Classification theorems for hermitian forms, the Rost kernel and Hasse principle over fields with [formula omitted

Abstract

In this paper we prove classification theorems for hermitian forms over some central simple algebras with involution over a field k with cd2(k)⩽3. We apply these results to show the triviality of the kernel of the Rost invariant for the classical algebraic groups associated to such hermitian forms over k. We also deduce a Hasse principle for algebraic groups defined over function fields of curves over p-adic fields thus proving a conjecture due to Colliot-Thélène–Parimala–Suresh for a large class of groups.