Lower bound for the canonical on Abelian varieties over totally $p$-adic extensions

TL;DR

This paper proves that for abelian varieties over number fields, the Néron-Tate height of points in certain infinite extensions is bounded below, establishing the Bogomolov property in new cases, especially for good reduction primes.

Contribution

It establishes the Bogomolov property for abelian varieties over specific infinite extensions, including those with good reduction at primes, a novel result in this context.

Findings

Height of points in certain extensions is bounded below by a constant.

The Bogomolov property holds for abelian varieties over these extensions.

First such result in the case of good reduction at a prime.

Abstract

Let $A$ be an abelian variety defined over a number field $\mathbb{Q}$, and let $\hat{h}$ be the N\'eron-Tate height on $A(\overline{\mathbb{Q}})$ corresponding to a symmetric ample line bundle on $A$. In this article, we prove that the N\'eron-Tate height of totally $p$-adic points is bounded below by an absolute constant depending only on $A$ for all but finitely many primes. In other words, if we denote by $\mathbb{Q}^{(p)}$ the maximal algebraic extension of $\mathbb{Q}$ in which $p$ is totally split, then $A(\mathbb{Q}^{(p)})$ satisfies the Bogomolov property for all but finitely many primes. In particular, if $A$ has good reduction at a prime $p$, we obtain the Bogomolov property $A(\Q^{(p)})$. This is the first instance where such a result has been obtained in the good reduction case. In a more general setting, if $A/K$ is an abelian variety and $\mathcal{K}/K$ is an asymptotically positive extension as defined in \cite{AB-SK}, which includes infinite Galois extensions with finite local degree at a non-archimedean place, then $A(\mathcal{K})$ satisfies the Bogomolov property.