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

设V是三次对称矩阵的向量空间。则群G = SL(3) x GL(2)作用于V，且（G，V）是一个预齐次向量空间。设L是V的晶格，它由所有带积分系数的矩阵对组成。对于任意一对x = (x_1,x_2)∈L，我们用Φ_x(u,v) = det（ux_1 + vs_2）定义了一个二元三次形式Φ_x（u,v）。这是一个二元三次积分形式。设Γ = SL（3，Ζ），将Γ视为g的子群，则Γ∈Γ对x = (x_1,x_2)的作用由Γ x = （Γ x_1^tx_1γ， Γ x_2^tx_1γ）给出。可见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，有多少对x∈L （Φ_x = Φ）的Γ-equivalence类？J. Morales推广了这个问题，并在一定的假设下得到了一些结果。在这个项目中，我们研究了没有他的假设的配对x。证明了对于n次的积分二元形式Φ，与Φ相关的阶在Frohlich意义上是弱自对偶的当且仅当Φ是本原的。应用这一结果，我们研究了L中Γ-equivalence类对的集合与n次代数数域的理想类群的2-扭转子群之间的关系，特别是在n = 2和n = 3的情况下，当Φ不是本元时，我们得到了一些结果。这些结果将发表在《算术学报》上。我还在2001年的Journees Arithmetiques上做了一个关于结果的演讲。

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.