Pure and Applied Mathematics Quarterly Volume 2, Number 2 (Special Issue: In honor of John H. Coates, Part 2 of 2 ) 617—636, 2006

The Manin Constant

Amod Agashe, Kenneth Ribet and William A. Stein

Abstract: The Manin constant of an elliptic curve is an invariant that arises in connection with the conjecture of Birch and Swinnerton-Dyer. One conjectures that this constant is 1; it is known to be an integer. After surveying what is known about the Manin constant, we establish a new sufficient condition that ensures that the Manin constant is an odd integer. Next, we generalize the notion of the Manin constant to certain abelian variety quotients of the Jacobians of modular curves; these quotients are attached to ideals of Hecke algebras. We also generalize many of the results for elliptic curves to quotients of the new part of J_{0}(N), and conjecture that the generalized Manin constant is 1 for newform quotients. Finally an appendix by John Cremona discusses computation of the Manin constant for all elliptic curves of conductor up to 130000.

1. Introduction

Let E be an elliptic curve over Q, and and let N be the conductor of E. By [BCDT01], we may view E as a quotient of the modular Jacobian J_{0}(N). 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).

618  Amod Agashe, Kenneth Ribet and William A. Stein

Let ω be the unique (up to sign) rational 1-form on a minimal Weierstrass model of E over Z that restricts to a nowhere-vanishing 1-form on the smooth locus. The pullback of ω is a rational multiple of the differential associated to the normalized new cuspidal eigenform f_{E} ∈ S_{2}(Γ_{0}(N)) associated to E. The Manin constant c_{E} of is E is the absolute value of this rational multiple. The Manin constant plays a role in the conjecture of Birch and Swinnerton-Dyer (see, e.g., [GZ86, p. 310]) and in work on modular parametrizations (see [Ste89, SW04, Vat05]). It is known that the Manin constant is an integer; this fact is important to Cremona’s computations of elliptic curves (see [Cre97, pg. 45]), and algorithms for computing special values of elliptic curve L-functions. Manin conjectured that c_{E} = 1. In Section 2, we summarize known results about c_{E}, and give the new result that 2 ∤ c_{E} if 2 is not a congruence prime and 4 ∤ N.

Received January 25, 2006.

Stein was supported by the National Science Foundation by Grant No. 0400386.

In Section 3, we generalize the definition of the Manin constant and many of the results mentioned so far to optimal quotients of J_{0}(N) and J_{1}(N) of any dimension associated to ideals of the Hecke algebra. The generalized Manin constant comes up naturally in studying the conjecture of Birch and Swinnerton-Dyer for such quotients (see [AS05, §4]), which is our motivation for studying the generalization. We state what we can prove about the generalized Manin constant, and make a conjecture that the constant is also 1 for quotients associated to newforms. The proofs of the theorems stated in Section 3 are in Section 4. Section 5 is an appendix written by J. Cremona about computational verification that the Manin constant is 1 for many elliptic curves.

Acknowledgments. The authors are grateful to A. Abbes, K. Buzzard, R. Coleman, B. Conrad, B. Edixhoven, A. Joyce, L. Merel, and R. Taylor for discussions and advice regarding this paper. The authors wish to thank the referee for helpful comments and suggestions.

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. Suppose f is 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}(f). 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 x to (∞) − (x) with the quotient map J_{0}(N) → E, we obtain a surjective morphism of curves ϕ_{E} : X_{0}(N) → E. The modular degree m_{E} of E is the degree of ϕ_{E}.

The Manin Constant  619

Let E_{Z} denote the Néron model of E over Z. A general reference for Néron models is [BLR90]; for the special case of elliptic curves, see, e.g., [Sil92, App. C, §15], and [Sil94]. Let ω be a generator for the rank 1 Z-module of invariant differential 1-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, the differential ϕ∗_{E}ω is a T-eigenvector with the same eigenvalues as f(z), so by [AL70] we have ϕ∗_{E}ω = c · 2πif(z)dz for some c ∈ Q^{∗} (see also [Man72, §5]). The Manin constant c_{E} of E is the absolute value of the rational number c defined above.

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

Conjecture 2.1 (Manin). We have c_{E} = 1.

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

Theorem 2.2 (Edixhoven [Edi91, Prop. 2]). The constant 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 theorem to quotients of arbitrary dimension in Theorem 3.4.

Theorem 2.3 (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 theorem in Corollary 3.7.

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

Theorem 2.4 (Raynaud [AU96, Prop. 3.1]). If 4 | c_{E}, then 4 | N.

Theorem 2.5 (Abbes-Ullmo [AU96, Thm. A]). If p | c_{E}, then p | N.

We generalize Theorem 2.4 in Theorem 3.10. However, it is not clear if Theorem 2.5 generalizes to dimension greater than 1. It would be fantastic if the theorem could be generalized. It would imply that the Manin constant is 1 for newform quotients A_{f} of J_{0}(N), with N odd and square free, which be useful for computations regarding the conjecture of Birch and Swinnerton-Dyer.

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 the construction of certain stable integral models for X_{0}(p^{2}).

See Section 5 for more details about the following computation:

Theorem 2.6 (Cremona). If E is an optimal elliptic curve over Q with conductor at most 130000, then c_{E} = 1.

To the above list of theorems we add the following:

Theorem 2.7. If p | c_{E} then p^{2} | N or p | m_{E}.

This theorem is a special case of Theorem 3.11 below. In view of Theorem 2.3, 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 optimal elliptic curve quotients of conductor ≤ 120000, only 56 of them satisfy the hypothesis.

3. Quotients of arbitrary dimension

For N ≥ 4, let Γ a subgroup of Γ_{1}(N) that contains Γ_{0}(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, so T/I is torsion free. Then A = A_{I} = J/IJ is an optimal quotient of J.

For a newform f = ∑ a_{n}(f)q^{n} ∈ S_{2}(Γ), consider the ring homomorphism T → Z[…, a_{n}(f), …] that sends T_{n} to a_{n}(f). The kernel I_{f} ⊂ T of this homomorphism is a saturated prime ideal of T. The newform quotient A_{f} associated to f is the quotient J/I_{f}J. Shimura introduced A_{f} in [Shi73] where he proved that A_{f} is an abelian variety over Q of dimension equal to the degree of the field Q(…, a_{n}(f), …). He also observed that there is a natural map T → End(A_{f}) with kernel I_{f}.

For the rest of this section, fix a quotient A associated to a saturated ideal I of T; note that A may or may not be attached to a newform.

