2m}], and J the Néron model of J_{0}(N) over R. Let X be the smooth locus a minimal proper regular model for X_{0}(N) over R. Consider the diagram

H^{0}(A, Ω_{A}) ⟶^{π^{∗}} H^{0}(J , Ω_{J}) ≅ H^{0}(X , Ω_{X}) ⟶^{q-exp} R[[q]]  (1)

The map π^{∗} must be an inclusion because H^{0}(A, Ω_{A}) is torsion-free and π^{∗} 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, Ω_{A}) in R[[q]] is saturated, in the sense that the cokernel is torsion free; indeed, the image of S_{2}(Γ_{0}(N); R)[I] is saturated and S_{2}(Γ_{0}(N); R)[I] ⊗ Q = H^{0}(A, Ω_{A}) ⊗ Q.

For the image of H^{0}(A, Ω_{A}) 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, Ω_{A}) ⟶^{q-exp} R[[q]] → D → 0

with F_{ℓ} we obtain

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

Here we have used 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 that D[ℓ] = 0, it suffices to prove that the map Θ : H^{0}(A, Ω_{A}) ⊗ F_{ℓ} → F_{ℓ}[[q]] is injective.

Since ℓ ≠ 2 and A is optimal, [Maz-iso, 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). In particular, H^{0}(B/Z_{ℓ}, Ω_{B/Z_{ℓ}}) is torsion free, so

H^{0}(A/Z_{ℓ}, Ω_{A/Z_{ℓ}}) ⊗ F_{ℓ} → H^{0}(J /Z_{ℓ}, Ω_{J /Z_{ℓ}}) ⊗ F_{ℓ} ≅ H^{0}(X /F_{ℓ}, Ω_{X /F_{ℓ}})

is injective. We also remark that

H^{0}(A, Ω_{A}) ⊗ F_{ℓ} ≅ H^{0}(A/Z_{ℓ}, Ω_{A/Z_{ℓ}}) ⊗ F_{ℓ},

because Z_{ℓ} is torsion-free, hence flat over R. Thus the map

H^{0}(A, Ω_{A}) ⊗ F_{ℓ} → H^{0}(X /F_{ℓ}, Ω_{X /F_{ℓ}})

is injective.

If ℓ ∤ N, then injectivity of Θ 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.

Suppose ℓ does divide N, and let ω ∈ ker(Θ). Since ℓ | N and ℓ ∤ 2m, we have that ℓ||N; thus X /F_{ℓ} breaks up into a union of two irreducible components, and the q-expansion principle implies only that ω vanishes on the irreducible component containing the cusp ∞. However, since A is new and corresponds to a singe eigenform, ω is an eigenvector for the involution W_{N} (since f and all of its conjugates are). Since W_{N} permutes the two components, ω must be 0 on all X /F_{ℓ}. Therefore ω = 0, and hence Θ is injective. This proves Thm. 2.3.

We remark that the argument of the proof works just as well for any new quotient of J_{0}(N); it is not necessary that the quotient correspond to a single conjugacy class of newforms.

Remark 3.1 (communicated by K. Ribet and E. DeShalit) One comes across results like Prop. 1.8 while proving multiplicity one theorems, as was the case in [Maz-Eis, §II.9]. More generally, one can ask for the order of the torsion part of EndJ/T (recall J = J_{0}(N)_{Q}). If a prime does not divide this order, then it does not divide r_{A}/n_{A} for any quotient A either. Again, following the ideas of [Maz-Eis, §II.9], to prove that a prime p does not divide the order of the torsion part of EndJ/T, it suffices to check that dim_{T/m}H^{0}(M_{0}(N)^{0}, Ω_{M_{0}(N)^{0}/Z}^{1})[m] = 1. This is easy to check if p ∤ N. If p | N, but p^{2} ∤ N, Wiles [Wiles, Lem 2.2] proves that this condition is satisfied if U_{p} ∉ m. We can have U_{p} ∈ m only if S_{2}(Γ_{0}(N/p)) is non-empty, i.e., there are oldforms from level N/p. So if a prime p divides the order of the torsion part of EndJ/T, but p^{2} ∤ N, then p | N, and S_{2}(Γ_{0}(N/p)) is non-empty. For example, the only prime that can divide EndJ/T when N = 33 is the prime 3, since S_{2}(Γ_{0}(33/11)) is empty.

