Counting torsion points on subvarieties of the algebraic torus

TL;DR

This paper investigates the growth of torsion points on algebraic subvarieties of algebraic tori, providing sharp bounds and characterizations, with applications in characteristic zero and finite fields.

Contribution

It establishes sharp upper bounds for torsion point counts and characterizes subvarieties with maximal growth, extending results to finite fields.

Findings

Sharp upper bounds for torsion points growth rate

Characterization of subvarieties with maximal torsion growth

New bounds for algebraic closures of finite fields

Abstract

We estimate the growth rate of the function which counts the number of torsion points of order at most $T$ on an algebraic subvariety of the algebraic torus $\mathbb G_m^n$ over some algebraically closed field. We prove a general upper bound which is sharp, and characterize the subvarieties for which the growth rate is maximal. For all other subvarieties there is a better bound which is power saving compared to the general one. Our result includes asymptotic formulas in characteristic zero where we use Laurent's Theorem, the Manin-Mumford Conjecture. However, we also obtain new upper bounds for $K$ the algebraic closure of a finite field.