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ZERO-LOCI OF BRAUER GROUP ELEMENTS ON SEMI-SIMPLE ALGEBRAIC GROUPS

Published online by Cambridge University Press:  29 November 2018

Daniel Loughran
Affiliation:
University of Manchester, School of Mathematics, Oxford Road, Manchester, M13 9PL, UK (daniel.loughran@manchester.ac.uk)
Ramin Takloo-Bighash
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S Morgan St (M/C 249), Chicago, IL 60202, USA (rtakloo@math.uic.edu)
Sho Tanimoto
Affiliation:
Department of Mathematics, Faculty of Science, Kumamoto University, Kurokami 2-39-1 Kumamoto 860-8555, Japan (stanimoto@kumamoto-u.ac.jp)

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 $\mathbb{Q}$ 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.

Type
Research Article
Copyright
© Cambridge University Press 2018

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