3.1. Generalization to quotients of arbitrary dimension. If R is a subring of C, let S_{2}(R) = S_{2}(Γ; R) denote the T-submodule of S_{2}(Γ; C) of modular forms whose Fourier expansions have all coefficients in R.

Lemma 3.1. The Hecke operators leave S_{2}(R) stable.

Proof. If Γ = Γ_{0}(N), then by the explicit description of the Hecke operators on Fourier expansions (e.g., see [DI95, Prop. 3.4.3]), it is clear that the Hecke operators leave S_{2}(R) stable. For a general Γ, by [DI95, (12.4.1)], one just has to check in addition that the diamond operators also leave S_{2}(R) stable, which in turn follows from [DI95, Prop. 12.3.11]. □

Lemma 3.2. We have S_{2}(R) ≅ S_{2}(Z) ⊗ R.

Proof. This is [DI95, Thm. 12.3.2] when our spaces S_{2}(R) and S_{2}(Z) are replaced by their algebraic analogues (see [DI95, pg. 111]). Our spaces and their algebraic analogues are identified by the natural q-expansion maps according to [DI95, Thm. 12.3.7]. □

If B is an abelian variety over Q and S is a Dedekind domain with field of fractions Q, then we denote by B_{S} the Néron model of B over S; also, for ease of notation, we will abbreviate H^{0}(B_{S}, Ω_{B_{S}/S}^{1}) by H^{0}(B_{S}, Ω_{B/S}^{1}).

The inclusion X ↪ J that sends the cusp ∞ to 0 induces an isomorphism

H^{0}(X, Ω_{X/Q}^{1}) ≅ H^{0}(J, Ω_{J/Q}^{1}).

Let ϕ_{2} be the optimal quotient map J → A. Then ϕ_{2}^{∗} induces an inclusion ψ : H^{0}(A_{Z}, Ω_{A/Z}^{1}) ↪ H^{0}(J, Ω_{J/Q}^{1})[I] ≅ S_{2}(Q)[I], and we have the following commutative diagram:

\xymatrix{H^{0}(A, Ω_{A/Q}^{1}) \ar@{^{(}->}[r]^{≅} & H^{0}(J, Ω_{J/Q}^{1})[I] \ar[r]^{≅} & S_{2}(Q)[I] \\ H^{0}(A_{Z}, Ω_{A/Z}^{1}) \ar@{^{(}->}[u] \ar@{^{(}->}[urr]_{ψ} &  & S_{2}(Z)[I] \ar@{^{(}->}[u]}

Definition 3.3. The Manin constant of A is the (lattice) index

c_{A} = [S_{2}(Z)[I] : ψ(H^{0}(A_{Z}, Ω_{A/Z}^{1}))].

Theorem 3.4 below asserts that c_{A} ∈ Z, so we may also consider the Manin module of A, which is the quotient M = S_{2}(Z)[I]/ψ(H^{0}(A_{Z}, Ω_{A/Z}^{1})), and the Manin ideal of A, which is the annihilator of M in T.

If A is an elliptic curve, then c_{A} is the usual Manin constant. The constant c as defined above was also considered by Gross [Gro82, 2.5, p.222] and Lang [Lan91, III.5, p.95]. The constant c_{A} was defined for the winding quotient in [Aga99], where it was called the generalized Manin constant. A Manin constant is defined in the context of Q-curves in [GL01].

3.2. Motivation: connection with the conjecture of Birch and Swinnerton-Dyer. On a Néron model, the global differentials are the same as the invariant differentials, so H^{0}(A_{Z}, Ω_{A/Z}^{1}) is a free Z-module of rank d = dim(A). The real measure Ω_{A} of A is the measure of A(R) with respect to the volume given by a generator of ⋀^{d} H^{0}(A_{Z}, Ω_{A/Z}^{1}) ≃ H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{d}). This quantity is of interest because it appears in the conjecture of Birch and Swinnerton-Dyer, which expresses the ratio L^{(r)}(A, 1)/Ω_{A}, in terms of arithmetic invariants of A, where r = ord_{s=1} L(A, s) (see, e.g., [Lan91, Chap. III, §5] and [AS05, §2.3]).

If we take a Z-basis of S_{2}(Z)[I] and take the inverse image via the top chain of arrows in the commutative diagram above, we get a Q-basis of H^{0}(A, Ω_{A/Q}^{1}); let Ω′_{A} denote the volume of A(R) with respect to the wedge product of the elements in the latter basis (this is independent of the choice of the former basis). In doing calculations or proving formulas regarding the ratio in the Birch and Swinnerton-Dyer conjecture mentioned above, it is easier to work with the volume Ω′_{A} instead of working with Ω_{A}. 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 conjecture of Birch and Swinnerton-Dyer, since Ω_{A} = c_{A} · Ω′_{A}. For example, see [AS05, §4.2] when r = 0 and [GZ86, p. 310–311] when r = 1; in each case, one gets a formula for computing the Birch and Swinnerton-Dyer conjectural order of the Shafarevich-Tate group, and the formula contains the Manin constant (see, e.g., [Mc91]).

The method of Section 5 for verifying that c_{A} = 1 for specific elliptic curves is of little use when applied to general abelian varieties, since there is no simple analogue of the minimal Weierstrass model (but see [GL01] for Qcurves). This emphasizes the need for general theorems regarding the Manin constant of quotients of dimension bigger than one.

3.3. Results and a conjecture. We start by giving several results regarding the Manin constant for quotients of arbitrary dimension. The proofs of most of the theorems are given in Section 4.

Let Γ be a subgroup of Γ_{0}(N) that contains Γ_{1}(N). We have the following generalization of Edixhoven’s Theorem 2.2.

Theorem 3.4. The Manin constant c_{A} is an integer. (In the notation of Section 3.1 we even have that ψ(H^{0}(A_{Z}, Ω_{A/Z}^{1})) ⊆ S_{2}(Z)[I].)

Proof. Let J = Jac(X_{Γ}) and J′ = J_{1}(N). Suppose A is an optimal quotient of J. We have natural maps H^{0}(J′_{Z}, Ω_{J′/Z}^{1}) ↪ H^{0}(J′, Ω_{J′/Q}^{1}) →^{≅} S_{2}(Γ_{1}(N); Q); from the proof of Lemma 6.1.6 of [CES03], the image of the composite is contained in S_{2}(Γ_{1}(N); Z). The maps J′ → J → A induce a chain of inclusions

H^{0}(A_{Z}, Ω_{A/Z}^{1}) ↪ H^{0}(J_{Z}, Ω_{J/Z}^{1}) ↪ H^{0}(J′_{Z}, Ω_{J′/Z}^{1}) ↪ S_{2}(Γ_{1}(N); Z) ↪ Z[[q]].

