Counting torsion points on subvarieties of the algebraic torus

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.

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Cited by in corpus (1)

• Algebraic relations over finite fields that preserve the endomorphism rings of CM $j$-invariants