Uniform Results for Serre’s Theorem for Elliptic Curves

Let E/K be an elliptic curve defined over a number field K and without complex multiplication (CM). For a rational prime , let K(E[ ]) be the th division field of E, which we know is a finite Galois extension of K. By a celebrated result of Serre [18], there exists a positive constant c(E,K), depending on E and K, such that Gal(K(E[ ])/K) GL2(Z/ Z) for all ≥ c(E,K). In [18, 19], Serre asked whether there exists a positive constant c(K), depending at most on K, such that Gal(K(E[ ])/K) GL2(Z/ Z) for all ≥ c(K). An affirmative answer to this question would have important diophantine applications, as illustrated in [14]. Currently, there exist few results related to Serre’s question. In [13], Mazur showed that for K = Q and for semistable elliptic curves E/Q without CM, one has Gal(Q(E[ ])/Q) GL2(Z/ Z) for any prime ≥ 11. In [2, 9, 12, 19], upper bounds in terms of invariants of E (height and conductor) were given for the exceptional primes of an elliptic curve E/Q, that is, for those primes for which Gal(Q(E[ ])/Q) GL2(Z/ Z). More ideas are still needed, however, to completely answer Serre’s question. Naturally, one can ask if Serre’s question is true “on average” or “over function fields.” The goal of our paper is to study these two questions. In [4], Duke gave an affirmative answer to the first question for a natural two-parameter family of elliptic curves E/Q which contains every elliptic curve over Q. One of our aims is to obtain a more refined average result; that is, to answer Serre’s question for “most” elements of a oneparameter family of elliptic curves. This is the content of Theorem 1.3 below. We will also