New evidence for R\'emond's generalisation of Lehmer's conjecture

TL;DR

This paper extends Rémond's generalization of Lehmer's conjecture to almost split semiabelian varieties, showing that certain rational point groups become free after specific quotients, with implications for number theory.

Contribution

It generalizes a result of Pottmeyer to a broader class of semiabelian varieties, providing new structural insights related to Rémond's conjecture.

Findings

The group of rational points becomes free after quotienting by the saturated closure of a finite rank subgroup.

The proof combines criteria from Pottmeyer, Pontryagin, and Kummer theory.

It offers new evidence supporting Rémond's generalization of Lehmer's conjecture.

Abstract

In this article, we generalise a result of Pottmeyer from the multiplicative group of the algebraic numbers to almost split semiabelian varieties defined over number fields. This concerns a consequence of R\'emond's generalisation of Lehmer's conjecture. Namely, for a finite rank subgroup $\Gamma$ of an almost split semiabelian variety $G$, we consider the group of rational points of $G$ over a finite extension of the field generated by the saturated closure of $\Gamma$, i.e. the division closure of the subgroup generated by $\Gamma$ and all its images under geometric endomorphisms of $G$. We show that this becomes a free group after one quotients out the saturated closure of $\Gamma$. The proof uses, amongst other ingredients, a criterion of Pottmeyer, which relies on a result of Pontryagin, together with a result from Kummer theory, of which we reproduce a proof by R\'emond.