Isometries of virtual quadratic spaces

TL;DR

This paper introduces virtual quadratic spaces as a generalization of quadratic spaces, unifying properties of finite fields and orthogonal groups across different characteristics.

Contribution

It presents the concept of virtual quadratic spaces and their isometry groups, providing a unified framework for quadratic forms over finite fields.

Findings

Unified enumeration of quadratic forms across characteristics

Derived properties of isometry groups in the virtual setting

Simplified classification of quadratic forms

Abstract

In this article, we introduce a new object, a virtual quadratic space, and its group of isometries. They are presented as natural generalizations of quadratic spaces and orthogonal groups. It is then shown that by replacing quadratic spaces with virtual quadratic spaces, we can unify certain enumerative properties of finite fields, without distinguishing between even and odd characteristics, such as the number of non-isomorphic non-degenerate quadratic forms, and the order of groups of isometries.