On the $p$-adic distribution of torsion values for a section of an abelian scheme

Abstract

Let $A\to S$ be an abelian scheme over a $p$-adic field, and let $s:S\to A$ be a section. We study the torsion locus $\bigcup _{n\ge 1}{s}^{-1}(A[n])$ on $S$, and we show that torsion points on $S$ of different orders stay away from each other.