Hensel's lemma for the norm principle for spinor groups

TL;DR

This paper extends the validity of the norm principle for spinor groups from residue fields to complete discretely valued fields, under certain assumptions about quaternion algebras and skew-hermitian forms.

Contribution

It proves that the norm principle for spinor groups over residue fields implies the same over complete discretely valued fields, assuming it holds over all finite extensions.

Findings

Norm principle holds for spinor groups over residue fields.

Norm principle extends to complete discretely valued fields.

Results depend on assumptions about quaternion algebras and skew-hermitian forms.

Abstract

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 $K$.