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

David Zywina

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 $ρ_E: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\widehat{\mathbb{Z}})$ that is unique up to isomorphism. A renowned theorem of Serre says that the image of $ρ_E$ is an open, and hence finite index, subgroup of $GL_2(\widehat{\mathbb{Z}})$. We describe an algorithm that computes the image of $ρ_E$ 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 $ρ_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))\cap SL_2(\widehat{\mathbb{Z}})$ and the indices $[GL_2(\widehat{\mathbb{Z}}):ρ_E(Gal(\overline{\mathbb{Q}}/\mathbb{Q}))]$ 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.

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.
Paper Structure (93 sections, 54 theorems, 164 equations, 1 algorithm)

This paper contains 93 sections, 54 theorems, 164 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Serre's open image theorem
  3. The Kronecker--Weber constraint on the image
  4. Modular curves
  5. Agreeable closures
  6. Finding the image of Galois
  7. Example: Serre curves
  8. An example with the largest known index
  9. An involved example
  10. Some related results
  11. Overview
  12. Implementation
  13. Notation
  14. Acknowledgements
  15. Cyclotomic constraints on the image of Galois
  16. ...and 78 more sections

Key Result

Theorem 1.1

Let $E$ be a non-CM elliptic curve defined over $\mathbb Q$. Then $\rho_E(\operatorname{Gal}_\mathbb Q)$ is an open subgroup of $\operatorname{GL}_2(\widehat{\mathbb Z})$. Equivalently, $\rho_E(\operatorname{Gal}_\mathbb Q)$ is a finite index subgroup of $\operatorname{GL}_2(\widehat{\mathbb Z})$.

Theorems & Definitions (130)

  • Theorem 1.1: Serre's open image theorem
  • Conjecture 1.2
  • Conjecture 1.3
  • Remark 1.4
  • Conjecture 1.5
  • Remark 1.6
  • Lemma 1.7
  • proof
  • Example 1.8
  • Theorem 1.9
  • ...and 120 more