Geometric Gauss-Dedekind

TL;DR

This paper extends Gauss-Dedekind's classical correspondence between quadratic forms and class groups to positive characteristic using étale cohomology, revealing new insights into genera and form composition.

Contribution

It introduces a novel analogue of the Gauss-Dedekind correspondence in positive characteristic via étale cohomology, expanding classical number theory results.

Findings

Describes the set of genera in positive characteristic.

Shows that forms composed with themselves belong to the principal genus.

Establishes an analogue of Gauss' result in a new setting.

Abstract

Gauss and Dedekind have shown a bijection between the set of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes of primitive positive definite binary quadratic $\mathbb{Z}$-forms of the discriminant of $\mathbb{Q}(\sqrt{\Delta<0})$ and the class group of its ring of integers. Using \'etale cohomology we show an analogue of this correspondence in the positive characteristic. This leads to the description of the set of genera and to another result analogous to Gauss' one by which any form composed with itself belongs to the principal genus.