Harrison's criterion, Witt equivalence and reciprocity equivalence

Abstract

Harrison's criterion characterizes the isomorphy of the Witt rings of two fields in terms of properties of these fields. In this article, we discuss about the existence of such characterizations for the isomorphism of Witt groups of hermitian forms over certain algebras with involution. In the cases where we consider the Witt group of a quadratic extension with its non-trivial automorphism or the Witt group of a quaternion division algebra with its canonical involution, such criteria are proved. In the framework of global fields, these criteria are reformulated in terms of properties involving certain real places of the considered fields.