Explicit open images for elliptic curves over $\mathbb{Q}$

TL;DR

This paper presents a practical algorithm to compute the Galois image of non-CM elliptic curves over , providing a near-complete classification of these images under certain conjectural assumptions, with applications to modular curve computations.

Contribution

It introduces an efficient algorithm for determining the Galois image of elliptic curves over  and offers a classification of these images assuming certain uniformity and rational point finiteness conjectures.

Findings

Algorithm successfully computes Galois images up to conjugacy.

Provides a classification of possible Galois images and indices.

Details methods for computing modular curves using Eisenstein series.

Abstract

For a non-CM elliptic curve $E$ defined over $\mathbb{Q}$, the Galois action on its torsion points gives rise to a Galois representation $\rho_E: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\widehat{\mathbb{Z}})$ that is unique up to isomorphism. A renowned theorem of Serre says that the image of $\rho_E$ is an open, and hence finite index, subgroup of $GL_2(\widehat{\mathbb{Z}})$. We describe an algorithm that computes the image of $\rho_E$ up to conjugacy in $GL_2(\widehat{\mathbb{Z}})$; this algorithm is practical and has been implemented. Up to a positive answer to a uniformity question of Serre and finding all the rational points on a finite number of explicit modular curves of genus at least $2$, we give a complete classification of the groups $\rho_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))\cap SL_2(\widehat{\mathbb{Z}})$ and the indices $[GL_2(\widehat{\mathbb{Z}}):\rho_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))]$ for non-CM elliptic curves $E/\mathbb{Q}$. Much of the paper is dedicated to the efficient computation of modular curves via modular forms expressed in terms of Eisenstein series.