Title: Tamagawa Numbers of Linear Algebraic Groups III: Exotic Groups

Abstract: The classification theory of pseudo-reductive groups developed by Conrad, Gabber, and Prasad shows that all pseudo-reductive groups are built in an explicit manner out of three basic building blocks: simply connected groups, commutative groups, and (only in characteristics 2 and 3) exotic groups. Simply connected groups are handled by Weil's conjecture, proved over function fields by Gaitsgory and Lurie, which says that they have Tamagawa number 1. Commutative groups are handled by a generalization of classical Tate duality to positive-dimensional groups, as discussed in the second talk. In this third talk, we discuss how to handle exotic groups. This involves some nice ideas, including the open cell decomposition associated to a cocharacter, deep theorems on the cohomology of simply connected groups, and the existence of large split tori inside simply connected groups over global fields. This talk is not dependent on the previous two, so feel free to come even if you weren't at those.