An explicit formula for Larmour's decomposition of hermitian forms

TL;DR

This paper provides explicit formulas for Larmour's decomposition of hermitian forms over quaternion algebras, clarifying the structure of these forms and their associated Witt groups.

Contribution

It offers an explicit description of the diagonalization elements in Larmour's decomposition for quaternion algebras, enhancing understanding of hermitian forms over such algebras.

Findings

Explicit formulas for diagonalization elements in quaternion cases

Detailed description of Larmour's isomorphism of Witt groups

Clarification of hermitian form decomposition structure

Abstract

Let $K$ be a complete discretely valued field whose residue field has characteristic different from $2$. Let $(D,\sigma)$ be a $K-$division algebra with involution of the first kind, and $h$ be a $K-$anisotropic $\epsilon$-hermitian form over $(D,\sigma)$. By a theorem due to Larmour, there is a decomposition $h=h_0 \perp h_1$ such that the elements in a diagonalization of $h_0$ are units, the elements in a diagonalization of $h_1$ are uniformizers, and $h_0$, $h_1$ are determined uniquely up to $K-$isometry. In this paper, we give an explicit description of the elements in the diagonalization of $h_0$ and $h_1$ in the case of quaternion algebras. Then we will derive explicit formulas for Larmour's isomorphism of Witt groups.