Combining this chain of inclusions with commutativity of the diagram

\xymatrix{ & S_{2}(Γ_{1}(N)) \ar[dr]^{F -exp} \\ S_{2}(Γ) \ar[ur]^{f(q)↦f(q)} \ar[rr]^{F -exp} &  & C[[q]],}

where F -exp is the Fourier expansion map, we see that the image of H^{0}(A_{Z}, Ω_{A/Z}^{1}) lies in S_{2}(Z)[I], as claimed. □

For the rest of the paper, we take Γ = Γ_{0}(N). For each prime ℓ | N with ord_{ℓ}(N) = 1, let W_{ℓ} be the ℓth Atkin-Lehner operator. Let J = J_{0}(N) and A = A_{I} = J/IJ be an optimal quotient of J attached to a saturated ideal I. If ℓ is a prime, then as usual, Z_{(ℓ)} will denote the localization of Z at ℓ.

Theorem 3.5. Suppose that ℓ is an odd prime such that ℓ^{2} ∤ N, and that if ℓ | N, then A^{∨} ⊂ J is stable under W_{ℓ}. Then ℓ | c_{A} if and only if ℓ | N and S_{2}(Z_{(ℓ)})[I] is not stable under the action of W_{ℓ}.

We will prove this theorem in Section 4.2.

Remark 3.6. The condition that S_{2}(Z_{(ℓ)})[I] is stable under W_{ℓ} can be verified using standard algorithms. Thus in light of Theorem 3.5, if A is stable under all Atkin-Lehner operators and N is square free, then one can compute the set of odd primes that divide c_{A}. It would be interesting to refine the arguments of this paper to find an algorithm to compute c_{A} exactly.

Let J_{old} denote the abelian subvariety of J generated by the images of the degeneracy maps from levels that properly divide N (see, e.g., [Maz78, §2(b)]) and let J^{new} denote the quotient of J by J_{old}. A new quotient is a quotient J → A that factors through the map J → J^{new}. The following corollary generalizes Mazur’s Theorem 2.3:

Corollary 3.7. If A = A_{f} is an optimal newform quotient of J_{0}(N) and ℓ | c_{A} is a prime, then ℓ = 2 or ℓ^{2} | N.

Proof. Since f is a newform, W_{ℓ} acts as either 1 or −1 on A hence on S_{2}(Z_{(ℓ)})[I]. Thus S_{2}(Z_{(ℓ)})[I] is W_{ℓ}-stable. □

Corollary 3.8. If A = J_{0}(N)_{new} is the new subvariety of J_{0}(N) and ℓ | c_{A} is a prime, then ℓ = 2 or ℓ^{2} | N. (In particular, if N is prime then the Manin constant of J_{0}(N) is a power of 2, since A = J_{0}(N)[I] for I = 0.)

Proof. We have W_{ℓ} = −T_{ℓ} on A (e.g., see the end of [DI95, §6.3]). Also the new subspace S_{2}(Z)_{new} of S_{2}(Γ_{0}(N)) is T_{ℓ}-stable. □

Remark 3.9. If A = J_{0}(33), then

W_{3} = ( \begin{matrix}1 & 0 & 0 \\ \frac{1}{3} & \frac{1}{3} & −\frac{4}{3} \\ \frac{1}{3} & −\frac{2}{3} & −\frac{1}{3} \end{matrix})

with respect to the basis

f_{1} = q − q^{5} − 2q^{6} + 2q^{7} + ⋯,
f_{2} = q^{2} − q^{4} − q^{5} − q^{6} + 2q^{7} + ⋯,
f_{3} = q^{3} − 2q^{6} + ⋯

for S_{2}(Z). Thus W_{3} does not preserve S_{2}(Z_{(3)}). In fact, the Manin constant of J_{0}(33) is not 1 in this case (see Section 3.4).

The hypothesis of Theorem 3.5 sometimes holds for non-new A. For example, take J = J_{0}(33) and ℓ = 3. Then W_{3} acts as an endomorphism of J, and a computation shows that the characteristic polynomial of W_{3} on S_{2}(33)_{new} is x − 1 and on S_{2}(33)_{old} is (x − 1)(x + 1), where S_{2}(33)_{old} is the old subspace of S_{2}(33). Consider the optimal elliptic curve quotient A = J/(W_{3} + 1)J, which is isogenous to J_{0}(11). Then A is an optimal old quotient of J, and W_{3} acts as −1 on A, so W_{3} preserves the corresponding spaces of modular forms. Thus Theorem 3.5 implies that 3 ∤ c_{A}.

The following theorem generalizes Raynaud’s Theorem 2.4 (see also [GL01] for generalizations to Q-curves).

Theorem 3.10. If f ∈ S_{2}(Γ_{0}(N)) is a newform and ℓ is a prime such that ℓ^{2} ∤ N, then ord_{ℓ}(c_{A_{f}}) ≤ dim A_{f}.

Note that in light of Theorem 3.5, this theorem gives new information only at ℓ = 2, when 2 ∥ N. We prove the theorem in Section 4.4

Let π denote the natural quotient map J → A. When we compose π with its dual A^{∨} → J^{∨} (identifying J^{∨} with J using the inverse of the principal polarization of J), we get an isogeny ϕ : A^{∨} → A. The modular exponent m_{A} of A is the exponent of the group ker(ϕ). When A is an elliptic curve, the modular exponent is just the modular degree of A (see, e.g., [AU96, p. 278]).

Theorem 3.11. If f ∈ S_{2}(Γ_{0}(N)) is a newform and ℓ | c_{A_{f}} is a prime, then ℓ^{2} | N or ℓ | m_{A}.

Again, in view of Corollary 3.7, this theorem gives new information only at ℓ = 2, when ord_{2}(N) ≤ 1. We prove the theorem in Section 4.3.

The theorems above suggest that the Manin constant is 1 for quotients associated to newforms of square-free level. 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 for 28 two-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 suggest the following conjecture, which generalizes Conjecture 2.1:

Conjecture 3.12. If f is a newform on Γ_{0}(N) then c_{A_{f}} = 1.

It is plausible that c_{A_{f}} = 1 for any newform on any congruence subgroup between Γ_{0}(N) and Γ_{1}(N). However, we do not have enough data to justify making a conjecture in this context.

