Zero-loci of Brauer group elements on semi-simple algebraic groups

TL;DR

This paper develops asymptotic formulas for counting rational points of bounded height on zero-loci of Brauer group elements in semi-simple algebraic groups, using automorphic forms, with applications to matrices over Q.

Contribution

It introduces new asymptotic counting formulas for rational points on these loci, advancing understanding of Serre's questions in this area.

Findings

Asymptotic formulas for rational points on zero-loci of Brauer elements

Application to counting matrices with determinant as sum of two squares

Positive progress on Serre's counting problem

Abstract

We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over Q whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems.