Equidistribution of rational points and the geometric sieve for toric varieties

TL;DR

This paper establishes the equidistribution of rational points of bounded height on smooth projective split toric varieties over rationals, using a refined torsor method and a geometric sieve, with applications to counting points under local conditions.

Contribution

It proves the Manin--Peyre equidistribution principle for toric varieties and develops an effective geometric sieve for counting rational points with local constraints.

Findings

Rational points are equidistributed with respect to Tamagawa measure.

Asymptotic formulas with effective error terms are provided for counting points.

An Ekedahl-type geometric sieve is developed for toric varieties.

Abstract

We prove the Manin--Peyre equidistribution principle for smooth projective split toric varieties over the rational numbers. That is, rational points of bounded anticanonical height outside of the boundary divisors are equidistributed with respect to the Tamagawa measure on the adelic space. Based on a refinement of the universal torsor method due to Salberger, we provide asymptotic formulas with effective error terms for counting rational points in arbitrary adelic neighbourhoods, and develop an Ekedahl-type geometric sieve for such toric varieties. Several applications to counting rational points satisfying infinitely many local conditions are derived.