On the relations among the class numbers of binary cubic forms
type: Item
instance of: scholarly article
title: On the relations among the class numbers of binary cubic forms
zbMATH Open document ID: 1016.11014
author: Jin Nakagawa
published in: Inventiones Mathematicae
publication date: 2003-05-19T00:00:00Z
review text: In the middle of the 19th century, G. Eisenstein initiated the study of class numbers of binary cubic forms. F. Arndt, C. Hermite, Mathews and Berwick made further contributions. In 1951, H. Davenport obtained asymptotic formuale for the sums of the class numbers of binary cubic forms of positive and negative discriminant. Using this result Davenport and Heilbronn obtained the density theorem of the discriminants of cubic fields of positive and negative discriminants. Using the theory of prehomogeneous vector spaces, \textit{T. Shintani} [J. Math. Soc. Japan 24, 132-188 (1972; Zbl 0223.10032)] introduced four Dirichlet series whose coefficients are class numbers of integral binary cubic forms and proved that the four series are analytically continued to meromorphic functions on the whole complex plane and satisfy certain functional equations. Recently, [Am. J. Math. 119, 1083-1094 (1997; Zbl 0893.11035)] \textit{Y. Ohno} calculated the first two hundred coefficients of all the four series and conjectured that two of the four Dirichlet series are essentially the same as the remining two series up to some elementary factors. To prove this conjecture, in this paper the author has obtained an expression of the class number of integral binary cubic forms of positive discriminants in terms of ideal class groups of orders of imaginary quadratic fields. On the other hand, \textit{B. Datskovsky} and \textit{D. J. Wright} [J. Reine Angew. Math. 386, 116-138 (1988; Zbl 0632.12007) see also Zbl 0619.12005] have obtained an expression of the series as infinite sums of known functions which are certain products of Dedekind zeta-functions of cubic fields and the Riemann zeta-function. This gives an expression of the class numbers of integral binary cubic forms of negative discriminants as a finite sum of the coefficients of the known functions over a certain set of cubic fields of negative discriminants. Using class field theory it is deduced that the two expressions coincide. As …
reviewed by: Ranjeet Sehmi
Mathematics Subject Classification ID: 11E41
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 1230016
zbMATH Keywords: Dirichlet series
zbMATH Keywords: Dedekind zeta-functions
zbMATH Keywords: class numbers
zbMATH Keywords: integral binary cubic forms
zbMATH Keywords: class field theory
MaRDI profile type: Publication
full work available at URL: s002220050259
OpenAlex ID: W2044956029
DOI: 10.1007/S002220050259
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description: scientific article; zbMATH DE number 1230016