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Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers

Open Access article
 Printed: Feb 2009
  • Kai Behrend
  • Ajneet Dhillon
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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.
MSC Classifications: 11E, 11R, 14D, 14H show english descriptions unknown classification 11E
unknown classification 11R
unknown classification 14D
unknown classification 14H
11E - unknown classification 11E
11R - unknown classification 11R
14D - unknown classification 14D
14H - unknown classification 14H

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