Classifying binary quadratic forms using Clifford invariants

TL;DR

This paper introduces a functorial method to classify binary quadratic forms via Clifford invariants, linking algebraic structures with geometric objects and extending classical composition laws.

Contribution

It provides a new functorial classification framework for quadratic forms using Clifford invariants and generalizes Gauss Composition in a geometric setting.

Findings

Classifies quadratic forms using Clifford invariants.

Generalizes Gauss Composition law.

Connects quadratic forms with Picard groups of quadratic algebras.

Abstract

We functorially identify similarity classes of line-bundle-valued quadratic forms on rank two vector bundles with isomorphism classes of pairs consisting of the degree zero and the degree one parts of the associated generalized Clifford algebras. As applications, we generalize the Gauss Composition and explore connections with Picard groups of quadratic algebras.