AI 中文总结
本文受西尔弗曼定理启发，研究数域上阿贝尔簇相关问题。先证特征\(p>3\)的整体函数域上椭圆曲线对于特定\(n\)的情况，后考虑\(\mathbb{C}\)上阿贝尔概型相对版本，通过贝蒂映射得出相关结论。
AI 中文摘要
受西尔弗曼定理启发，考虑如下问题：设\(A\)是数域\(K\)上的阿贝尔簇，给定非挠点\(P\in A(K)\)，对充分大正整数\(n\)，是否存在\(K\)的位\(v\)使得\(P\)模\(v\)的约化阶为\(n\)？本文首先证明对于特征\(p>3\)的整体函数域上的椭圆曲线及与\(p\)互素的充分大正整数\(n\)成立。第二部分考虑其在\(\mathbb{C}\)上的相对版本，设\(\pi:\mathcal{A}\to S\)是\(\mathbb{C}\)上某个簇\(S\)上的阿贝尔概型，\(P\)是\(\pi\)的非挠截面，若与\(P\)相关的贝蒂映射一般是浸没的，则对每个充分大\(n\)，存在\(s\in S(\mathbb{C})\)使得\(P(s)\)在相应纤维中是阶为\(n\)的点。
英文摘要
Motivated by a theorem of Silverman, we consider the following problem. Let $A$ be an abelian variety over a global field $K$. Given a non-torsion point $P \in A(K)$, for a sufficiently large positive integer $n$, whether there exists a place $v$ of $K$ such that the order of the reduction of $P$ modulo $v$ is $n$? In this article, we first show that this holds for an elliptic curve over a global function field of positive characteristic $p>3$ and for sufficiently large positive integers $n$ coprime to $p$. In the second part of the paper, we consider its relative version over $\mathbb C$. More precisely, let $π: \mathcal{A} \rightarrow S$ be an abelian scheme over some variety $S$ over $\mathbb C$, and let $P$ be a non-torsion section of $π$. If the Betti map associated to $P$ is generically submersive, then for every sufficiently large $n$, there is a point $s$ in $S(\mathbb C)$ such that $P(s)$ is a point of order $n$ in the corresponding fiber.
Journal refThe Ramanujan Journal (2026) 70:54
DOI：10.1007/s11139-026-01430-5
