Title:  Finiteness for class numbers for groups over global fields

Abstract:  For any smooth affine algebraic group over a global field
one can define a concept of "class number" that generalizes
the classical notion of size of generalized ideal class groups
for number fields (corresponding to the case of the multiplicative  
group).
In characteristic 0 it was proved by Borel that such class numbers are
always finite.  Over global function fields the finiteness problem for
semisimple groups was settled by Borel and Prasad.
We show how to settle the finiteness problem in the affirmative over
global function fields for any group whose unipotent radical is
defined and split over the base field (and so for any reductive group)
by reducing the problem to the known semisimple case.
The main ingredients are class field theory and a close
study of tori and purely inseparable isogenies.