On the possible images of the mod ell representations associated to elliptic curves over Q

Abstract

Consider a non-CM elliptic curve $E$ defined over $\mathbb{Q}$. For each prime $\ell$, there is a representation $\rho_{E,\ell}: G \to GL_2(\mathbb{F}_\ell)$ that describes the Galois action on the $\ell$-torsion points of $E$, where $G$ is the absolute Galois group of $\mathbb{Q}$. A famous theorem of Serre says that $\rho_{E,\ell}$ is surjective for all large enough $\ell$. We will describe all known, and conjecturally all, pairs $(E,\ell)$ such that $\rho_{E,\ell}$ is not surjective. Together with another paper, this produces an algorithm that given an elliptic curve $E/\mathbb{Q}$, outputs the set of such exceptional primes $\ell$ and describes all the groups $\rho_{E,\ell}(G)$ up to conjugacy. Much of the paper is dedicated to computing various modular curves of genus $0$ with their morphisms to the $j$-line.