On the relations among the class numbers of binary cubic forms
On the relations among the class numbers of binary cubic forms
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论二元三次形式的类数关系
DOI：
10.1007/s002220050259
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
Jin Nakagawa
中科院分区：
文献类型：
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作者：
Jin Nakagawa
二元三次型类数的研究是由G.爱森斯坦[7]，[8]在世纪中期。F.做出了进一步的贡献。Arndt [1]，C. [9]《易经》云：“人之道也，人之道也。Arndt [2]、马修斯和贝里克[11]对类数进行了显式计算。1951年，H. Davenport在[4]中得到的二元三次型正负判别式类数之和的渐近公式。Davenport和海尔布龙在[5]中利用这个结果得到了正、负判别式的三次域的判别式的稠密性定理。Shintani应用M. 1960年代的Sato（cf. [16]，[18]）。Shintani定义了与二进制三次形式的预齐次向量空间相关的zeta函数。他介绍了四个狄利克雷级数的系数是类号码的整体二进制三次形式。利用理论的预齐次向量空间，他证明了四个狄利克雷级数解析继续亚纯函数在整个复平面和满足一定的功能方程。最近，Y.大野计算了前200个系数的所有四个系列，并提出了在[13]一个新的猜想，其中指出，两个四狄利克雷级数基本上是相同的，其余两个系列的一些基本因素。本文的目的是证明
The study of class numbers of binary cubic forms was initiated by G. Eisenstein [7],[8] in the middle of the 19th century. Further contributions were made by F. Arndt [1], C. Hermite [9] and others. Explicit computation of the class numbers was done by Arndt [2] and Mathews and Berwick [11]. In 1951, H. Davenport obtained in [4] asymptotic formulae for the sums of the class numbers of binary cubic forms of positive and negative discriminants. Using this result, Davenport and Heilbronn obtained in [5] the density theorems of the discriminants of cubic fields of positive and negative discriminants.In 1972, T. Shintani made a remarkable contribution to the study of class numbers of binary cubic forms by applying the theory of prehomogeneous vector spaces which was founded by M. Sato in 1960's (cf.[16],[18]). Shintani defined the zeta functions associated with the prehomogeneous vector space of binary cubic forms. He introduced four Dirichlet series whose coefficients are class numbers of integral binary cubic forms. Using the theory of prehomogeneous vector spaces, he proved that the four Dirichlet series are analytically continued to meromorphic functions on the whole complex plane and satisfy certain functional equations. Recently, Y. Ohno calculated the first two hundred coefficients of all of the four series and presented in [13] a new conjecture which states that the two of the four Dirichlet series are essentially the same as the remaining two series up to some elementary factors. The purpose of this paper is to prove