4 Proofs of Theorems 1.6, 1.8, 2.5 and Prop. 2.7

We continue using the notation introduced so far. In particular, A is the quotient of J_{0}(N) under the action of an ideal I of the Hecke algebra. Also we will be using notation similar to the one in [AbbUll] since we will be following their techniques closely. Recall we had the maps Â →^{ϕ_{1}} J →^{ϕ_{2}} A.

Pullling back differentials along ϕ_{2} and ϕ_{1} gives us the maps:

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

Let m denote the largest square that divides the level N. Let S = Spec(Z[\frac{1}{m}]). Then M_{0}(N)_{S} is smooth and semistable over S. Let Ω be the relative dualizing sheaf of M_{0}(N)_{S} over S. We have an injection

q-exp : H^{0}(M_{0}(N)_{S}, Ω) ↪ S_{2}(S).

This 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})

for each prime p that does not divide N.

Thus we get

H^{0}(M_{0}(N)_{S}, Ω) ↪ S_{2}(S) ↪ 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[\frac{1}{m}])). Denote the cokernel by C. It is a finite group and the only primes that can divide its order are the primes that divide N. An easy generalization of [AbbUll, Prop 3.2] gives ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ≅ H^{0}(Â_{S}, Ω_{Â/S}^{1}).

Quotienting by the pullback of H^{0}(A_{S}, Ω_{A/S}^{1}) under ϕ_{2} ∘ ϕ_{1}, we get

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

Now

# (\frac{ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}]))}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1})}) = (c_{A})_{m}# (\frac{ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}]))}{ϕ_{1}^{∗}(S_{2}(Z)[I] ⊗ Z[\frac{1}{m}])}) ,  (3)

where, if i is an integer, i_{m} denotes the largest divisor of i prime to m.

Then, as in the proof of [AbbUll, Prop 3.3], we have the isomorphisms

(\frac{S_{2}(Z)}{S_{2}(Z)[I] ⊕ (W (I) ∩ S_{2}(Z))})⊗Z[\frac{1}{m}] →^{≃} \frac{S_{2}(Z[\frac{1}{m}])}{(S_{2}(Z)[I] ⊗ Z[\frac{1}{m}]) ⊕ (W (I) ∩ S_{2}(Z[\frac{1}{m}]))} →^{≃} \frac{ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])))}{ϕ_{1}^{∗}(S_{2}(Z)[I] ⊗ Z[\frac{1}{m}])}.

Thus

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

Putting this in (3) and then using (2), we get

(c_{A})_{m}(r_{A})_{m} = # (\frac{H^{0}(Â_{S}, Ω_{Â/S}^{1})}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1})}) #C,  (4)

Recall that n_{A} was the smallest integer that annihilates the kernel of the composite isogeny Â →^{ϕ_{1}} J →^{ϕ_{2}} A. Hence there exists an isogeny ϕ′ : A → Â such that the composite Â →^{ϕ_{1}} J →^{ϕ_{2}} A →^{ϕ′} Â is multiplication by n_{A}. So it is easy to see that some power of n_{A} kills H^{0}(Â_{S}, Ω_{Â/S}^{1})/ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1}). Thus we have

Lemma 4.1. If p is a prime such that p | # (\frac{H^{0}(Â_{S},Ω_{Â/S}^{1})}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S},Ω_{A/S}^{1})}), then p | n_{A}.

We know that a prime can divide #C only if it divides N. The main addition to the techniques of [AbbUll] is the following result further controlling the primes that can divide C:

Proposition 4.2. If A factors through J^{new} and p is a prime such that p^{2} ∤ N, but p | #C, then p | n_{A}.

