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 \(\pi : \mathcal {A} \rightarrow S\) be an abelian scheme over some variety \(S\) over \(\mathbb {C}\) , and let \(P\) be a non-torsion section of \(\pi \) . 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.
