On the Ohno-Nakagawa Theorem

TL;DR

This paper provides a new proof of the Ohno-Nakagawa Theorem by connecting $L$-series techniques with class field theory, simplifying the theorem to an identity involving quadratic order $L$-series.

Contribution

It introduces a novel proof of the Ohno-Nakagawa Theorem using $L$-series and class field theory, offering a different perspective from previous proofs.

Findings

Established a general relation among $L$-series of quadratic orders.

Reduced the theorem to an identity involving $L$-series and truncated $L$-series.

Provided a new proof technique for the Ohno-Nakagawa Theorem.

Abstract

In this paper we give a new proof of the Ohno-Nakagawa Theorem using the techniques of $L$-series. By applying Eisenstein's parametrization of binary cubic forms on the one hand, and a class field theory interpretation of Datskovsky \& Wright's Theorem on the other, we reduce the Ohno-Nakagawa Theorem to an identity involving the $L$-series and the truncated $L$-series of quadratic orders. We prove this identity by establishing a general relation among these two types $L$-series.