Investigation of prehomogeneous vector spaces and ideal class groups of algebraic number fields

预齐次向量空间和代数数域理想类群的研究

基本信息

批准号：
12640018
负责人：
NAKAGAWA Jin
金额：
$ 1.86万
依托单位：
Joetsu University of Education
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640018/
关键词：
prehomogeneous vector space ideal class group number field 概均的ベクトル空間ゼータ関数

项目摘要

Let V be the vector space of symmetric matrices of degree three. Then the group G = SL(3) x GL(2) acts on V and (G,V) is a prehomogeneous vector space. Let L be the lattice of V consisting of all pairs of matrices with integral coefficients. For any pair x = (x_1,x_2) ∈ L, we define a binary cubic form Φ_x(u,v) by Φ_x(u,v) = det(ux_1 + vs_2). This is an integral binary cubic form. Put Γ = SL(3,Ζ) and consider Γ as a subgroup of G. Then the action of γ∈Γ on x = (x_1,x_2) is given by γx = (γx_1^tx_1γ, γx_2^tx_1γ). It is obvious that Φ_<γx> = Φ_x. So we can consider the following problem: For a given binary form Φ, how many Γ-equivalence classes of pairs x ∈ L with Φ_x = Φ are there? J. Morales generalized this problem and obtained some results under certain assumptions. In this project, we have studied pairs x without his assumptions. We proved that for an integral binary form Φ of degree n, the order associated with Φ is weakly self dual in the meaning of Frohlich if and only if Φ is primitive. Applying this result, we studied the relations between the set of Γ-equivalence classes of pairs in L and the 2-torsion subgroups of ideal class groups of algebraic number fields of degree n. In particular, we obtained some results in the case of n = 2 and n = 3 when Φ is not primitive. These results are to be published in Acta Arithemetica. I also gave a talk on the results at Journees Arithmetiques 2001.

令V为三分之二的对称物质的矢量空间。然后，G = SL（3）X GL（2）对V和（G，V）起作用是一个前均匀的矢量空间。令l为V的晶格，由所有成对的具有积分系数的物质组成。对于任何一对x =（x_1，x_2）∈L，我们通过φ_x（u，v）= det（ux_1 + vs_2）定义了二进制立方体形式φ_x（u，v）。这是一种整体的二进制立方体形式。 pUTγ= sl（3，ζ），并将γ视为G的亚组。然后γ∈γ对x =（x_1，x_2）的作用由γx=（γx_1^tx_1γ，γx_2^tx_1γ）给出。显然，φ_<γx>=φ_x。因此，我们可以考虑以下问题：对于给定的二进制形式φ，有多少个与φ_x=φ的对x∈L的γ等效类别有？ J. Morales概括了这个问题，并在某些假设下获得了一些结果。在这个项目中，我们没有他的假设X对X。我们证明，对于n度n的积分二进制形式φ，与φ相关的顺序在弗洛希里奇的含义上是微弱的，并且仅当φ是原始的。在应用此结果时，我们为L中的γ级等级类别与n的理想级别组的2个扭转亚组讲述了n的代数数n。特别是，当φ不是原始的时，我们在n = 2和n = 3的情况下获得了一些结果。这些结果将发表在Acta arithemetica中。我还在《 Journes Arithmetiques 2001》中就结果进行了演讲。