On 7-adic Galois representations for elliptic curves over Q

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 Q. Currently, the classification is only complete for p ∈ \2,3,13,17\. The main difficulty for other primes arises from the need to understand elliptic curves whose mod-pⁿ 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 Xₙₛ⁺(pⁿ). Here, we consider the case $p=7$ and show that the modular curve Xₙₛ⁺(49), of genus 69, has no non-CM rational points. To achieve this, we establish a correspondence between the rational points on Xₙₛ⁺(49) and the primitive integer solutions of the generalised Fermat equation a² + 28b³ = 27 c⁷, 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.