 
Summary: 1
The Modular Degree, Congruence Primes, and Multiplicity One
Amod Agashe 1
Kenneth A. Ribet William A. Stein
Abstract.
The modular degree and congruence number are two fundamental in
variants of an elliptic curve over the rational field. Frey and M¨uller
have asked whether these invariants coincide. We find that the ques
tion has a negative answer, and show that in the counterexamples,
multiplicity one (defined below) does not hold. At the same time, we
prove a theorem about the relation between the two invariants: the
modular degree divides the congruence number, and the ratio is divis
ible only by primes whose squares divide the conductor of the elliptic
curve. We discuss the ratio even in the case where the square of a
prime does divide the conductor, and we study analogues of the two
invariants for modular abelian varieties of arbitrary dimension.
1 Introduction
Let E be an elliptic curve over Q. By [BCDT01], we may view E as an
abelian variety quotient over Q of the modular Jacobian J0(N), where N is
the conductor of E. We assume that the kernel of the map J0(N) E is
