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Finiteness theorems for algebraic groups over function fields

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  30 November 2011

Brian Conrad
Brian Conrad*
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA (email: [email protected])
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Abstract

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We prove the finiteness of class numbers and Tate–Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and Godement’s compactness criterion (and a local analogue) for all such groups that are smooth and connected. This builds on the known cases of solvable and semi-simple groups via systematic use of the recently developed structure theory and classification of pseudo-reductive groups.

Keywords

class numbersTamagawa numbersTate–Shafarevich setspseudo-reductive groups

MSC classification

Secondary: 20G30: Linear algebraic groups over global fields and their integers 20G25: Linear algebraic groups over local fields and their integers
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 2 , March 2012 , pp. 555 - 639
Copyright
Copyright © Foundation Compositio Mathematica 2011

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