An orthogonal perspective on Gauss composition

TL;DR

This paper offers a new categorical perspective on Gauss composition over general base schemes, connecting algebraic and geometric approaches through Clifford and norm functors, and relates orthogonal eigenforms to Hecke characters.

Contribution

It introduces a discriminant-preserving equivalence of categories using Clifford and norm functors, unifying prior algebraic and geometric constructions in Gauss composition.

Findings

Clifford and norm functors establish an equivalence of categories.

Reconciliation of algebraic and geometric approaches to Gauss composition.

Binary orthogonal eigenforms correspond to Hecke characters.

Abstract

We revisit Gauss composition over a general base scheme, with a focus on orthogonal groups. We show that the Clifford and norm functors provide a discriminant-preserving equivalence of categories between binary quadratic modules and pseudoregular modules over quadratic algebras. This perspective synthesizes the constructions of Kneser and Wood, reconciling algebraic and geometric approaches and clarifying the role of orientations and the natural emergence of narrow class groups. As an application, we restrict to lattices and show that binary orthogonal eigenforms correspond to Hecke characters.