On the flat cohomology of binary norm forms

Let O be√an order of index m in the maximal order of a quadratic number field k = Q( d). Let O_d,m be the orthogonal Z-group of the associated norm form q_d,m. We describe the structure of the pointed set H_fl¹(Z, O_d,m), which classifies quadratic forms isomorphic (properly or improperly) to q_d,m in the flat topology. Gauss classified quadratic forms of fundamental discriminant and showed that the composition of any binary Z-form of discriminant ∆_k with itself belongs to the principal genus. Using cohomological language, we extend these results to forms of certain non-fundamental discriminants.