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

Part of: (Co)homology theory Diophantine equations Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  29 November 2018

Daniel Loughran Open the ORCID record for Daniel Loughran [Opens in a new window] ,
Ramin Takloo-Bighash  and
Sho Tanimoto Open the ORCID record for Sho Tanimoto [Opens in a new window]
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)
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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.

Keywords

heightscounting rational pointssemi-simple groupsBrauer groups

MSC classification

Primary: 14G05: Rational points
Secondary: 11D45: Counting solutions of Diophantine equations 14F22: Brauer groups of schemes
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 5 , September 2020 , pp. 1467 - 1507
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Copyright
© Cambridge University Press 2018

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