$7$-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 Xns+(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 w7 of Furio and Lombardo, in the cases 7 | 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 $7$.
Related
- Complete classification of $7$-adic Galois images for non-CM elliptic curves over Q
- $2$-adic Representations of Elliptic Curves over ℚ2
- Lifting mod pn Galois representations
- Revisiting the infinite Galois correspondence via group representations
- On the modularity of residually reducible Galois representations of rank 2
BibTeX
@misc{arangopineros20267adic,
title = {{$7$-adic Galois representations of elliptic curves over the rationals via Kummer descent}},
author = {Santiago Arango-Piñeros and David Zureick-Brown},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06521},
eprint = {2610.06521},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06521},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - $7$-adic Galois representations of elliptic curves over the rationals via Kummer descent
AU - Santiago Arango-Piñeros
AU - David Zureick-Brown
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06521
UR - https://arxiv.org/abs/2610.06521
AB - 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^+(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^7 of Furio and Lombardo, in the cases 7 | 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 $7$.
ER -