# Optimal Elliptic Curve Quotients

Let  be a positive integer and let be the modular curve over  that classifies isomorphism classes of elliptic curves with a cyclic subgroup of order  . The Hecke algebra  of level  is the subring of the ring of endomorphisms of generated by the Hecke operators for all . Suppose  is a newform of weight  for  with integer Fourier coefficients, and let be kernel of the homomorphism that sends to . Then the quotient is an elliptic curve over  . We call  the optimal quotient associated to  . Composing the embedding that sends to with the quotient map , we obtain a surjective morphism of curves . The modular degree of  is the degree of  .

Let denote the Néron model of  over  . 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  be a generator for the rank -module of invariant differential -forms on  . The pullback of  to  is a differential on  . The newform  defines another differential  on  . Because the action of Hecke operators is compatible with the map , the differential is a -eigenvector with the same eigenvalues as , so by [AL70] we have for some (see also [Man72, §5]). The Manin constant of  is the absolute value of the rational number  defined above.

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

Conjecture 2.1 (Manin)   We have .

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

Theorem 2.2 (Edixhoven [][Prop. 2)   edix:manin] The constant is an integer.

Edixhoven proved this using an integral -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, [][Cor. 4.1)   maziso] If  , then .

Mazur proved this by applying theorems of Raynaud about exactness of sequences of differentials, then using the  -expansion principle'' in characteristic  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 .

Theorem 2.4 (Raynaud [][Prop. 3.1)   abbull] If , then .

Theorem 2.5 (Abbes-Ullmo [][Thm. A)   abbull] If , then .

We generalize Theorem 2.4 in Theorem 3.10. However, it is not clear if Theorem 2.5 generalizes to dimension greater than  . It would be fantastic if the theorem could be generalized. It would imply that the Manin constant is  for newform quotients of , with  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  are , , , and ; he also gives bounds that are independent of  on the valuations of  at , , , and . His arguments rely on the construction of certain stable integral models for .

See Section 5 for more details about the following computation:

Theorem 2.6 (Cremona)   If is an optimal elliptic curve over with conductor at most , then .

theorem_type[defi][lem][][definition][][] [Congruence Number] The congruence number tex2html_wrap_inline$r_E$ of tex2html_wrap_inline$E$ is the largest integer tex2html_wrap_inline$r$ such that there exists a cusp form tex2html_wrap_inline$g&isin#in;S_2(&Gamma#Gamma;_0(N))$ that has integer Fourier coefficients, is orthogonal to tex2html_wrap_inline$f$ with respect to the Petersson inner product, and satisfies tex2html_wrap_inline$g &equiv#equiv;f r$. The congruence primes of tex2html_wrap_inline$E$ are the primes that divide tex2html_wrap_inline$r_E$.

To the above list of theorems we add the following:

Theorem 2.7   If then or .

This theorem is a special case of Theorem 3.11 below. In view of Theorem 2.3, our new contribution is that if is odd and , then is odd. This hypothesis is very stringent--of the optimal elliptic curve quotients of conductor , only  of them satisfy the hypothesis. In the notation of [Cre], 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, 31214c, 38158a, 39086a, 40366a, 41774a, 42638a, 45134a, 48878a, 50894b, 53678a, 54286a, 56558f, 58574b, 59918a, 61454b, 63086a, 63694a, 64366b, 64654b, 65294a, 65774b, 71182b, 80942a, 83822a, 93614a

Each of the curves in this list has conductor tex2html_wrap_inline$2p$ with tex2html_wrap_inline$p&equiv#equiv; 34$ prime. The situation is similar to that of [SW04, Conj. 4.2], which suggests there are infinitely many such curves. See also [CE05] for a classification of elliptic curves with odd modular degree.

William Stein 2006-06-25