7-adic Galois representations and generalised Fermat equations

The study of Galois representations attached to elliptic curves is a highly fruitful branch of number theory, leading to the resolution of deep problems such as Fermat’s Last Theorem. In 1972, Serre proved his foundational Open Image Theorem, which states that for every non-CM elliptic curve defined over a number field, the image of the adelic Galois representation on its torsion points has finite index. This result soon inspired Mazur to propose his famous Program B, aiming to classify all possible images of such representations.

In recent years, substantial progress has been made toward Mazur’s Program B, with several authors undertaking a systematic classification of all possible images of $p$-adic Galois representations attached to elliptic curves over $\mathbb{Q}$. At present, the classification is complete only for $p \in \{2, 3, 13, 17\}$. The main obstacle for other primes arises from the difficulty of understanding elliptic curves whose mod-$p^n$ Galois representations are contained in the normalizer of a non-split Cartan subgroup. Equivalently, this amounts to determining the rational points on the modular curves $X_{ns}^+(p^n)$.

In this talk, we focus on the case $p=7$ and show that the modular curve $X_{ns}^+(49)$, which has 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 generalized Fermat equation $a^2 + 28b^3 = 27c^7$, the resolution of which can be reduced to determining the rational points on several genus-three curves. Furthermore, we reduce the complete classification of $7$-adic images to the determination of the rational points on a single plane quartic.

This is joint work with Davide Lombardo.