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# Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers

Published:2009-02-01
Printed: Feb 2009
• Kai Behrend
• Ajneet Dhillon
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## Abstract

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.
 MSC Classifications: 11E - unknown classification 11E11R - unknown classification 11R14D - unknown classification 14D14H - unknown classification 14H