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Summary

We describe the basic properties of algebraic groups, starting with tori and diagonalisable groups, then going on to solvable groups and Borel and parabolic subgroups, then going on to semi-simple and reductive groups, radical and unipotent radical, and finally giving as examples the classical linear, symplectic and orthogonal groups.

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.

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1 - Basic Results on Algebraic Groups

Summary

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

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2 results

1 - Basic Results on Algebraic Groups

Summary

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