On the Surjectivity of Galois Representations Associated to Elliptic Curves over Number Fields

TL;DR

This paper establishes new bounds on the largest exceptional prime for Galois representations of elliptic curves over number fields, showing under GRH that it is proportional to the logarithm of the conductor's norm.

Contribution

It provides a new bound on the largest exceptional prime and the product of all exceptional primes for non-CM elliptic curves over number fields, affirming a question of Serre.

Findings

Conditional bound on the largest exceptional prime proportional to log of the conductor's norm

New bounds on the product of all exceptional primes

Affirmative answer to Serre's question under GRH

Abstract

Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$. A famous theorem of Serre states that as long as $E$ has no Complex Multiplication (CM), the map $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$ is surjective for all but finitely many $\ell$. We say that a prime number $\ell$ is exceptional (relative to the pair $(E,K)$) if this map is not surjective. Here we give a new bound on the largest exceptional prime, as well as on the product of all exceptional primes of $E$. We show in particular that conditionally on the Generalized Riemann Hypothesis (GRH), the largest exceptional prime of an elliptic curve $E$ without CM is no larger than a constant (depending on $K$) times $\log N_E$, where $N_E$ is the absolute value of the norm of the conductor. This answers affirmatively a question of Serre.