Torsion groups of rational elliptic curves over $\mathbb{Z}_p$-extensions of quadratic fields: the $p\le 5$ case
Year: 2026
Type: preprint
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 $E(K)_{\mathrm{tors}} = E(L)_{\mathrm{tors}}$ holds for every $\mathbb{Z}_p$-extension $L/K$. In this paper, we consider the setting where the $\mathbb{Z}_p$-extension $L$ is replaced by the compositum $K_{\infty}$ of all $\mathbb{Z}_p$-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 $E(K_\infty)_{\mathrm{tors}}$ is infinite.
Source: arXiv (Cornell University)
Author: Li, Haidong
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7168921625