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Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers

Published online by Cambridge University Press:  20 November 2018

Kai Behrend  and
Ajneet Dhillon
Kai Behrend
Affiliation:
Department of Mathematics, University of Britich Columbia, Vancouver, BC, V6T 1Z2, behrend@math.ubc.ca
Ajneet Dhillon
Affiliation:
Department of Mathematics, Middlesex College, University of Western Ontario, London, ON, N6A 5B7, adhill3@uwo.ca
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Abstract

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Let $X$ be a smooth projective geometrically connected curve over a finite field with function field $K$. Let $g$ be a connected semisimple group scheme over $X$. Under certain hypotheses we prove the equality of two numbers associated with $g$. The first is an arithmetic invariant, its Tamagawa number. The second is a geometric invariant, the number of connected components of the moduli stack of $g$-torsors on $X$. Our results are most useful for studying connected components as much is known about Tamagawa numbers.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 1 , 01 February 2009 , pp. 3 - 28
Copyright
Copyright © Canadian Mathematical Society 2009

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