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

Abstract

Let $\Gamma$ be a finite rank subgroup of ${\overline{\mathbb{Q}}}^{\ast }$. We prove that the multiplicative group of 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 Rémond's generalized Lehmer conjecture is satisfied.

A serious error was found in the proof of the former Lemma 5.1 which could not be fixed. Since the result concerning elliptic curves in the former Theorem 1.2 was dependent on this lemma, this result and Sections 5, 6, and 7, were removed. The new version now only contains the proof of the result on the linear torus

References in corpus (2)

• A Mordell-Weil theorem for abelian varieties over fields generated by torsion points
• Remarks on Rémond's generalized Lehmer problems