Reflection theorems for number rings generalizing the Ohno-Nakagawa identity

TL;DR

This paper introduces a new Fourier analysis approach on adelic cohomology to prove reflection theorems for number rings, generalizing the Ohno-Nakagawa identity to arbitrary number fields and other forms.

Contribution

It develops a novel method using adelic cohomology Fourier analysis to establish reflection theorems for cubic and quadratic forms over number fields, extending previous identities.

Findings

Established reflection theorems for cubic forms over arbitrary number fields.

Extended reflection identities to quadratic forms with specific invariants.

Connected local identities to global reflection theorems using adelic Fourier analysis.

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.