Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers

LetX 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 hypothesis 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.