3.4. Examples of nontrivial Manin constants. We present two sets of examples in which the Manin constant is not 1.

Using results of [Kil02], Adam Joyce [Joy05] proves that there is a new optimal quotient of J_{0}(431) with Manin constant 2. Joyce’s methods also produce examples with Manin constant 2 at levels 503 and 2089. For the convenience of the reader, we breifly discuss his example at level 431. There are exactly two elliptic curves E_{1} and E_{2} of prime conductor 431, and E_{1} ∩E_{2} = 0 as subvarieties of J_{0}(431), so A = E_{1} × E_{2} is an optimal quotient of J_{0}(431) attached to a saturated ideal I. If f_{i} is the newform corresponding to E_{i}, then one finds that f_{1} ≡ f_{2} (mod 2), and so g = (f_{1} − f_{2})/2 ∈ S_{2}(Z)[I]. However g is not in the image of H^{0}(A_{Z}, Ω_{A/Z}^{1}). Thus the Manin constant of A is divisible by 2.

As another class of examples, one finds by computation for each prime ℓ ≤ 100 that W_{ℓ} does not leave S_{2}(Γ_{0}(11ℓ); Z_{(ℓ)}) stable. Theorem 3.5 (with I = 0) then implies that the Manin constant of J_{0}(11ℓ) is divisible by ℓ for these values of ℓ.

4. Proofs of some of the Theorems

In Sections 4.2, 4.3, and 4.4, we prove Theorems 3.5, 3.11, and 3.10 respectively. In Section 4.1, we state two lemmas that will be used in these proofs. The proofs of the theorems may be read independently of each other, after reading Section 4.1.

4.1. Two lemmas. The following lemma is a standard fact; we state it as a lemma merely because it is used several times.

Lemma 4.1. Suppose i : A ↪ B is an injective homomorphism of torsion-free abelian groups. If p is a prime, then B/i(A) has no nonzero p-torsion if and only if the induced map A ⊗ F_{p} → B ⊗ F_{p} is injective.

Proof. Let Q denote the quotient B/i(A). Tensor the exact sequence 0 → A → B → Q → 0 with F_{p}. The associated long exact sequences reveal that ker(A ⊗ F_{p} → B ⊗ F_{p}) ≅ Q_{tor}[p]. □

Suppose ℓ is a prime such that ℓ^{2} ∤ N. In what follows, we will be stating some standard facts taken from [Maz78, §2(e)] (which in turn relies on [DR73]). Let X_{Z_{(ℓ)}} be the minimal regular resolution of the coarse moduli scheme associated to Γ_{0}(N) (as in [DR73, § VI.6.9]) over Z_{(ℓ)}, and let Ω_{X /Z_{(ℓ)}} denote the relative dualizing sheaf of X_{Z_{(ℓ)}} over Z_{(ℓ)}. The Tate curve over Z_{(ℓ)}[[q]] gives rise to a morphism from Spec Z_{(ℓ)}[[q]] to the smooth locus of X_{Z_{(ℓ)}} → Spec Z_{(ℓ)}. Since the module of completed Kahler differentials for Z_{(ℓ)}[[q]] over Z_{(ℓ)} is free of rank 1 on the basis dq, we obtain a map q-exp : H^{0}(X_{Z_{(ℓ)}}, Ω_{X /Z_{(ℓ)}}) → Z_{(ℓ)}[[q]].

The natural morphism Pic_{X /Z_{(ℓ)}}^{0} → J_{Z_{(ℓ)}} identifies Pic_{X /Z_{(ℓ)}}^{0} with the identity component of J_{Z_{(ℓ)}} (see, e.g., [BLR90, §9.4–9.5]). Passing to tangent spaces along the identity section over Z_{(ℓ)}, we obtain an isomorphism H^{1}(X_{Z_{(ℓ)}}, O_{X_{Z_{(ℓ)}}} ) ≅

Tan(J_{Z_{(ℓ)}}). Using Grothendieck duality, one gets an isomorphism Cot(J_{Z_{(ℓ)}}) →^{≅} H^{0}(X_{Z_{(ℓ)}}, Ω_{X /Z_{(ℓ)}}), where Cot(J_{Z_{(ℓ)}}) is the cotangent space at the identity section. On the Néron model J_{Z_{(ℓ)}}, the group of global differentials is the same as the group of invariant differentials, which in turn is naturally isomorphic to Cot(J_{Z_{(ℓ)}}). Thus we obtain an isomorphism H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) ≅ H^{0}(X_{Z_{(ℓ)}}, Ω_{X /Z_{(ℓ)}}).

Let G be a T-module equipped with an injection G ↪ H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) of T-modules such that G is annihilated by I. If ℓ | N, assume moreover that G is a T[W_{ℓ}]-module and that the inclusion in the previous sentence is a homomorphism of T[W_{ℓ}]-modules. As a typical example, G = H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}), with the

injection π^{∗} : H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}) ↪ H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}). Let Φ be the composition of the inclusions

(1)  G ↪ H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) ≅ H^{0}(X_{Z_{(ℓ)}}, Ω_{X /Z_{(ℓ)}}) ⟶^{q-exp} Z_{(ℓ)}[[q]],

and let ψ′ be the composition of

G ↪ H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1})[I] ↪ S_{2}(Z_{(ℓ)})[I],

where the last inclusion follows from a “local” version of Theorem 3.4. The maps Φ and ψ′ are related by the commutative diagram

(2)  \xymatrix{ & S_{2}(Z_{(ℓ)})[I] \ar[dr]^{F -exp} \\ G \ar[ur]^{ψ′} \ar[rr]^{Φ} &  & Z_{(ℓ)}[[q]],}

where F -exp is the Fourier expansion map (at infinity), as before.

We say that a subgroup B of an abelian group C is saturated (in C) if the quotient C/B is torsion free.

Lemma 4.2. Recall that ℓ is a prime such that ℓ^{2} ∤ N. If ℓ divides N, suppose that S_{2}(Z_{(ℓ)})[I] is stable under the action of W_{ℓ}; if ℓ = 2 assume moreover that W_{ℓ} acts as a scalar on A. Consider the map

G ⊗ F_{ℓ} → H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) ⊗ F_{ℓ},

which is obtained by tensoring the inclusion G ↪ H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) with F_{ℓ}. If this map is injective, then the image of G under the map Φ of (2) is saturated in Z_{(ℓ)}[[q]].

Proof. By Lemma 4.1, it suffices to prove that the map

Φ_{ℓ} : G ⊗ F_{ℓ} → Z_{(ℓ)}[[q]] ⊗ F_{ℓ} = F_{ℓ}[[q]]

obtained by tensoring (1) with F_{ℓ} is injective. Let X_{F_{ℓ}} denote the special fiber of X_{Z_{(ℓ)}} and let Ω_{X /F_{ℓ}} denote the relative dualizing sheaf of X_{F_{ℓ}} over F_{ℓ}.