We will prove this shortly, but let us use this first to get:

Proofs of Theorems 2.5, 1.6, 1.8 and Prop. 2.7. Thm 2.5 (respectively Prop. 2.7) follows easily, since if p satisfies the hypothesis of the theorem, then p | (c_{A})_{m} (respectively p | (r_{A})_{m}), so by equation (4), Lem 4.1 and Prop 4.2, p divides n_{A}. Also if A = E is an elliptic curve, as in §1,

then n_{E} = deg ϕ = # (\frac{H^{0}(Ê_{S},Ω_{Ê/S}^{1})}{ϕ_{2}^{∗}ϕ_{1}^{∗}H^{0}(E_{S},Ω_{E/S}^{1})}) (e.g., see [AbbUll]). Equation (4) becomes (c_{E})_{m}r_{m} =

(deg ϕ)_{m}#C or, using the fact that deg ϕ | r, (c_{E})_{m}(r/deg ϕ)_{m} = #C. Using Prop 4.2, we again easily get Thm. 1.6 and we get Thm. 1.8.

It remains to give:

Proof of Prop 4.2. We have the exact sequence

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

We know that the only primes that can divide #C are those that divide N. So let p be a prime dividing N. Then tensoring with F_{p} gives us:

0 → C[p] → ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p} → ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])) ⊗ F_{p} → C ⊗ F_{p} → 0,

where we used that Tor^{1}(C, F_{p}) = C[p] and Tor^{1} is trivial for the other groups since they are torsion-free. Suppose p ∤ n_{A}. Then to show that p ∤ #C, i.e., that C[p] is trivial, all we have to show is that the map

\begin{matrix}ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p} \\ ↓ q-exp \\ ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])) ⊗ F_{p} \end{matrix}  (5)

is injective.

Recall the definition of ϕ′ given just after equation (4). Let ϕ′′ = ϕ′ ∘ ϕ_{2}. Then we have maps

Â →^{ϕ_{1}} J →^{ϕ′′} Â such that the composite is multiplication by n_{A}. Pulling back differentials, we get the maps

H^{0}(Â_{C}, Ω_{Â/C}^{1}) →^{ϕ′′^{∗}} H^{0}(J_{C}, Ω_{J/C}^{1}) →^{ϕ_{1}^{∗}} H^{0}(Â_{C}, Ω_{Â/C}^{1}),

where the composite is again multiplication by n_{A}. Note that these maps extend to the Néron models over S and preserve the Z[\frac{1}{m}]-integral structure. Hence, applying these maps to equation (5), we get:

ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p} →^{ϕ′′^{∗}⊗F_{p}} H^{0}(M_{0}(N)_{S}, Ω) ⊗ F_{p} →^{ϕ_{1}^{∗}⊗F_{p}} ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p}
↓ q-exp  ↓ q-exp  ↓ q-exp
ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])) ⊗ F_{p}  →^{ϕ′′^{∗}⊗F_{p}}  S_{2}(Z[\frac{1}{m}]) ⊗ F_{p}  →^{ϕ_{1}^{∗}⊗F_{p}}  ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])) ⊗ F_{p}

Now if an element x of ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p} (the top left group in the diagram above) is in the kernel of the map in (5), i.e., the map q-exp above, then its image y = (ϕ′′^{∗} ⊗ F_{p})(x) in H^{0}(M_{0}(N)_{S}, Ω) ⊗ F_{p} maps to 0 in S_{2}(Z[\frac{1}{m}]) ⊗ F_{p} under q-exp (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). Suppose p^{2} ∤ N. Then M_{0}(N)_{F_{p}} consists 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 containing the cusp ∞. Since A is a quotient of J^{new}, the element x is an eigenvector for the Atkin-Lehner involution W_{N}, and hence so is y. But W_{N} 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. Looking at the top line in the diagram above, we find that x maps to 0 under the composite. But its image is n_{A}x, so n_{A}x = 0. If p ∤ n_{A}, then that means x = 0, i.e., the map (5) is injective, which is what was left to prove.

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