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].

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

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.

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 .

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B. Edixhoven also has unpublished results (see [Edi89]) which assert that the only primes that can divide are , , , and ; he also gives bounds that are independent of on the valuations of at ,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_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