Title:

Abstract: Generalized ideal class groups of number fields can be described adelically in terms of a coset space for the group

I will explain how to use a mixture of arithmetic and geometric techniques to prove the finiteness of class numbers for arbitrary affine group varieties over global function fields in odd characteristic. As one application, we get a proof of finiteness of Tate-Shafarevich "groups" (really pointed sets) for all affine algebraic group schemes over such function fields (going far beyond the reductive case). The crucial new ingredient is a structure theorem for a general class of group varieties over imperfect fields. This structure theorem has been recently established in joint work with G. Prasad and O. Gabber, so I will say something about that part of the story if time permits.