Journal of the Ramanujan Mathematical Society

Volume 41, Issue 1, March 2026 pp. 61–73.

An explicit formula for Larmour's decomposition of Hermitian forms

Authors: Amin Soofiani
Author institution:Department of Mathematics, University of British Columbia, Vancouver, BC Canada, V6T 1Z4.

Summary: 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 = h{0} ⊥ 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 derive explicit formulas for Larmour's isomorphism of Witt groups.

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