Explicit open images for elliptic curves over $\mathbb{Q}$
Abstract
For a non-CM elliptic curve defined over , the Galois action on its torsion points gives rise to a Galois representation that is unique up to isomorphism. A renowned theorem of Serre says that the image of is an open, and hence finite index, subgroup of . We describe an algorithm that computes the image of up to conjugacy in ; 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 , we give a complete classification of the groups and the indices for non-CM elliptic curves . Much of the paper is dedicated to the efficient computation of modular curves via modular forms expressed in terms of Eisenstein series.
Cite
@article{arxiv.2206.14959,
title = {Explicit open images for elliptic curves over $\mathbb{Q}$},
author = {David Zywina},
journal= {arXiv preprint arXiv:2206.14959},
year = {2024}
}
Comments
Several minor corrections. Added a conjecture with motivation to the last section