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Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers

2009; Cambridge University Press; Volume: 61; Issue: 1 Linguagem: Inglês

10.4153/cjm-2009-001-5

1496-4279

Kai Behrend, Ajneet Dhillon,

Finite Group Theory Research

Abstract. Let X be a smooth projective geometrically connected curve over a finite field with function field K . Let Gbe 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 Gtorsors on X . Our results are most useful for studying connected components as much is known about Tamagawa numbers.