Remarks on a theorem of Silverman

Abstract

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 $\ddot{I}€:\mathcal{A}\to S$ be an abelian scheme over some variety $S$ over $\mathbb{C}$, and let $P$ be a non-torsion section of $\ddot{I}€$. 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.