First suppose that ℓ does not divide N. Then X_{Z_{(ℓ)}} is smooth and proper over Z_{(ℓ)}. Thus the formation of H^{0}(X_{Z_{(ℓ)}}, Ω_{X_{Z_{(ℓ)}}} ) is compatible with any base change on Z_{(ℓ)} (such as reduction modulo ℓ). The injectivity of Φ_{ℓ} now follows since by hypothesis the induced map G ⊗ F_{ℓ} → H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) ⊗ F_{ℓ} is injective, and

H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) ⊗ F_{ℓ} ≅ H^{0}(X_{Z_{(ℓ)}}, Ω_{X /Z_{(ℓ)}}) ⊗ F_{ℓ} ≅ H^{0}(X_{F_{ℓ}}, Ω_{X /F_{ℓ}}) → F_{ℓ}[[q]]

is injective by the q-expansion principle (which is easy in this setting, since X_{F_{ℓ}} is a smooth and geometrically connected curve).

Next suppose that ℓ divides N. First we verify that ker(Φ_{ℓ}) is stable under W_{ℓ}. Suppose ω ∈ ker(Φ_{ℓ}). Choose ω′ ∈ G such that the image of ω′ in G ⊗ F_{ℓ} is ω, and let f = ψ′(ω′). Because Φ_{ℓ}(ω) = 0 in F_{ℓ}[[q]], there exists h ∈ Z_{(ℓ)}[[q]] such that ℓh = F -exp(f). Let f′ = f/ℓ ∈ S_{2}(Q); then f′ is actually in S_{2}(Z_{(ℓ)}) (since F -exp(f/ℓ) = h ∈ Z_{(ℓ)}[[q]]). Now ℓf′ = f is annihilated by every element of I, hence so is f′; thus f′ ∈ S_{2}(Z_{(ℓ)})[I]. By hypothesis, W_{ℓ}(f′) ∈ S_{2}(Z_{(ℓ)})[I]. Then

Φ(W_{ℓ}ω′) = F -exp(W_{ℓ}f) = ℓ · F -exp(W_{ℓ}f′) ∈ ℓZ_{(ℓ)}[[q]].

Reducing modulo ℓ, we get Φ_{ℓ}(W_{ℓ}ω) = 0 in F_{ℓ}[[q]]. Thus W_{ℓ}ω ∈ ker(Φ_{ℓ}), which proves that ker(Φ_{ℓ}) is stable under W_{ℓ}.

Since W_{ℓ} is an involution, and by hypothesis either ℓ is odd or W_{ℓ} is a scalar, the space ker(Φ_{ℓ}) breaks up into a direct sum of eigenspaces under W_{ℓ} with eigenvalues ±1. It suffices to show that if ω ∈ ker(Φ_{ℓ}) is an element of either eigenspace, then ω = 0. For this, we use a standard argument that goes back to Mazur (see, e.g., the proof of Prop. 22 in [MR91]); we give some details to clarify the argument in our situation.

Following the proof of Prop. 3.3 on p. 68 of [Maz77], we have

H^{0}(X_{Z_{(ℓ)}}, Ω_{X /Z_{(ℓ)}}) ⊗ F_{ℓ} ≅ H^{0}(X_{F_{ℓ}}, Ω_{X /F_{ℓ}}).

In the following, we shall think of G⊗F_{ℓ} as a subgroup of H^{0}(X_{F_{ℓ}}, Ω_{X /F_{ℓ}}), which we can do by the hypothesis that the induced map G ⊗ F_{ℓ} → H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) ⊗ F_{ℓ} is injective and that

H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) ⊗ F_{ℓ} ≅ H^{0}(X_{Z_{(ℓ)}}, Ω_{X /Z_{(ℓ)}}) ⊗ F_{ℓ} ≅ H^{0}(X_{F_{ℓ}}, Ω_{X /F_{ℓ}}).

Suppose ω ∈ ker(Φ_{ℓ}) is in the ±1 eigenspace (we will treat the cases of +1 and −1 eigenspaces together). We will show that ω is trivial over X_{\overline{F}_{ℓ}}, the base change of X_{F_{ℓ}} to an algebraic closure \overline{F}_{ℓ}, which suffices for our purposes. Since ℓ^{2} ∤ N, we have ℓ || N, and so the special fiber X_{\overline{F}_{ℓ}} is as depicted on p. 177 of [Maz77]: it consists of the union of two copies of X_{0}(N/ℓ)_{\overline{F}_{ℓ}} identified transversely at the supersingular points, and some copies of P^{1}, each of which intersects exactly one of the two copies of X_{0}(N/ℓ)_{\overline{F}_{ℓ}} and perhaps another P^{1}, all of them transversally. All the singular points are ordinary double points, and the cusp ∞ lies on one of the two copies of X_{0}(N/ℓ)_{\overline{F}_{ℓ}}.

In particular, X_{\overline{F}_{ℓ}} → Spec \overline{F}_{ℓ} is locally a complete intersection, hence Gorenstein, and so by [DR73, § I.2.2, p. 162], the sheaf Ω_{X /\overline{F}_{ℓ}} = Ω_{X /F_{ℓ}} ⊗ \overline{F}_{ℓ} is invertible. Since ω ∈ ker(Φ_{ℓ}), the differential ω vanishes on the copy of X_{0}(N/ℓ)_{\overline{F}_{ℓ}} containing the cusp ∞ by the q-expansion principle (which is easy in this case, since all that is being invoked here is that on an integral curve, the natural map from the group of global sections of an invertible sheaf into the completion of the sheaf’s stalk at a point is injective). The two copies of X_{0}(N/ℓ)_{\overline{F}_{ℓ}} are swapped under the action of the Atkin-Lehner involution W_{ℓ}, and thus W_{ℓ}(ω) vanishes on the other copy that does not contain the cusp ∞. Since W_{ℓ}(ω) = ±ω, we see that ω is zero on both copies of X_{0}(N/ℓ)_{\overline{F}_{ℓ}}. Also, by the description of the relative dualizing sheaf in [DR73, § I.2.3, p. 162], if π : X̃_{\overline{F}_{ℓ}} → X_{\overline{F}_{ℓ}} is a normalization, then ω correponds to a meromorphic differential ω̃ on X̃_{\overline{F}_{ℓ}} which is regular outside the inverse images (under π) of the double points on X_{\overline{F}_{ℓ}} and has at worst a simple pole at any point that lies over a double point on X_{\overline{F}_{ℓ}}. Moreover, on the inverse image of any double point on X_{\overline{F}_{ℓ}}, the residues of ω̃ add to zero. For any of the P^{1}’s, above a point of intersection of the P^{1} with a copy of X_{0}(N/ℓ)_{\overline{F}_{ℓ}}, the residue of ω̃ on the inverse image of the copy of X_{0}(N/ℓ)_{\overline{F}_{ℓ}} is zero (since ω is trivial on both copies of X_{0}(N/ℓ)_{\overline{F}_{ℓ}}), and thus the residue of ω̃ on the inverse image of P^{1} is zero. Thus ω̃ restricted to the inverse image of P^{1} is regular away from the inverse image of any point where the P^{1} meets another P^{1} (recall that there can be at most one such point). Hence the restriction of ω̃ to the inverse image of the P^{1} is either regular everywhere or is regular away from one point where it has at most a simple pole; in the latter case, the residue is zero by the residue theorem. Thus in either case, ω̃ restricted to the inverse image of the P^{1} is regular, and therefore is zero. Thus ω is trivial on all the copies of P^{1} as well. Hence ω = 0, as was to be shown. □

