New evidence for R\'emond's generalisation 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.