 
Summary: The Manin Constant, Congruence Primes, and the
Modular Degree #
Amod Agashe Kenneth A. Ribet William A. Stein
November 11, 2004
Abstract. We show that if E is an optimal elliptic curve quotient of J 0 (N) and
p is a prime such that p 2 does not divide N and p does not divide
the congruence number of E, then p does not divide the Manin
constant of E either (in light of what is known before, the only
new information is when p = 2). We also answer a question raised
in a paper of Frey and M˜uller about whether or not the congruence
number and modular degree of elliptic curves are always equal (they
are not). Finally, we generalize the notion of the Manin constant
to arbitrary quotients of J 1 (N) and J 0 (N) by ideals of the Hecke
algebra, prove several results about it, and make some conjectures.
Let N be a positive integer and E be an optimal elliptic curve quo
tient of J 0 (N) (optimal means that ker(J 0 (N) # E) is connected). The
Manin constant of E is an invariant associated to E that plays a role in the
Birch and SwinnertonDyer conjecture (see Section 2.2.1). Manin conjec
tured that it is always equal to 1. In Section 1.1, we recall the definition
of the Manin constant of an optimal elliptic curve quotient E of J 0 (N) and
