Title: Local-global principles for homogeneous spaces of linear<br>
algebraic groups over function fields of p-adic curves<br>
(joint work with Parimala and Suresh)


Abstract: (From the corresponding preprint)

``Let F = K(X) be the function field of a smooth projective curve over a p-adic
field K. To each rank one discrete valuation of F one may associate the
completion F_v . 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
F_v-points on Y for each v is enough to ensure that Y has an F-point. 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.''