4.2. Proof of Theorem 3.5. We continue to use the notation of Section 4.1.

First suppose that ℓ | N and S_{2}(Z_{(ℓ)})[I] is not stable under the action of W_{ℓ}. Relative differentials and Néron models are functorial, so H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}) is

W_{ℓ}-stable. Thus the map H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}) → S_{2}(Z_{(ℓ)})[I] is not surjective. But c_{A} is the order of the cokernel, so ℓ | c_{A}.

Next we prove the other implication, namely that if ℓ | c_{A}, then ℓ | N and S_{2}(Z_{(ℓ)})[I] is not stable under W_{ℓ}. We will prove this by proving the contrapositive, i.e., that if either ℓ ∤ N or S_{2}(Z_{(ℓ)})[I] is stable under W_{ℓ}, then ℓ ∤ c_{A}.

We now follow the discussion preceding Lemma 4.2, taking G = H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}). To show that ℓ ∤ c_{A}, we have to show that c_{A} is a unit in Z_{(ℓ)}.

For this, it suffices to check that in diagram (2), the image of H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}) in Z_{(ℓ)}[[q]] under Φ is saturated, since the image of S_{2}(Γ_{0}(N); Z_{(ℓ)})[I] under F -exp is saturated in Z_{(ℓ)}[[q]]. In view of Lemma 4.2, it suffices to show that the map

H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}) ⊗ F_{ℓ} → H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) ⊗ F_{ℓ}

is injective.

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

0 → H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}) → H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) → H^{0}(B_{Z_{(ℓ)}}, Ω_{B/Z_{(ℓ)}}^{1}) → 0

where B = ker(J → A). Since H^{0}(B_{Z_{(ℓ)}}, Ω_{B/Z_{(ℓ)}}^{1}) is torsion free, by Lemma 4.1

the map H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}) ⊗ F_{ℓ} → H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) ⊗ F_{ℓ} is injective, as was to be shown.

4.3. Proof of Theorem 3.11. We continue to use the notation and hypotheses of Section 4.1 (so ℓ^{2} ∤ N) and assume in addition that A is a newform quotient, and that ℓ ∤ m_{A}. We have to show that then ℓ ∤ c_{A}. Just as in the previous proof, it suffices to check that the image of H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}) in Z_{(ℓ)}[[q]] is saturated. Since A is a newform quotient, if ℓ | N, then W_{ℓ} acts as a scalar on A and on S_{2}(Γ_{0}(N); Z_{(ℓ)})[I]. So again, using Lemma 4.2, it suffices to show that the map H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}) ⊗ F_{ℓ} → H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) ⊗ F_{ℓ} is injective.

The composition of pullback and pushforward in the following diagram is multiplication by the modular exponent of A:

\xymatrix{ & H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) \ar@{-->}[dr]^{π_{∗}} \\ H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}) \ar@{-->}[ur]^{π^{∗}} \ar[rr]^{m_{A}} &  & H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1})}

Since m_{A} ∈ Z_{(ℓ)}^{×}, the map π^{∗} is a section to the map π_{∗} up to a unit and hence its reduction modulo ℓ is injective, which is what was left to be shown.

4.4. Proof of Theorem 3.10. Theorem 3.10 asserts that if A = A_{f} is a quotient of J = J_{0}(N) attached to a newform f, and ℓ is a prime such that ℓ^{2} ∤ N, then ord_{ℓ}(c_{A}) ≤ dim(A). Our proof follows [AU96], except at the end we argue using lattice indices 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_{(ℓ)}} → J_{Z_{(ℓ)}} → A_{J_{Z_{(ℓ)}}} of Néron models. Because J_{Z_{(ℓ)}} has semiabelian reduction (since ℓ^{2} ∤ 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_{(ℓ)}}) → Tan(J_{Z_{(ℓ)}}) → Tan(A_{Z_{(ℓ)}}) → (Z/ℓZ)^{r} → 0.

Here Tan is the tangent space at the 0 section; it is a finite free Z_{(ℓ)}-module of rank equal to the dimension. In particular, we have r ≤ d = dim(A). Note that Tan is Z_{(ℓ)}-dual to the cotangent space, and the cotangent space is isomorphic to the space of global differential 1-forms. The theorem of Raynaud mentioned above is the generalization to e = ℓ − 1 of [Maz78, Cor. 1.1], which we used above in the proof of Theorem 3.5.

Let C be the cokernel of Tan(B_{Z_{(ℓ)}}) → Tan(J_{Z_{(ℓ)}}). We have a diagram
(3) \xymatrix{0 \ar[r] & Tan(B_{Z_{(ℓ)}}) \ar[r] & Tan(J_{Z_{(ℓ)}}) \ar[rr] \ar@{->>}[dr] &  & Tan(A_{Z_{(ℓ)}}) \ar[r] & (Z/ℓZ)^{r} \ar[r] & 0. \\  &  &  & C \ar@{^{(}->}[ur]}

Since C ⊂ Tan(A_{Z_{(ℓ)}}), so C is torsion free, we see that C is a free Z_{(ℓ)}-module of rank d. Let C^{∗} = Hom_{Z_{(ℓ)}}(C, Z_{(ℓ)}) be the Z_{(ℓ)}-linear dual of C. Applying the Hom_{Z_{(ℓ)}}(−, Z_{(ℓ)}) functor to the two short exact sequences in (3), we obtain exact sequences

0 → C^{∗} → H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) → H^{0}(B_{Z_{(ℓ)}}, Ω_{B/Z_{(ℓ)}}^{1}) → 0,

and

(4)  0 → H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}) → C^{∗} → (Z/ℓZ)^{r} → 0.

