Brian Conrad (Michigan and Stanford)

Title: Finiteness of class numbers over global function fields

Abstract: Generalized ideal class groups of number fields can be described adelically in terms of a coset space for the group GL1, and this in turns leads to a notion of "class number" (as the size of a certain set, if finite!) for an arbitrary smooth affine group variety over a global field. Over number fields the finiteness of these numbers was proved by Borel via reduction theory. The situation for the analogous finiteness problem over global function fields was not as well-understood in general.
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.