二次域的\(\mathbb{Z}_p\)-扩张上有理椭圆曲线的挠群：\(p\leq5\)的情形
Torsion groups of rational elliptic curves over $\mathbb{Z}_p$-extensions of quadratic fields: the $p\le 5$ case
浏览论文内容
中文总结 AI 辅助
研究二次域的\(\mathbb{Z}_p\)-扩张上有理椭圆曲线挠群问题，推广相关定理，将\(\mathbb{Z}_p\)-扩张换为合成域\(K_{\infty}\)，证明\(p = 5\)时类似结论并给出\(p = 3\)、\(p = 2\)的部分结果。
中文摘要 AI 辅助
设\(E\)为有理椭圆曲线。我们推广了Avcı的一个定理，该定理表明对于任意二次域\(K\)和素数\(p>5\)，对于\(K\)的每个\(\mathbb{Z}_p\)-扩张\(L\)，等式\(E(K)_{\mathrm{tors}} = E(L)_{\mathrm{tors}}\)成立。本文考虑用\(K\)的所有\(\mathbb{Z}_p\)-扩张的合成域\(K_{\infty}\)代替\(\mathbb{Z}_p\)-扩张\(L\)的情形。在此新情形下，我们证明了\(p = 5\)时的类似结论，并进一步给出了\(p = 3\)和\(p = 2\)时的部分结果。
英文摘要
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.
发表机构
- Jiangsu Police Institute（江苏警官学院）
机构由 AI 辅助整理，请以论文原文为准。
