An orthogonal perspective on Gauss composition

TL;DR

The paper reframes Gauss composition over arbitrary base schemes through orthogonal groups, proving that Clifford and norm functors yield a discriminant-preserving equivalence between oriented binary quadratic modules and pseudoregular modules, unifying Kneser and Wood and clarifying orientations and narrow class groups. It develops a robust framework around the universal Clifford center and a norm functor that acts as a quasi-inverse to the Clifford construction, showing that composition is governed by Picard-group data and orientational structures. The results extend to lattices, giving a correspondence between lattice classes and ideal classes of quadratic orders, and provide a pathway to interpret binary orthogonal modular forms as algebraic modular forms via Hecke characters. This framework harmonizes algebraic and geometric viewpoints, enabling explicit descriptions of genera, class sets, and modular forms in the orthogonal setting with potential applications to Hilbert modular forms and related arithmetic geometry.

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.

Theorems & Definitions (91)

• Theorem 1.3.1
• Corollary 1.3.2
• Corollary 1.3.3
• Theorem 1.4.1
• Corollary 1.4.2
• Theorem 1.5.1
• Corollary 1.5.2
• Corollary 1.6.1
• Lemma 2.1.4
• proof