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].
Significant progress has been made towards this conjecture. In the following theorems, denotes a prime and denotes the conductor of .
-expansion map, whose existence and properties follow from results in [KM85]. We generalize his theorem to quotients of arbitrary dimension in Theorem 3.4.
-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 .
. 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B. Edixhoven also has unpublished results (see [Edi89]) which assert that the only primes that can divide are , , , and ; he also gives bounds that are independent of on the valuations of at ,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_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:
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