Let O be an order of index m in the maximal order of a
quadratic number field k=Q(d). Let
Od,m be the orthogonal Z-group of the
associated norm form qd,m. We describe the structure of the pointed set
Hfl1(Z,Od,m), which classifies
quadratic forms isomorphic (properly or improperly) to qd,m in the flat
topology. Gauss classified quadratic forms of fundamental discriminant and
showed that the composition of any binary Z-form of discriminant
Δk with itself belongs to the principal genus. Using cohomological
language, we extend these results to forms of certain non-fundamental
discriminants.Comment: 24 pages, submitted. Comments are welcom