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

Abstract

In (Letter to J.-P. Serre, 12 June 1991) Colliot-Thélène 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)→∏ v∈Ω k H 1(F v,G) has trivial kernel, Ω k denoting the set of places of k. The conjecture is true if G is of type 1A ∗ , i.e., isomorphic to SL 1( A) for a central simple algebra A over F of square free index, as pointed out by Colliot-Thélène, being an immediate consequence of the theorems of Merkurjev–Suslin [S1] 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 2A ∗ , B n , C n , D n ( D 4 nontrialitarian), G 2 or F 4; a group is said to be of type 2A ∗ , if it is isomorphic to SU( B, τ) for a central simple algebra B of square free index over a quadratic extension k′ of k with a unitary k′| k involution τ.