On the relations among the class numbers of binary cubic forms
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 a consequence a theorem on the 3-rank of the ideal class group of orders of quadratic fields is obtained. This gives certain relations on the number of cubic fields of positive and negative discriminants.
scientific article; zbMATH DE number 1230016
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