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.Comment: Version 2, 16 pages, to appear in Compositi

Cohen

Darmon

Frank Thorne

Fröhlich

Greenberg

Henri Cohen

Nakagawa

Poitou

Scholz

Serre

Simon Rubinstein-Salzedo

Tate

English

arXiv

Simon Rubinstein-salzedo

CiteSeerX

IDENTITIES FOR FIELD EXTENSIONS GENERALIZING THE OHNO–NAKAGAWA RELATIONS

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.</jats:p

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Compositio Mathematica

Identities for field extensions generalizing the Ohno–Nakagawa relations

Abstract. In previous work, Ohno [Ohn97] conjectured, and Nakagawa [Nak98] proved, relations between the counting functions of certain cubic fields. These relations may be viewed as comple-ments 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 ` fields with Galois groups D ` and F ` respectively, for any odd prime `, and in particular we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms. 1

In previous work, Ohno [Ohn97] conjectured, and Nakagawa [Nak98] 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 fields with Galois groups D and F respectively, for any odd prime, and in particular we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms.Algorithmic Number Theory in Computer Scienc

COHEN, Henri

RUBINSTEIN-SALZEDO, Simon

THORNE, Frank

Oskar Bordeaux

Identitites for Field Extensions Generalizing the Ohno–Nakagawa Relations

International audienceIn previous work, Ohno [Ohn97] conjectured, and Nakagawa [Nak98] 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 fields with Galois groups D and F respectively, for any odd prime, and in particular we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms

Identities for field extensions generalizing the Ohno-Nakagawa relations

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