$7$-adic Galois representations of elliptic curves over the rationals via Kummer descent

Abstract: We show that the two modular curves of level $49$ and genus $9$ left open by the work of Rouse, Sutherland and Zureick-Brown, and of Furio and Lombardo, have no non-CM rational points. Together with the theorem of Furio and Lombardo on $X_{ns}<sup>{+}(49)$, this completes the classification of the $7$-adic images of Galois of non-CM elliptic curves over $Q$. The proof is a Kummer descent on the superelliptic equations $F(x,y) = k\,w<sup>{7}$ of Furio and Lombardo, in the cases $7 \mid k$ that they left open. The covering curves are twists of the Fermat septic, they map to twists of the Klein quartic, and a $7$-adic computation shows that the twist attached to a solution is trivial. The rational points of the Klein quartic were determined by Hurwitz, who reduced the question to Fermat's Last Theorem for exponent $7$, proved by Lamé. Thus, the last open case of the $7$-adic part of Mazur's Program B rests on Fermat's Last Theorem for exponent $7$.