Modular Parametrizations of Elliptic Curves

Abstract

Many — conjecturally all — elliptic curves E / have a "modular parametrization," i.e. for some N there is a map φ from the modular curve X 0 ( N ) to E such that the pull-back of a holomorphic differential on E is a modular form (newform) f of weight 2 and level N . We describe an algorithm for computing the degree of φ as a branched covering, discuss the relationship of this degree to the "congruence primes" for f (the primes modulo which there are congruences between f and other newforms), and give estimates for the size of this degree as a function of N .