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Identities for field extensions generalizing the Ohno–Nakagawa relations

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  30 June 2015

Henri Cohen ,
Simon Rubinstein-Salzedo  and
Frank Thorne
Henri Cohen
Affiliation:
Université de Bordeaux, Institut de Mathématiques, UMR 5251 du CNRS, 351 Cours de la Libération, 33405 Talence Cedex, France email [email protected]
Simon Rubinstein-Salzedo
Affiliation:
Department of Statistics, Stanford University, 390 Serra Mall, Stanford, CA 94305, USA email [email protected]
Frank Thorne
Affiliation:
Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, SC 29208, USA email [email protected]
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Abstract

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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-$\ell$ fields to Galois groups $D_{\ell }$ and $F_{\ell }$, respectively, for any odd prime $\ell$; in particular, we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms.

Keywords

Ohno–Nakagawa relationsScholz reflectionfield extensions

MSC classification

Primary: 11R21: Other number fields
Secondary: 11R29: Class numbers, class groups, discriminants
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 11 , November 2015 , pp. 2059 - 2075
Copyright
© The Authors 2015 

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