Representation fields for quaternionic skew-hermitian forms

Abstract

Classes of indefinite quadratic forms in a genus are in correspondence with the Galois group of an abelian extension called the spinor class field (Estes and Hsia, Japanese J. Math. 16, 341–350 (1990)). Hsia has proved (Hsia et al., J. Reine Angew. Math. 494, 129–140 (1998)) the existence of a representation field F with the property that a lattice in the genus represents a fixed given lattice if and only if the corresponding element of the Galois group is trivial on F. This far, the corresponding result for skew-hermitian forms was known only in some special cases, e.g., when the ideal (2) is square free over the base field. In this work we prove the existence of representation fields for quaternionic skew-hermitian forms in complete generality.