$p$-adic equidistribution and an application to $S$-units

TL;DR

This paper establishes a $p$-adic Galois equidistribution theorem for torsion points in algebraic tori, providing quantitative decay estimates and applying the results to prove Ih's Conjecture for certain divisors.

Contribution

It introduces a $p$-adic equidistribution result with explicit decay rates and applies it to verify Ih's Conjecture for specific divisors of algebraic tori.

Findings

Proved a $p$-adic Galois equidistribution theorem for torsion points.

Provided quantitative estimates on the decay rate of equidistribution.

Confirmed Ih's Conjecture for a class of divisors in $\

Abstract

We prove a Galois equidistribution result for torsion points in $\mathbb G_m^n$ in the $p$-adic setting for test functions of the form $\log |F|_p$ where $F$ is a nonzero polynomial with coefficients in the $p$-adic numbers. Our result includes a power saving quantitative estimate of the decay rate rate of the equidistribution. As an application we show that Ih's Conjecture is true for a class of divisors of $\mathbb G_m^n$.