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Algebraic groups with good reduction

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Andrei Rapinchuk
University of Virginia, Charlottesville
Mit, 14/03/2018 - 11:00 - 12:00
MPIM Lecture Hall

Let $G$ be an absolutely almost simple algebraic group over a field $K$. Assume that $K$ is equipped with a "natural" set $V$ of discrete valuations. We are interested in the $K$-forms of $G$ that have good reduction at all $v$ in $V$. In the case $K$ 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 the 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. I will also indicate 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 $G$), the Hasse principle, weakly commensurable Zariski-dense subgroups, etc. (Joint work with V. Chernousov and I. Rapinchuk.)

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