The (Z/ℓZ)^{r} on the right in (4) occurs as Ext_{Z_{(ℓ)}}^{1}((Z/ℓZ)^{r}, Z_{(ℓ)}).

Since H^{0}(B_{Z_{(ℓ)}}, Ω_{B/Z_{(ℓ)}}^{1}) is torsion free, by Lemma 4.1, the induced map
C^{∗} ⊗ F_{ℓ} → H^{0}(J_{Z_{(ℓ)}}, Ω_{J/Z_{(ℓ)}}^{1}) ⊗ F_{ℓ}

is injective. Since A is a newform quotient, if ℓ | N then W_{ℓ} acts as a scalar on C^{∗} and on S_{2}(Γ_{0}(N); Z_{(ℓ)})[I]. Using Lemma 4.2, with G = C^{∗}, we see that the image of C^{∗} in Z_{(ℓ)}[[q]] under the composite of the maps in (1) is saturated. The Manin constant for A at ℓ is the index of the image via q-expansion of H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}) in Z_{(ℓ)}[[q]] in its saturation. Since the image of C^{∗} in Z_{(ℓ)}[[q]] is saturated, the image of C^{∗} is the saturation of the image of H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}), so the Manin

constant at ℓ is the index of H^{0}(A_{Z_{(ℓ)}}, Ω_{A/Z_{(ℓ)}}^{1}) in C^{∗}, which is ℓ^{r} by (4), hence is

at most ℓ^{d}.

5. Appendix by J. Cremona: Verifying that c = 1

Let f be a normalised rational newform for Γ_{0}(N). Let Λ_{f} be its period lattice; that is, the lattice of periods of 2πif(z)dz over H_{1}(X_{0}(N), Z).

We know that E_{f} = C/Λ_{f} is an elliptic curve E_{f} defined over Q and of conductor N. This is the optimal quotient of J_{0}(N) associated to f. Our goal is two-fold: to identify E_{f} (by giving an explicit Weierstrass model for it with integer coeffients); and to show that the associated Manin constant for E_{f} is 1. In this section we will give an algorithm for this; our algorithm applies equally to optimal quotients of J_{1}(N).

As input to our algorithm, we have the following data:

(1) a Z-basis for Λ_{f}, known to a specific precision;

(2) the type of the lattice Λ_{f} (defined below); and

(3) a complete isogeny class of elliptic curves {E_{1}, …, E_{m}} of conductor N, given by minimal models, all with L(E_{j}, s) = L(f, s).

So E_{f} is isomorphic over Q to E_{j_{0}} for a unique j_{0} ∈ {1, …, m}.

The justification for this uses the full force of the modularity of elliptic curves defined over Q: we have computed a full set of newforms f at level N, and the same number of isogeny classes of elliptic curves, and the theory tells us that there is a bijection between these sets. Checking the first few terms of the Lseries (i.e., comparing the Hecke eigenforms of the newforms with the traces of Frobenius for the curves) allows us to pair up each isogeny class with a newform.

We will assume that one of the E_{j}, which we always label E_{1}, is such that Λ_{f} and Λ_{1} (the period lattice of E_{1}) are approximately equal. This is true in practice, because our method of finding the curves in the isogeny class is to compute the coefficients of a curve from numerical approximations to the c_{4} and c_{6} invariants of C/Λ_{f}; in all cases these are very close to integers which are the invariants of the minimal model of an elliptic curve of conductor N, which we call E_{1}. The other curves in the isogeny class are then computed from E_{1}. For the algorithm described here, however, it is irrelevant how the curves E_{j} were obtained, provided that Λ_{1} and Λ_{f} are close (in a precise sense defined below).

Normalisation of lattices: every lattice Λ in C which defined over R has a unique Z-basis ω_{1}, ω_{2} satisfying one of the following:

• Type 1: ω_{1} and (2ω_{2} − ω_{1})/i are real and positive; or

• Type 2: ω_{1} and ω_{2}/i are real and positive.

For Λ_{f} we know the type from modular symbol calculations, and we know ω_{1}, ω_{2} to a certain precision by numerical integration; modular symbols provide us with cycles γ_{1}, γ_{2} ∈ H_{1}(X_{0}(N), Z) such that the integral of 2πif(z)dz over γ_{1}, γ_{2} give ω_{1}, ω_{2}.

For each curve E_{j} we compute (to a specific precision) a Z-basis for its period lattice Λ_{j} using the standard AGM method. Here, Λ_{j} is the lattice of periods of the Néron differential on E_{j}. The type of Λ_{j} is determined by the sign of the discriminant of E_{j}: type 1 for negative discriminant, and type 2 for positive discriminant.

For our algorithm we will need to know that Λ_{1} and Λ_{f} are approximately equal. To be precise, we know that they have the same type, and also we verify, for a specific positive ε, that

(*)  |\frac{ω_{1,1}}{ω_{1,f}} − 1| < ε  and  |\frac{im(ω_{2,1})}{im(ω_{2,f})} − 1| < ε.

Here ω_{1,j}, ω_{2,j} denote the normalised generators of Λ_{j}, and ω_{1,f}, ω_{2,f} those of Λ_{f}.

Pulling back the Néron differential on E_{j_{0}} to X_{0}(N) gives c·2πif(z)dz where c ∈ Z is the Manin constant for f. Hence

cΛ_{f} = Λ_{j_{0}}.

Our task is now to

(1) identify j_{0}, to find which of the E_{j} is (isomorphic to) the “optimal” curve E_{f}; and

(2) determine the value of c.

Our main result is that j_{0} = 1 and c = 1, provided that the precision bound ε in (*) is sufficiently small (in most cases, ε < 1 suffices). In order to state this precisely, we need some further definitions.

A result of Stevens says that in the isogeny class there is a curve, say E_{j_{1}}, whose period lattice Λ_{j_{1}} is contained in every Λ_{j}; this is the unique curve in the class with minimal Faltings height. (It is conjectured that E_{j_{1}} is the Γ_{1}(N)- optimal curve, but we do not need or use this fact. In many cases, the Γ_{0}(N)- and Γ_{1}(N)-optimal curves are the same, so we expect that j_{0} = j_{1} often; indeed, this holds for the vast majority of cases.)

For each j, we know therefore that a_{j} = ω_{1,j_{1}}/ω_{1,j} ∈ N and also b_{j} = im(ω_{2,j_{1}})/im(ω_{2,j}) ∈ N. Let B be the maximum of a_{1} and b_{1}.

