One-parameter families of elliptic curves over ℚ with maximal Galois representations

Abstract

Let E be an elliptic curve over ℚ and let ℚ(E[n]) be its nth division field. In 1972, Serre showed that if E is without complex multiplication, then the Galois group of ℚ(E[n])/ℚ is as large as possible, that is, GL2(ℤ/nℤ), for all integers n coprime to a constant integer m(E, ℚ) depending (at most) on E/ℚ. Serre also showed that the best one can hope for is to have |GL2(ℤ/nℤ) : Gal(ℚ(E[n])/ℚ)| ⩽ 2 for all positive integers n. We study the frequency of this optimal situation in a one-parameter family of elliptic curves over ℚ, and show that in essence, for almost all one-parameter families, almost all elliptic curves have this optimal behavior.