Patching and Weak Approximation in Isometry Groups

TL;DR

This paper investigates the classification of quadratic spaces over semilocal principal ideal domains, establishing finiteness results for their genus classes and proving a weak approximation theorem for isometry groups.

Contribution

It introduces a finite power of 2 bound for the number of isomorphism classes in the genus and proves a weak approximation theorem for isometry groups over various base rings.

Findings

Number of isomorphism classes in the genus is a finite power of 2.

Under certain conditions, the genus contains only one class.

Weak approximation holds for isometry groups over semilocal rings.

Abstract

Let $R$ be a semilocal principal ideal domain. Two algebraic objects over $R$ in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all completions of $R$ and its fraction field. We prove that the number of isomorphism classes in the genus of unimodular quadratic spaces over (non necessarily commutative) $R$-orders is always a finite power of $2$, and under further assumptions, e.g. that the order is hereditary, this number is $1$. The same result is also shown for related objects, e.g. systems of sesquilinear forms. A key ingredient in the proof is a weak approximation theorem for groups of isometries, which is valid over any (topological) base field, and even over semilocal base rings. The appendix proves that the isometry group of a quadratic space over an $R$-order with involution can be regarded as a smooth affine group scheme under mild assumptions.