Patching and local-global principles for homogeneous spaces over function fields of \(p\)-adic curves
Summary: 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 [\textit{D. Harbater} et al., Invent. Math. 178, No. 2, 231--263 (2009; Zbl 1259.12003)]. There is a connection to injectivity properties of the Rost invariant and a result of \textit{K. Kato} [J. Reine Angew. Math. 366, 142--183 (1986; Zbl 0576.12012)].
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