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.

令V为三元二次形式的空间，G = GL（3）XGL（2）对V和（G，V）作用是维度12的均匀矢量空间。此空间与D.J.Wright和A.yukie的工作与四分之一的场扩展密切相关。更确切地说，半稳定有理点的一组有理等效类别几乎对应于四分之一场扩展。但是，未研究半稳定积分点的一组积分等效类别。我们有两个与此空间相关的数字理论主题。一个是成对的成对三元二次形式的晶格。另一个是第3度的一对成对的对称材料对。根据J.Morales的结果，L'中半稳定点的一组积分等效类别与理想的立方场的理想类别组的2个小数亚组密切相关。另一方面，L中半稳定点的一组积分等效类别尚不清楚。作为这项研究的结果，我们规定它与四分之一领域的同构类别类别密切相关。在结果提交期刊之前，我们知道M.Bhuargava在2004年底发表了相同的结果。我们用l（1，n），l（2，n），l（3，n），l'（1，n），l'（2，n）和l'（3，n）分别分别为4、2和0真实的无限次数。然后，我们有一个猜想，即6个Zeta函数之间存在某些关系，其系数是L（i，n）'s and L'的积分等效类的数量（j，j，256）。这项研究证明了一些猜想的特殊情况。我在2006年6月在莱顿大学举行的“低级戒指”研讨会上就此结果进行了演讲。