Galois orbits of torsion points near atoral sets

TL;DR

This paper extends Galois equidistribution results for torsion points on algebraic tori to include certain singular test functions, providing quantitative decay estimates and confirming a conjecture on torsion points for specific divisors.

Contribution

It generalizes Galois equidistribution to singular functions and offers a Diophantine proof, also confirming Ih's integrality conjecture for atoral divisors.

Findings

Extended equidistribution to singular test functions of the form log|P|

Provided power-saving decay estimates for the equidistribution

Confirmed Ih's integrality finiteness conjecture for certain atoral divisors

Abstract

We prove that the Galois equidistribution of torsion points of the algebraic torus $\mathbb{G}_{m}^d$ 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 $\mathbb{G}_m^d$ 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 $\mathbb{G}_m^d$.