Gauss composition over an arbitrary base

TL;DR

This paper generalizes the classical correspondence between binary quadratic forms and ideal classes to arbitrary base rings and schemes, providing a comprehensive and base-change compatible framework.

Contribution

It establishes a complete, base-change compatible relationship between binary quadratic forms and modules over quadratic algebras for any base ring or scheme, extending classical results.

Findings

Includes all binary quadratic forms in the correspondence

Works over arbitrary base rings and schemes

Provides explicit geometric and local descriptions

Abstract

The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have been of tremendous importance in mathematics, and many authors have given extensions of these theorems to rings other than the integers. However, such extensions have always included hypotheses on the rings, and the theorems involve only binary quadratic forms satisfying further hypotheses. We give a complete statement of the relationship between binary quadratic forms and modules for quadratic algebras over any base ring, or in fact base scheme. The result includes all binary quadratic forms, and commutes with base change. We give global geometric as well as local explicit descriptions of the relationship between forms and modules.