Geometric Gauss-Dedekind

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.