Fields Generated by Finite Rank Subgroups of Tori and Elliptic Curves

Lukas Pottmeyer

Published 2019-10-25Version 1

Let $\Gamma$ be a finite rank subgroup of $G$, where $G$ is either the linear torus or an CM-elliptic curve defined over a number field. We prove that the group of points in $G$ which are rational over the field generated by all elements in the divisible hull of $\Gamma$, is free abelian modulo this divisible hull. This proves that a necessary condition for R\'emond's generalized Lehmer conjecture is satisfied.