Amin Soofiani | Searcharxiv

arXiv2024

Hensel's lemma for the norm principle for spinor groups

Let $K$ be a complete discretely valued field with residue field $k$ with $char \ k \ne 2$. Assuming that the norm principle holds for spinor groups $Spin(\mathfrak{h})$ for every regular skew-hermitian form $\mathfrak{h}$ over every quaternion algebra $\mathfrak{D}$ (with respect to the canonical involution on $\mathfrak{D}$) defined over any finite extension of $k$, we show that the norm principle holds for spinor groups $Spin(h)$ for every regular skew-hermitian form $h$ over every quaternion algebra $D$ (with respect to the canonical involution on $D$) defined over $K$.

math.GR↗

arXiv2024

An explicit formula for Larmour's decomposition of hermitian forms

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.

math.RA↗