PreprintCC BY-NC-SA 4.0arXiv
Hopf-Galois Structures on Extensions Arising from Division Points on Elliptic Curves
Abstract
We provide a general construction of infinite Hopf-Galois extensions from a direct system of finite Hopf-Galois extensions. We then apply this framework to extensions generated by division points on elliptic curves. Specifically, for a number field K and an elliptic curve E/K of rank at least $1$ with P∈ E(K) of infinite order, we consider for each integer n≥ 2 the extension Ln=K(Qn), where Qn∈ E(K̅) is such that nQn=P. We show that the finite-level extensions Ln/K need not be Hopf-Galois, and we give sufficient conditions for them to be so. We then prove that the infinite extension L∞=⋃n≥ 2Ln of K is potentially Hopf-Galois, meaning it becomes Hopf-Galois after a finite extension of the base field.
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BibTeX
@misc{nkuete2026hopfgalois,
title = {{Hopf-Galois Structures on Extensions Arising from Division Points on Elliptic Curves}},
author = {Leolin Nkuete},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06253},
eprint = {2611.06253},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06253},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Hopf-Galois Structures on Extensions Arising from Division Points on Elliptic Curves
AU - Leolin Nkuete
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06253
UR - https://arxiv.org/abs/2611.06253
AB - We provide a general construction of infinite Hopf-Galois extensions from a direct system of finite Hopf-Galois extensions. We then apply this framework to extensions generated by division points on elliptic curves. Specifically, for a number field K and an elliptic curve E/K of rank at least $1$ with P∈ E(K) of infinite order, we consider for each integer n≥ 2 the extension L_n=K(Q_n), where Q_n∈ E(K̅) is such that nQ_n=P. We show that the finite-level extensions L_n/K need not be Hopf-Galois, and we give sufficient conditions for them to be so. We then prove that the infinite extension L_∞=⋃_n≥ 2L_n of K is potentially Hopf-Galois, meaning it becomes Hopf-Galois after a finite extension of the base field.
ER -