K-theory of rational quadratic forms

TL;DR

This paper links the genus of rational quadratic forms to K-theory of associated C*-algebras, leading to a higher composition law and illustrating with Gauss composition.

Contribution

It introduces a novel connection between the genus of quadratic forms and K-theory, providing a new perspective and tools for understanding composition laws.

Findings

Computed the genus of rational quadratic forms via K-theory.

Established a higher composition law for rational quadratic forms.

Applied the theory to classical Gauss composition.

Abstract

We compute the genus of a rational quadratic form in terms of the K-theory of a C*-algebra attached to the adelic orthogonal group of the form. As a corollary, one gets a higher composition law for the rational quadratic forms. As an illustration, we consider the Gauss composition of the binary quadratic forms.