A Dual Description of Integral Binary Cubic Forms and the Ohno-Nakagawa Identities

Abstract

Given a nonzero integer D, we analyze the action of SL2(Z) on the space of integral binary cubic forms of discriminant D, and on the space of integer-matrix binary cubic forms of reduced discriminant D. Ohno conjectured that the ratio of SL2(Z)-equivalence classes is 1-to-1, when D is negative, and 1-to-3, when D is positive. The current proof, due to Nakagawa, offers an analytical perspective by studying these class numbers as coefficients in four Dirichlet series; using a result of Datskovsky and Wright, Nakagawa relates these series to certain products of Dedekind zeta functions of cubic fields and the Riemann zeta function. Our goal is to present a proof that does not characterize these class numbers in terms of the Dirichlet series, but only in terms of quadratic and cubic orders. In this respect, we construct a correspondence between the SL2(Z)-classes of integral binary cubic forms and isomorphism classes of oriented cubic rings of discriminant D, and a correspondences between SL2(Z)-classes of integer-matrix binary cubic forms and 3-torsion ideal classes in the unique quadratic order of discriminant D.