Abstract. In a recent paper, Colliot-Thélène, Parimala and Suresh conjectured that a local-global principle holds for projective homogeneous spaces under connected linear algebraic groups over function fields of p-adic curves. In this paper, we show that the conjecture is true for any linear algebraic group whose almost simple factors of its semisimple part are isogenous to unitary groups or special unitary groups of hermitian or skew-hermitian spaces over central simple algebras with involutions. The proof implements patching techniques of Harbater, Hartmann and Krashen. As an application, we obtain a Springer-type theorem for isotropy of hermitian spaces over odd degree extensions of function fields of p-adic curves.

Let p p be an odd prime. Let F F be the function field of a p p -adic curve. Let A A be a central simple algebra of period 2 over F F with an involution σ \sigma . There are known upper bounds for the u u -invariant of hermitian forms over ( A , σ ) (A, \sigma ) . In this article we compute the exact values of the u u -invariant of hermitian forms over ( A , σ ) (A, \sigma ) .

Let p be an odd prime. Let F be the function field of a p-adic curve. Let A be a central simple algebra of period 2 over F with an involution σ. There are known upper bounds for the u-invariant of hermitian forms over (A, σ). In this article we compute the exact values of the u-invariant of hermitian forms over (A, σ).

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