Counting torsion points on subvarieties of the algebraic torus

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 $\mathbb G_{\rm m}^n$ 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 $K$ equal to the algebraic closure of a finite field.