Fields of definition of p-torsion points of elliptic curves and their ramification
Abstract
Let E be an elliptic curve over ℚp. We study the field ℚp(E[p]) generated by the p-torsion points of E. When E has good reduction, we determine not only ℚp(E[p]) but also the field ℚp(P) for every point P∈ E[p]. This classification yields criteria for the existence of p-torsion over unramified and Lubin--Tate extensions of ℚp. As a global application, let E/ℚ be an elliptic curve and let p be an odd prime of good reduction. Define fp(E) to be the multiplicative order of ap(E) modulo p in the ordinary case, and set fp(E)=p2−1 in the supersingular case. We prove that E(K)[p]=0 for every number field K/ℚ with [K:ℚ]<fp(E). When E has bad reduction, we describe the ramified part of ℚp(E[p])/ℚp. For every reduction type, we determine the maximal upper ramification break of ℚp(E[p])/ℚp.
Related
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- On the Ramification of Quadratic Fields where there is a torsion growth
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BibTeX
@misc{ozeki2026fields,
title = {{Fields of definition of $p$-torsion points of elliptic curves and their ramification}},
author = {Yoshiyasu Ozeki and Manabu Yoshida},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06554},
eprint = {2609.06554},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06554},
}
RIS
TY - UNPB
TI - Fields of definition of p-torsion points of elliptic curves and their ramification
AU - Yoshiyasu Ozeki
AU - Manabu Yoshida
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06554
UR - https://arxiv.org/abs/2609.06554
AB - Let E be an elliptic curve over ℚ_p. We study the field ℚ_p(E[p]) generated by the p-torsion points of E. When E has good reduction, we determine not only ℚ_p(E[p]) but also the field ℚ_p(P) for every point P∈ E[p]. This classification yields criteria for the existence of p-torsion over unramified and Lubin--Tate extensions of ℚ_p. As a global application, let E/ℚ be an elliptic curve and let p be an odd prime of good reduction. Define f_p(E) to be the multiplicative order of a_p(E) modulo p in the ordinary case, and set f_p(E)=p^2−1 in the supersingular case. We prove that E(K)[p]=0 for every number field K/ℚ with [K:ℚ]<f_p(E). When E has bad reduction, we describe the ramified part of ℚ_p(E[p])/ℚ_p. For every reduction type, we determine the maximal upper ramification break of ℚ_p(E[p])/ℚ_p.
ER -