Proposition 5.1. Suppose that (*) holds with ϵ = B^{−1}; then j_{0} = 1 and c = 1. That is, the curve E_{1} is the optimal quotient and its Manin constant is 1.

Proof. Let ε = B^{−1} and λ = \frac{ω_{1,1}}{ω_{1,f}}, so |λ − 1| < ε. For some j we have cΛ_{f} = Λ_{j}. The idea is that lcm(a_{1}, b_{1})Λ_{1} ⊆ Λ_{j_{1}} ⊆ Λ_{j} = cΛ_{f}; if a_{1} = b_{1} = 1, then the closeness of Λ_{1} and Λ_{f} forces c = 1 and equality throughout. To cover the general case it is simpler to work with the real and imaginary periods separately.

Firstly,

\frac{ω_{1,j}}{ω_{1,f}} = c ∈ Z.

Then

c = \frac{ω_{1,1}}{ω_{1,f}} \frac{ω_{1,j}}{ω_{1,1}} = \frac{a_{1}}{a_{j}} λ.

Hence

0 ≤ |λ − 1| = \frac{|a_{j}c − a_{1}|}{a_{1}} < ε.

If λ ≠ 1, then ε > |λ − 1| ≥ a_{1}^{−1} ≥ B^{−1} = ε, contradiction. Hence λ = 1, so ω_{1,1} = ω_{1,f}. Similarly, we have

\frac{im(ω_{2,j})}{im(ω_{2,f})} = c ∈ Z

and again we can conclude that im(ω_{2,1}) = im(ω_{2,f}), and hence ω_{2,1} = ω_{2,f}.

Thus Λ_{1} = Λ_{f}, from which the result follows. □

Theorem 5.2. For all N < 60000, every optimal elliptic quotient of J_{0}(N) has Manin constant equal to 1. Moreover, the optimal curve in each class is the one whose identifying number on the tables [Cre] is 1 (except for class 990h where the optimal curve is 990h3).

Proof. For all N < 60000 we used modular symbols to find all newforms f and their period lattices, and also the corresponding isogeny classes of curves. In all cases we verified that (*) held with the appropriate value of ε. (The case of 990h is only exceptional on account of an error in labelling the curves several years ago, and is not significant.) □

Remark 5.3. In the vast majority of cases, the value of B is 1, so the precision needed for the computation of the periods is very low. For N < 60000, out of 258502 isogeny classes, only 136 have B > 1: we found a_{1} = 2 in 13 cases, a_{1} = 3 in 29 cases, and a_{1} = 4 and a_{1} = 5 once each (for N = 15 and N = 11 respectively); b_{1} = 2 in 93 cases; and no larger values. Class 17a is the only one for which both a_{1} and b_{1} are greater than 1 (both are 2).

Finally, we give a slightly weaker result for 60000 < N < 130000; in this range we do not know Λ_{f} precisely, but only its projection onto the real line. (The reason for this is that we can find the newforms using modular symbols for H_{1}^{+}(X_{0}(N), Z), which has half the dimension of H_{1}(X_{0}(N), Z); but to find the exact period lattice requires working in H_{1}(X_{0}(N), Z).) The argument is similar to the one given above, using B = a_{1}.

Theorem 5.4. For all N in the range 60000 < N < 130000, every optimal elliptic quotient of J_{0}(N) has Manin constant equal to 1.

Proof. We continue to use the notation above. We do not know the lattice Λ_{f} but only (to a certain precision) a positive real number ω_{1,f}^{+} such that either Λ_{f} has type 1 and ω_{1,f} = 2ω_{1,f}^{+}, or Λ_{f} has type 2 and ω_{1,f} = ω_{1,f}^{+}. Curve E_{1} has lattice Λ_{1}, and the ratio λ = ω_{1,1}^{+}/ω_{1,f}^{+} satisfies |λ − 1| < ε. In all cases this holds with ε = \frac{1}{3}, which will suffice.

First assume that a_{1} = 1.

If the type of Λ_{f} is the same as that of Λ_{1} (for example, this must be the case if all the Λ_{j} have the same type, which will hold whenever all the isogenies between the E_{j} have odd degree) then from cΛ_{f} = Λ_{j} we deduce as before that λ = 1 exactly, and c = a_{1}/a_{j} = 1/a_{j}, hence c = a_{j} = 1. So in this case we have that c = 1, though there might be some ambiguity in which curve is optimal if a_{j} = 1 for more than one value of j.

Assume next that Λ_{1} has type 1 but Λ_{f} has type 2. Now λ = ω_{1,1}/2ω_{1,f}. The usual argument now gives ca_{j} = 2. Hence either c = 1 and a_{j} = 2, or c = 2 and a_{j} = 1. To see if the latter case could occur, we look for classes in which a_{1} = 1 and Λ_{1} has type 1, while for some j > 1 we also have a_{j} = 1 and Λ_{j} of type 2. This occurs 28 times for 60000 < N < 130000, but for 15 of these the level N is odd, so we know that c must be odd. The remaining 13 cases are

62516a, 67664a, 71888e, 72916a, 75092a, 85328d, 86452a, 96116a,
106292b, 111572a, 115664a, 121168e, 125332a;

we have been able to eliminate these by carrying out the extra computations necessary as in the proof of Theorem 5.2. We note that in all of these 13 cases, the isogeny class consists of two curves, E_{1} of type 1 and E_{2} of type 2, with [Λ_{1} : Λ_{2}] = 2, so that E_{2} rather than E_{1} has minimal Faltings height.

Next suppose that Λ_{1} has type 2 but Λ_{f} has type 1. Now λ = 2ω_{1,1}/ω_{1,f}. The usual argument now gives 2ca_{j} = 1, which is impossible; so this case cannot occur.

Finally we consider the cases where a_{1} > 1. There are only three of these for 60000 < N < 130000: namely, 91270a, 117622a and 124973b, where a_{1} = 3. In each case the Λ_{j} all have the same type (they are linked via 3-isogenies) and the usual argument shows that ca_{j} = 3. But none of these levels is divisible by 3, so c = 1 in each case. □

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Amod Agashe Department of Mathematics Florida State University 208 Love Building Tallahassee FL 32306 U.S.A. E-mail: agashe@mail.math.fsu.edu

Kenneth A. Ribet Department of Mathematics, M/C 3840 University of California Berkeley, CA 94720-3840, USA. E-mail: ribet@math.berkeley.edu

William A. Stein Department of Mathematics Harvard University Cambridge, MA 02138 E-mail: was@math.harvard.edu