On the \(p\)-adic distribution of torsion values for a section of an abelian scheme (Q2192699)

In the article under review, the authors study the torsion locus, depending on a fixed section \(s \colon S \rightarrow A\), of an abelian scheme \(A \rightarrow S\) over a \(p\)-adic field. To do so, they use Tate's theory of rigid analytic spaces to formulate a concept of accumulation for torsion points. The authors' main result is that torsion points fail to accumulate above points of good reduction. One aspect of the proof is a structural result for abelian schemes over \(p\)-adic fields. Further, the authors show, by way of example, that torsion points may accumulate above points of bad reduction. Finally, motivated by related work of \textit{T. Scanlon} [Int. Math. Res. Not. 1999, No. 17, 909--914 (1999; Zbl 0986.11038)], the authors pose a question about torsion accumulation for subschemes \(X\) of \(A\) and quasicompact rigid subspaces \(Z\) of \(S\). For instance, they ask if there exists a constant \(\epsilon\) such that for each torsion section \(s \colon Z \rightarrow A\) whose image is disjoint from \(X\), the minimum \(p\)-adic distance, with respect to a fixed integral structure on \(A\), from \(s(Z)\) to \(X\) is at least equal to \(\epsilon\).