Canad. J. Math. 61(2009), 3-28
Printed: Feb 2009
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.
11E - unknown classification 11E
11R - unknown classification 11R
14D - unknown classification 14D
14H - unknown classification 14H