On Bertrand's and Rodriguez Villegas' higher-dimensional Lehmer conjecture

Ted Chinburg, Eduardo Friedman and James Sundstrom

Vol. 321 (2022), No. 1, 119–165

DOI: 10.2140/pjm.2022.321.119

Abstract

Let $L$ be a number field and let $E\subset {\mathcal{𝒪}}_L^{\ast }$ be any subgroup of the units of $L$. If ${\mathrm{rank}<!--FUNCTION\; APPLICATION-->}_{\mathbb{Z}}(E)=1$, Lehmer’s conjecture predicts that the height of any nontorsion element of $E$ is bounded below by an absolute positive constant. If ${\mathrm{rank}<!--FUNCTION\; APPLICATION-->}_{\mathbb{Z}}(E)={\mathrm{rank}<!--FUNCTION\; APPLICATION-->}_{\mathbb{Z}}({\mathcal{𝒪}}_L^{\ast }$), Zimmert proved a lower bound on the regulator of $E$ which grows exponentially with $[L:\mathbb{Q}]$. By sharpening a 1997 conjecture of Daniel Bertrand’s, Fernando Rodriguez Villegas “interpolated” between these two extremes of rank with a new higher-dimensional version of Lehmer’s conjecture. Here we prove a high-rank case of the Bertrand–Rodriguez Villegas conjecture. Namely, it holds if $L$ contains a subfield $K$ for which $[L:K]\gg [K:\mathbb{Q}]$ and $E$ contains the kernel of the norm map from ${\mathcal{𝒪}}_L^{\ast }$ to ${\mathcal{𝒪}}_K^{\ast }$.

Keywords

Lehmer's conjecture, Mahler measure, units

Mathematical Subject Classification

Primary: 11R06

Secondary: 11R27

Milestones

Received: 11 September 2020

Revised: 4 August 2022

Accepted: 25 October 2022

Published: 3 March 2023