Date : April 11, 15, 18, 22, 25     Time：18:40-20:30     Venue: Room 77201, Jingchunyuan78

Date：April 17, 24     Time : 13:00-14:50     Venue: Lecture Hall, Jiayibing Building, Jingchunyuan 82

Let $G$ be an affine algebraic $k$-group defined over a local field of characteristic zero. Borel and Serre have shown in 1964 that there are finitely many isomorphism classes of $G$-torsors. Also if $f: X\rightarrow Y$ is a $G$-torsor, then the image of the map $X(k)\rightarrow Y(k)$ is locally closed. The starting point of the lecture is the investigation of the same issue for local fields of positive characteristic. It turns out that the two preceding results are false. The main result will be that the image of the map $X(k)\rightarrow Y(k)$ is locally closed. It has consequences of the topology of the set of isomorphism classes of $G$-torsors. Our setting is actually wider, it involves Henselian valued fields and algebraic spaces.

The goal of the lecture is to cover the proof of the above statement. On the way, we shall revisit actions of algebraic groups on homogeneous spaces, Galois cohomology, topological properties for algebraic varieties defined over a local field, and Gabber’s compactifications of algebraic groups.