Galois Cohomology of Classical Groups

Abstract

In this article, we survey recent results of Eva Bayer-Fluckiger and the author on the Galois cohomology of classical groups over fields of virtual cohomological dimension 2. Number fields are examples of such fields. We begin by describing a well-known classification theorem for quadratic forms over number fields in terms of the so-called classical invariants (§ 2). We explain in § 3 how this classification leads to Hasse principle for principal homogeneous spaces for Spin q over number fields. In § 4 and § 7, we state the conjecture of Serre concerning the triviality of principal homogeneous spaces under semi-simple, simply connected, linear algebraic groups over perfect fields of cohomological dimension 2 and its real analogue due to Colliot-Thelene and Scheiderer in the form of a Hasse principle, if the field has virtual cohomological dimension ≤ 2. As in the case of Spin q over number fields, a main step in the proof of these conjectures is a classification theorem of hermitian forms over involutorial division algebras defined over fields of virtual cohomological dimension ≤ 2, which is described in § 6 and § 7.