On a remarkable identity in class numbers of cubic rings

Abstract

In 1997, Y. Ohno empirically stumbled on an astoundingly simple identity relating the number of cubic rings h(Δ) of a given discriminant Δ, over the integers, to the number of cubic rings hˆ(Δ) of discriminant −27Δ in which every element has trace divisible by 3:(1)hˆ(Δ)={3h(Δ)if Δ>0h(Δ)if Δ<0, where in each case, rings are weighted by the reciprocal of their number of automorphisms. This allows the functional equations governing the analytic continuation of the Shintani zeta functions (the Dirichlet series built from the functions h and hˆ) to be put in self-reflective form. In 1998, J. Nakagawa verified (1). We present a new proof of (1) that uses the main ingredients of Nakagawa's proof (binary cubic forms, recursions, and class field theory), as well as one of Bhargava's celebrated higher composition laws, while aiming to stay true to the stark elegance of the identity.