On the flat cohomology of binary norm forms

In this talk, we will interpret some classical results of Gauss in the language of flat cohomology and extend them.  Given a quadratic number field k = Q(\sqrt{d}) with narrow class number h_d^+, let O_d be the orthogonal Z-group of the associated norm form q_k.  We will describe the structure of the pointed set H^1_fl(Z, O_d), which classifies quadratic forms isomorphic to q_k in the flat topology, and express its cardinality via h_d^+ and h_{-d}^+.  Furthermore, if N_d is the connected component of O_d, we show that any N_d - torsor tensored with itself belongs to the principal genus.