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

TL;DR

This paper investigates the possible images of Galois representations associated with elliptic curves over Q, identifying exceptional primes where the representation is not surjective, and providing an algorithm to determine these for any given curve.

Contribution

It classifies all known and conjecturally all non-surjective Galois representations for elliptic curves over Q and develops an algorithm to compute exceptional primes and their images.

Findings

Classification of non-surjective Galois representations for elliptic curves over Q.

Development of an algorithm to identify exceptional primes for any elliptic curve.

Analysis of modular curves of genus 0 related to these representations.

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.