Hasse Principle for Simply Connected Groups Over Function Fields of Surfaces

Abstract

Let K be the function field of a p-adic curve, G a semi-simple simply connected group over K and X a G-torsor over K. A conjecture of Colliot-Thel`ene, Parimala and Suresh predicts that if for every discrete valuation v of K, X has a point over the completion Kv, then X has a K-rational point. The main result of this paper is the proof of this conjecture for groups of some classical types. In particular, we prove the conjecture when G is of one of the following types- (1)^2 A*n, i.e. G = SU(h) is the special unitary group of some hermitian form h over a pair (D, τ), where D is a central division algebra of square-free index over a quadratic extension L of K and τ is an involution of the second kind on D such that L^τ = K; (2) Bn, i.e., G = Spin(q) is the spinor group of quadratic form of odd dimension over K (3) Dna, i.e., G = Spin(h) is the spinor group of a hermitian form h over a quaternion K-algebra D with an orthogonal involution. Our method actually yields a parallel local-global result over the fraction field of a 2-dimensional, henselian, excellent local domain with finite residue field, under suitable assumption on the residue characteristic.