Uniform results for Serre's theorem for elliptic curves

Abstract

A celebrated theorem of Serre from 1972 asserts that if E is an elliptic curve defined over ℚ and without complex multiplication, then its associated mod l representation is surjective for all sufficiently large primes l. In this paper we address the question of what “sufficiently large” means in Serre's theorem. More precisely, we obtain a uniform version of Serre's theorem for nonconstant elliptic curves defined over function fields, and a uniform version of Serre's theorem for one-parameter families of elliptic curves defined over ℚ.