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

TL;DR

This paper investigates the distribution of torsion points in a $p$-adic setting for sections of abelian schemes, revealing that torsion points of different orders are well-separated on the base scheme.

Contribution

It provides a new understanding of the $p$-adic distribution of torsion points for sections of abelian schemes, showing their separation based on order.

Findings

Torsion points of different orders are separated on the base scheme.

The torsion locus exhibits a structured distribution in the $p$-adic context.

Abstract

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