Bogomolov Property of some infinite nonabelian extensions of a totally $v$-adic field

TL;DR

This paper proves a lower bound for the heights of elements in certain infinite nonabelian extensions of a totally v-adic field, with explicit bounds in specific cases, extending the Bogomolov property.

Contribution

It establishes the Bogomolov property for infinite nonabelian extensions generated by torsion points of elliptic curves over totally v-adic fields, with explicit bounds in the toric case.

Findings

Height of elements is either zero or bounded below by a positive constant.

Explicit lower bounds are provided in the case of the rational field.

The results extend the Bogomolov property to new infinite nonabelian extensions.

Abstract

Let $E$ be an elliptic curve defined over a number field $K$ and let $v$ be a finite place of $K$. Write $K^{tv}$ the maximal extension of $K$ in which $v$ is totally split and $L$ the field generated over $K^{tv}$ by all torsion points of $E$. Under some conditions, we will show that the absolute logarithmic Weil height (resp. N\'eron-Tate height) of any element of $L^*$ (resp. $E(L)$) is either $0$ or bounded from below by a positive constant depending only on $E, K$ and $v$. This lower bound will be explicit in the toric case when $K=\mathbb{Q}$.