English

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

Number Theory 2024-03-25 v2

Abstract

For a non-CM elliptic curve EE defined over Q\mathbb{Q}, the Galois action on its torsion points gives rise to a Galois representation ρE:Gal(Q/Q)GL2(Z^)\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 ρE\rho_E is an open, and hence finite index, subgroup of GL2(Z^)GL_2(\widehat{\mathbb{Z}}). We describe an algorithm that computes the image of ρE\rho_E up to conjugacy in GL2(Z^)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 22, we give a complete classification of the groups ρE(Gal(Q/Q))SL2(Z^)\rho_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))\cap SL_2(\widehat{\mathbb{Z}}) and the indices [GL2(Z^):ρE(Gal(Q/Q))][GL_2(\widehat{\mathbb{Z}}):\rho_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))] for non-CM elliptic curves E/QE/\mathbb{Q}. Much of the paper is dedicated to the efficient computation of modular curves via modular forms expressed in terms of Eisenstein series.

Keywords

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

R2 v1 2026-06-24T12:09:01.562Z