Action of the group of classes on representations of numbers by a ternary quadratic form: The formulas of Siegel and Kneser-Hsia-Peters

Abstract

We specify in explicit form the action of the group of classes of binary quadratic forms of determinant dm on the set of primitive representations of the number m by a ternary quadratic form. As a consequence, we obtain an elementary proof of Siegel's formulas for the weighted number of representations by a genus and formulas for the weighted number of representations by a spinor genus of ternary forms.