Spinor genera of binary quadratic forms (Q758505)

The authors show how the modern theory of spinor genera, of binary quadratic forms, can be made to fit in with the Gaussian theory of genera. Using the letters \(f,g,h,k\) to denote forms with integer coefficients and fixed non-zero discriminant \(d\), let \(gh\) denote the Gaussian composition of \(g\) and \(h\). Then the authors show that \(f\) and \(g\) are in the same spinor genus if and only if \(f = gk^4\) for some \(k\); the corresponding Gaussian result being that \(f,g\) are in the same genus if and only if \(f = gh^2\) for some \(h\). They construct spinor generic characters, and investigate in detail some cases in which the discriminant \(d\) has a simple arithmetical structure.