The modular degree, congruence primes and multiplicity one
Amod Agashe, Ken Ribet, William Stein
We answer a question of Frey and M uller about whether or not the modular degree and congruence number of elliptic curves are equal. We give examples in which they are not, prove a theorem relating them, and make a conjecture about the extent to which they differ. We also obtain relations between analogues of the modular degree and congruence number for modular abelian varieties, and give new examples of failure of multiplicity one.