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

TL;DR

This paper investigates 7-adic Galois representations of elliptic curves over rationals, proving that a specific modular curve has no rational points outside CM cases, thus advancing the classification of possible Galois image structures.

Contribution

It establishes the non-existence of rational points on a high-genus modular curve related to 7-adic Galois representations, reducing the classification problem to a single quartic curve.

Findings

No non-CM rational points on X_{ns}^+(49)

Reduction of classification to a single plane quartic

Connection to solutions of a generalized Fermat equation

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.