J.-L. Colliot-Thélène, R. Parimala, V. Suresh: Patching and local-global principles for homogeneous spaces over function fields of p-adic curves

jlct@math.u-psud.fr, parimala@mathcs.emory.edu, vssm@oohyd.ernet.in

Submission: 2009, Jun 20, revised: 2009, Dec 26

Let F=K(X) be the function field of a smooth projective curve over a p-adic field K. Given an F-variety Y which is a homogeneous space of a connected reductive group G over F, one may wonder whether the existence of points on Y in each completion of F with respect to a discrete rank one valuation is enough to ensure that Y has a point rational over F. In this paper we prove such a result in two cases :

(i) Y is a smooth projective quadric and p is odd.

(ii) The group G is the extension of a reductive group over the ring of integers of K, and Y is a principal homogeneous space of G.

An essential use is made of recent patching results of Harbater, Hartmann and Krashen. There is a connection to injectivity properties of the Rost invariant and a result of Kato.

2000 Mathematics Subject Classification: 11G99, 14G99, 14G05, 11E72, 11E12, 20G35

Keywords and Phrases: Field patching; arithmetic curves; homogeneous spaces; linear algebraic groups; Galois cohomology; quadratic forms; local-global principles.

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