An explicit formula for Larmour's decomposition of hermitian forms

2024/12/15 by Amin Soofiani, Soofiani, Amin
Mathematics · #16K20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2412.11110

openalex publication_date 2024/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a complete discretely valued field whose residue field has characteristic different from 2. Let (D,σ) be a K-division algebra with involution of the first kind, and h be a K-anisotropic ε-hermitian form over (D,σ). By a theorem due to Larmour, there is a decomposition h=h₀ ⊥ h₁ such that the elements in a diagonalization of h₀ are units, the elements in a diagonalization of h₁ are uniformizers, and h₀, h₁ are determined uniquely up to K-isometry. In this paper, we give an explicit description of the elements in the diagonalization of h₀ and h₁ in the case of quaternion algebras. Then we will derive explicit formulas for Larmour's isomorphism of Witt groups.

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