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
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
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
prove the finiteness of class numbers for arbitrary affine group
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
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.