Topological properties of torsors and homogeneous spaces over valued fields

Philippe Gille
University of Lyon, France, and IMAR, Bucharest, Romania

This is a report on joint work with Ofer Gabber and Laurent Moret-Bailly. Let $K$ be the fraction field of a henselian valuation ring $R$ of positive characteristic $p$. Let $Y$ be a $K$-variety, $H$ an algebraic group over K, and $f \colon X \to Y$ an $H$-torsor over $Y$. We consider the induced map $X(K) \to Y(K)$ which is continuous for the topologies coming from the valuation. If $I$ denotes the image of this map, we investigate the following questions:
(a) Is $I$ locally closed (resp., closed) in $Y(K)$?
(b) Is the continuous bijection $X(K)/H(K) \to I$ a homeomorphism?

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