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 ${\rho }_{E,\ell }:G\to G{L}_2({\mathbb{F}}_{\ell })$ 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.

References in corpus (1)

• Rational points on X_0^+ (p^r)

Cited by in corpus (18)

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• On the minimal degree of definition of p-primary torsion subgroups of elliptic curves
• On the surjectivity of mod $\ell$ representations associated to elliptic curves
• Elliptic curves with non-abelian entanglements
• Double Covers of Cartan Modular Curves
• Groups of generalized $G$-type and applications to torsion subgroups of rational elliptic curves over infinite extensions of $\mathbb{Q}$
• A Classification of Genus 0 Modular Curves with Rational Points
• Torsion of elliptic curves with rational $j$-invariant defined over number fields of prime degree
• Lang--Trotter Conjecture for CM Elliptic Curves
• Cartan images and $\ell$-torsion points of elliptic curves with rational $j$-invariant
• Torsion groups of Mordell curves over number fields of higher degree
• Congruences of elliptic curves arising from non-surjective mod $N$ Galois representations