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Yisheng Tian (Harbin Institute of Technology, China)
2024-06-25  10:00-12:00
Room 202, Astronomy and Mathematics Building

Let k be a discrete valuation field with a perfect residue field of dimension at most 1. This series of talks aims to prove the following two theorems via the Bruhat--Tits theory.

(1) Let G be an absolutely almost-simple k-anisotropic group over k. Then it is of type A.

(2) Let G be a semi-simple simply connected group over k. Then any G-torsor over k is trivial.

The usual demonstrations of these theorems use deep Lie-theoretic case-by-case arguments (see, for example, "Algebraic Groups and Number Theory" by Platonov and Rapinchuk, Chapter 6). This approach, utilizing the Bruhat--Tits theory, will provide a much simpler and uniform proof.

More precisely, let K be the maximal unramified extension contained in a fixed algebraic closure of k. In the first talk, we shall study the basic Bruhat--Tits building theory for connected reductive groups over K. The second talk will focus on the subgroups of G(k) via the Bruhat--Tits building theory. Finally, after providing some preliminaries, we will provide complete proofs of the promised two theorems.