Galois orbits of torsion points near atoral sets

Dimitrov, Vesselin; Habegger, Philipp

doi:10.2140/ant.2024.18.1945

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Galois orbits of torsion points near atoral sets

Vesselin Dimitrov and Philipp Habegger

Vol. 18 (2024), No. 11, 1945–2001

DOI: 10.2140/ant.2024.18.1945

Abstract

We prove that the Galois equidistribution of torsion points of the algebraic torus $𝔾_{m}^{d}$ extends to the singular test functions of the form $\log | P |$ , where $P$ is a Laurent polynomial having algebraic coefficients that vanishes on the unit real $d$ -torus in a set whose Zariski closure in $𝔾_{m}^{d}$ has codimension at least $2$ . Our result includes a power-saving quantitative estimate of the decay rate of the equidistribution. It refines an ergodic theorem of Lind, Schmidt, and Verbitskiy, of which it also supplies a purely Diophantine proof. As an application, we confirm Ih’s integrality finiteness conjecture on torsion points for a class of atoral divisors of $𝔾_{m}^{d}$ .

Keywords

number theory, Diophantine approximation, Mahler measure

Mathematical Subject Classification 2010

Primary: 11J83

Secondary: 11R06, 14G40, 37A45, 37P30

Milestones

Received: 1 October 2019

Revised: 25 May 2023

Accepted: 27 November 2023

Published: 18 October 2024

Authors

Vesselin Dimitrov
School of Mathematics Georgia Institute of Technology Atlanta, GA United States

Philipp Habegger
Departement Mathematik und Informatik Universität Basel Basel Switzerland

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Vol. 18, No. 11, 2024