Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers

Abstract

Abstract.LetXbe a smooth projective geometrically connected curve over a finite field with function fieldK. Let Gbe a connected semisimple group scheme overX. 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 onX. Our results are most useful for studying connected components as much is known about Tamagawa numbers.