Hasse principle for classical groups over function fields of curves over number fields

In (Letter to J.-P. Serre, 12 June 1991) Colliot-Thelene conjectures the following: Let F be a function field in one variable over a number field, with field of constants k and G be a semisimple simply connected linear algebraic group defined over F. Then the map H-1(F, G) --> Pi(nuis an element ofOmegak) H'(F-nu, G) has trivial kernel, Omega(k) denoting the set of places of k. The conjecture is true if G is of type (1)A*, i.e., isomorphic to SL1 (A) for a central simple algebra A over F of square free index, as pointed out by Colliot-Thelene, being an immediate consequence of the theorems of Merkurjev-Suslin [Sl] and Kato [K]. Gille [G] proves the conjecture if G is defined over k and F = k(t), the rational function field in one variable over k. We prove that the conjecture is true for groups G defined over k of the types (2)A*, B, C, D, (D-4 nontrialitarian), G(2) or F-4; a group is said to be of type (2)A*, if it is isomorphic to SU(B, tau) for a central simple algebra B of square free index over a quadratic extension k' of k with a unitary k'/k involution tau. (C) 2003 Elsevier Science (USA). All rights reserved.