Previous issue ·  Next issue ·  Recently posted articles ·  Most recent issue · All issues   
Home Overview Authors Editorial Contact Subscribe

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.

Contents   Full-Text PDF