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BIRS Workshop Lecture Videos

Algebraic groups with good reduction and unramified cohomology Rapinchuk, Igor


Let $G$ be an absolutely almost simple algebraic group over a field $K$, which we assume to be equipped with a natural set $V$ of discrete valuations. In this talk, our focus will be on the $K$-forms of $G$ that have good reduction at all $v$ in $V$ . When $K$ is the fraction field of a Dedekind domain, a similar question was considered by G. Harder; the case where $K=\mathbb{Q}$ and $V$ is the set of all $p$-adic places was analyzed in detail by B.H. Gross and B. Conrad. I will discuss several emerging results in the higher-dimensional situation, where $K$ is the function field $k(C)$ of a smooth geometrically irreducible curve $C$ over a number field $k$, or even an arbitrary finitely generated field. These problems turn out to be closely related to finiteness properties of unramified cohomology, and I will present available results over various classes of fields. I will also highlight some connections with other questions involving the genus of $G$ (i.e., the set of isomorphism classes of $K$-forms of $G$ having the same isomorphism classes of maximal $K$-tori as $G$), Hasse principles, etc. The talk will be based in part on joint work with V. Chernousov and A. Rapinchuk.

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