A remarkable identity in class numbers of cubic rings

TL;DR

This paper proves a surprising identity relating counts of cubic rings with different discriminants, using class field theory, binary cubic forms, and Bhargava's composition laws, confirming Ohno's conjecture.

Contribution

The paper provides a new proof of Ohno's conjecture on the identity between counts of cubic rings, employing classical and modern algebraic tools.

Findings

Confirmed Ohno's conjecture mathematically.

Connected the identity to functional equations of Shintani zeta functions.

Enhanced understanding of class numbers of cubic rings.

Abstract

In 1997, Y. Ohno empirically stumbled on an astoundingly simple identity relating the number of cubic rings $h(D)$ of a given discriminant $D$, over the integers, to the number of cubic rings $h'(D)$ of discriminant $-27D$ in which every element has trace divisible by 3: $h'(D) = h(D)$, if $D < 0$, and $h'(D) = 3h(D)$ if $D > 0$, 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 values of $h$ and $h'$) to be put in self-reflective form. In 1998, J. Nakagawa verified Ohno's conjecture. We present a new proof of the identity 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.