# Journal of the Ramanujan Mathematical Society

Volume 29, Issue 2, June 2014 pp. 155–199.

Hasse principle for simply connected groups over function fields of surfaces

**Authors**:
Yong HU

**Author institution:**Universite de Caen, Campus 2, Laboratoire de Mathematiques Nicolas Oresme 14032, Caen Cedex France

**Summary: **
Let K be the function field of a p-adic curve, G a
semisimple simply connected group over K and X a G-torsor
over K. A conjecture of Colliot-Thélène, 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) 2An*, 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) Dn*, 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.

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