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 H¹(F,G)→∏_v∈_ΩkH¹(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 SL₁(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 (D₄ nontrialitarian), G₂ or F₄; 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.