A study on prehomogeneous vector spaces and extensions of algebraic number fields

预齐次向量空间与代数数域的延拓研究

• 批准号: 16540015
• 负责人: NAKAGAWA Jin
• 金额: $ 1.34万
• 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: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540015/
• 关键词: prehomogeneous vector space; algebraic number field; order; quartic field

Let V be the space of pairs of ternary quadratic forms, The group G=GL(3)XGL(2) acts on V and (G, V) is a prehomogeneous vector space of dimension 12. This space was closely related to quartic field extensions by D.J.Wright and A.Yukie's work. To be more precise, the set of rational equivalence classes of semi-stable rational points corresponds almost one to one to the set of quartic field extensions. However, the set of integral equivalence classes of semi-stable integral points was not investigated. We have two number theoretic subjects related to this space. One is the lattice L of pairs of integral ternary quadratic forms. The other is the lattice L' of pairs of integral symmetric matrices of degree 3. By the result of J.Morales, the set of integral equivalence classes of semi-stable points in L' is closely related to the 2-torsion subgroup of the ideal class groups of cubic fields. On the other hand, the set of integral equivalence classes of semi-stable points in L was not known. As the result of this research, we have proved that it is closely related to the set of isomorphism classes of orders of quartic fields. Just before the submission of the result to a journal, we know that the same result was published by M.Bhuargava in late 2004.Now let n be a non zero integer, and denote by L(n) and L'(n) the set of points x in L with Δ (x)=n, and the set of points x in L' with Δ (x)=n, respectively. We denote by L(1,n), L(2,n), L(3,n), L'(1,n), L'(2,n) and L'(3,n) the subset of L(n) or L'(n) corresponding to quartic fields with 4, 2 and 0 real infinite primes, respectively. Then we have a conjecture that there exist certain relations between the 6 zeta functions whose coefficients are the numbers of integral equivalence classes of L(i, n)'s and L'(j, 256)'s. Some special cases of the conjecture are proved by this research. I gave a lecture on this result at the workshop "Rings of Low Rank" held at Leiden University in June, 2006.

设V是三元二次型对空间，群G=GL（3）XGL（2）作用于V，（G，V）是12维预齐次向量空间.这个空间与D. J. Wright和A.Yukie的四次域扩张密切相关。更准确地说，半稳定有理点的有理等价类的集合几乎一一对应于四次域扩张的集合。然而，半稳定积分点的积分等价类的集合没有被研究。我们有两个数论课题与这个空间有关。一个是整数三元二次型对的格L。另一类是三次整数对称矩阵对构成的格L'。根据J.Morales的结果，L'中半稳定点的积分等价类集与三次域理想类群的2-挠子群密切相关。另一方面，L中半稳定点的积分等价类的集合是未知的。作为研究的结果，我们证明了它与四次域的级同构类的集合密切相关。在此结果发表之前，我们知道M.Bhuargava在2004年底发表了同样的结果。设n为非零整数，分别用L（n）和L '（n）表示L中的点集x，Δ（x）=n，以及L'中的点集x，Δ（x）=n。用L（1，n），L（2，n），L（3，n），L '（1，n），L'（2，n）和L '（3，n）分别表示L（n）或L'（n）中对应于具有4，2和0个真实的无限素数的四次域的子集。由此猜想，以L（i，n）和L '（j，256）的整数等价类数为系数的6个zeta函数之间存在一定的关系.本文还证明了该猜想的一些特殊情况。2006年6月，我在莱顿大学举行的研讨会“低秩环”上发表了关于这一结果的演讲。