Reflection theorems for number rings generalizing the Ohno-Nakagawa identity

Abstract

The Ohno-Nakagawa (O-N) reflection theorem is an unexpectedly simple identity relating the number of $\mathrm{GL}_2 \mathbb{Z}$-classes of binary cubic forms (equivalently, cubic rings) of two different discriminants $D$, $-27D$; it generalizes cubic reciprocity and the Scholz reflection theorem. In this paper, we present a new approach to this theorem using Fourier analysis on the adelic cohomology $H^1(\mathbb{A}_K, M)$ of a finite Galois module, modeled after the celebrated Fourier analysis on $\mathbb{A}_K$ used in Tate's thesis. This method reduces reflection theorems of O-N type to local identities. We establish reflection theorems of O-N type for cubic forms and rings over arbitrary number fields, and also for quadratic forms counting by a peculiar invariant $a(b^2 - 4ac)$. We also find relations for the number of forms over $\mathbb{Z}[1/N]$ and for forms of highly non-squarefree discriminant (discriminant reduction). In a sequel to this paper, we will deal with reflection theorems for quartic rings, $2\times 3\times 3$ symmetric boxes, and binary quartic forms. In these cases the local step is much more involved.