PreprintCC BY 4.0arXiv
Complete classification of $7$-adic Galois images for non-CM elliptic curves over Q
Abstract
We solve Conjecture 1.6 of Furio and Lombardo [On $7$-adic Galois representations for elliptic curves over Q, Proc.\ London Math.\ Soc.\ (3) \textbf{133} (2026), e70193.]. As a consequence, we complete the classification of $7$-adic Galois images for non-CM elliptic curves over Q. This is the first solved case in Problem 6.2 from Balakrishnan [\emph{Chabauty and beyond: Explicit p-adic methods for rational points}, Proceedings of the International Congress of Mathematicians 2026 - Volume 3: Invited Lectures (Sections 1-4), 321-341, 2026]. The proof is the result of a long AI-training period by the author from June 2026 to September 2026. The author has been working on this project alone without the help of any other.
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BibTeX
@misc{xuan2026complete,
title = {{Complete classification of $7$-adic Galois images for non-CM elliptic curves over $\Q$}},
author = {Tho Nguyen Xuan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35460},
eprint = {2609.35460},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35460},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Complete classification of $7$-adic Galois images for non-CM elliptic curves over Q
AU - Tho Nguyen Xuan
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.35460
UR - https://arxiv.org/abs/2609.35460
AB - We solve Conjecture 1.6 of Furio and Lombardo [On $7$-adic Galois representations for elliptic curves over Q, Proc.\ London Math.\ Soc.\ (3) \textbf{133} (2026), e70193.]. As a consequence, we complete the classification of $7$-adic Galois images for non-CM elliptic curves over Q. This is the first solved case in Problem 6.2 from Balakrishnan [\emph{Chabauty and beyond: Explicit p-adic methods for rational points}, Proceedings of the International Congress of Mathematicians 2026 - Volume 3: Invited Lectures (Sections 1-4), 321-341, 2026]. The proof is the result of a long AI-training period by the author from June 2026 to September 2026. The author has been working on this project alone without the help of any other.
ER -