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On 7-adic Galois representations for elliptic curves over $\mathbb{Q}$

Lorenzo Furio, Davide Lombardo

Abstract

In recent years, significant progress has been made on Mazur's Program B, with many authors beginning a systematic classification of all possible images of $p$-adic Galois representations attached to elliptic curves over $\mathbb{Q}$. Currently, the classification is only complete for $p \in \{2,3,13,17\}$. The main difficulty for other primes arises from the need to understand elliptic curves whose mod-$p^n$ Galois representations are contained in the normaliser of a non-split Cartan subgroup. Equivalently, this amounts to determining the rational points on the modular curves $X_{ns}^+(p^n)$. Here, we consider the case $p=7$ and show that the modular curve $X_{ns}^+(49)$, of genus 69, has no non-CM rational points. To achieve this, we establish a correspondence between the rational points on $X_{ns}^+(49)$ and the primitive integer solutions of the generalised Fermat equation $a^2 + 28b^3 = 27 c^7$, the resolution of which can be reduced to determining the rational points of several genus-three curves. Furthermore, we reduce the complete classification of $7$-adic images to the determination of the rational points of a single plane quartic.

On 7-adic Galois representations for elliptic curves over $\mathbb{Q}$

Abstract

In recent years, significant progress has been made on Mazur's Program B, with many authors beginning a systematic classification of all possible images of -adic Galois representations attached to elliptic curves over . Currently, the classification is only complete for . The main difficulty for other primes arises from the need to understand elliptic curves whose mod- Galois representations are contained in the normaliser of a non-split Cartan subgroup. Equivalently, this amounts to determining the rational points on the modular curves . Here, we consider the case and show that the modular curve , of genus 69, has no non-CM rational points. To achieve this, we establish a correspondence between the rational points on and the primitive integer solutions of the generalised Fermat equation , the resolution of which can be reduced to determining the rational points of several genus-three curves. Furthermore, we reduce the complete classification of -adic images to the determination of the rational points of a single plane quartic.

Paper Structure

This paper contains 19 sections, 37 theorems, 70 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Possible extensions
  3. Computer-assisted calculations.
  4. Acknowledgements.
  5. Preliminaries
  6. $7$-adic images of Galois representations
  7. From rational points on modular curves to Fermat equations
  8. From Fermat equations to elliptic curves via modularity
  9. Level $2^2 \cdot 7^2 = 196$
  10. Level $2^3 \cdot 7^2 = 392$
  11. Level $2^4 \cdot 7^2 = 784$
  12. Ruling out modular forms
  13. From elliptic curves with given $7$-torsion to rational points on genus-3 curves
  14. Rational points on the curves $X_{E_i}$
  15. Known rational points and proof of \ref{['thm: solutions twisted Fermat']}
  16. ...and 4 more sections

Key Result

Theorem 1.3

Let $p$ be a prime, let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication, and let $G:=\operatorname{Im}\rho_{E,p^\infty}$. Exactly one of the following is true:

Theorems & Definitions (84)

  • Theorem 1.3: Rouse--Sutherland--Zureick-Brown
  • Theorem 1.4
  • Corollary 1.5
  • proof
  • Conjecture 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Theorem 2.1: Zywina
  • Remark 2.2
  • Lemma 2.3
  • ...and 74 more