Fields Generated by Finite Rank Subgroups of $\overline{\mathbb{Q}}^*$

Abstract

Let Γ be a finite rank subgroup of the linear torus or an elliptic curve defined over a number field with complex multiplication. We prove that the group of points which are rational over the field generated by all elements in the divisible hull of Γ, is free abelian modulo this divisible hull. This proves that a necessary condition for Rémond's generalized Lehmer conjecture is satisfied.