On the $p$-adic closure of a subgroup of rational points on an Abelian variety

TL;DR

This paper investigates the $p$-adic closure of rational point subgroups on Abelian varieties, establishing a lower bound on the $p$-adic rank relative to the expected value, extending earlier work on algebraic groups.

Contribution

It provides a new lower bound for the $p$-adic rank of rational points on simple Abelian varieties over rationals, advancing understanding of their $p$-adic closure properties.

Findings

The $p$-adic rank is at least one-third of the expected rank for simple Abelian varieties.

The work extends previous studies on $p$-adic closures of algebraic groups.

Addresses a question posed by Bella"iche regarding Abelian varieties.

Abstract

In 2007, B. Poonen (unpublished) studied the $p$--adic closure of a subgroup of rational points on a commutative algebraic group. More recently, J. Bella\"iche asked the same question for the special case of Abelian varieties. These problems are $p$--adic analogues of a question raised earlier by B. Mazur on the density of rational points for the real topology. For a simple Abelian variety over the field of rational numbers, we show that the actual $p$--adic rank is at least the third of the expected value.