Identities for field extensions generalizing the Ohno-Nakagawa relations

TL;DR

This paper generalizes the Ohno-Nakagawa relations for cubic fields to degree l fields with specific Galois groups, providing new identities and alternative proofs without relying on binary cubic forms.

Contribution

It extends the known relations to a broader class of fields with Galois groups D_l and F_l, introducing new identities and proof techniques.

Findings

Established identities for degree l fields with Galois groups D_l and F_l

Provided an alternative proof of the Ohno-Nakagawa relation

Generalized the relations beyond cubic fields

Abstract

In previous work, Ohno conjectured, and Nakagawa proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of `extra functional equations' involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper we generalize their result by proving a similar identity relating certain degree l fields with Galois groups D_l and F_l respectively, for any odd prime l, and in particular we give another proof of the Ohno-Nakagawa relation without appealing to binary cubic forms.