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Torsion groups of rational elliptic curves over ℤp-extensions of quadratic fields: the p≤ 5 case
Abstract
Let E be a rational elliptic curve. We generalize a theorem due to Avcı\cite{AVCI2026153}, which asserts that for any quadratic field K and prime p>5, the equality E(K)tors = E(L)tors holds for every ℤp-extension L/K. In this paper, we consider the setting where the ℤp-extension L is replaced by the compositum K∞ of all ℤp-extensions of K. Under this new setting, we prove the analogous statement for p=5, and further provide some results for the remaining primes p=3 and p=2, where we also try to study the growth pattern of torsion groups. Moreover, we completely classify all triples (E,K,p) for which E(K∞)tors is infinite.
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BibTeX
@misc{li2026torsion,
title = {{Torsion groups of rational elliptic curves over $\mathbbZ_p$-extensions of quadratic fields: the $p\le 5$ case}},
author = {Haidong Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.13514},
eprint = {2607.13514},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.13514},
}
RIS
TY - UNPB
TI - Torsion groups of rational elliptic curves over ℤ_p-extensions of quadratic fields: the p≤ 5 case
AU - Haidong Li
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2607.13514
UR - https://arxiv.org/abs/2607.13514
AB - Let E be a rational elliptic curve. We generalize a theorem due to Avcı\cite{AVCI2026153}, which asserts that for any quadratic field K and prime p>5, the equality E(K)_tors = E(L)_tors holds for every ℤ_p-extension L/K. In this paper, we consider the setting where the ℤ_p-extension L is replaced by the compositum K_∞ of all ℤ_p-extensions of K. Under this new setting, we prove the analogous statement for p=5, and further provide some results for the remaining primes p=3 and p=2, where we also try to study the growth pattern of torsion groups. Moreover, we completely classify all triples (E,K,p) for which E(K_∞)_tors is infinite.
ER -