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Identities for field extensions generalizing the Ohno–Nakagawa relations
- Part of
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- Journal:
- Compositio Mathematica / Volume 151 / Issue 11 / November 2015
- Published online by Cambridge University Press:
- 30 June 2015, pp. 2059-2075
- Print publication:
- November 2015
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- Article
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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.
