Strong approximation, equidistribution and geometric sieve

Zoom ID: 919 6497 4060. For password please contact Pieter Moree (moree@mpim...).

The property of strong approximation for an algebraic variety over a number field measures the density of rational points in the adelic space. If it is satisfied, a natural quantitative aspect is to equip the variety with an appropriate height function and study the growth of rational points of bounded height. We expect to read out infinite products of local densities from the asymptotic formulas (if they exist), and this gives sense to equidistribution. Modern conjectural formulations are pioneered by Manin and his collaborators in the setting of projective varieties and respectively by Borovoi—Rudnick for affine varieties around the 90s. In this talk I’ll present results on strong approximation on open subvarieties of affine quadrics (joint with Yang Cao) and projective toric varieties. Ingredients include equidistribution with effective error terms and generalisations of Ekedahl's geometric sieve to these varieties.