Title: Tamagawa Numbers of Linear Algebraic Groups II

Abstract: In this second talk, I plan to take a tour through several topics related to the results on Tamagawa numbers presented in the first talk: (1) Tate Duality in Positive Dimension: this is a generalization of classical Tate duality from the case of finite Galois modules to all affine commutative group schemes of finite type over local and global fields, and it plays a crucial role in understanding Tamagawa numbers of commutative groups, which in turn play a central role in understanding Tamagawa numbers of more general groups. (2) Tamagawa numbers of exotic pseudo-reductive groups: these are very special pseudo-reductive groups that only show up in characteristics 2 and 3. Computing their Tamagawa numbers involves some interesting ideas that in turn depend upon very beautiful (but not very well-known) results on the arithmetic of semisimple groups over global fields. (3) Pathological behaviors of unipotent groups: time permitting, we will discuss some strange behaviors exhibited by unipotent groups over function fields, especially their Tamagawa numbers and Tate-Shafarevich sets.