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

设V是三元二次型对的空间，群G=GL(3)XGL(2)作用于V， （G， V）是一个12维的预齐次向量空间。这个空间与D.J.Wright和A.Yukie的四次场扩展密切相关。更确切地说，半稳定有理点的有理等价类集几乎与四次域扩展集一一对应。然而，对半稳定积分点的积分等价类集没有进行研究。我们有两个数论主题与这个空间有关。一个是积分三元二次型对的晶格L。另一种是由3次的积分对称矩阵对组成的晶格L'。J.Morales的结果表明，L'上的半稳定点的积分等价类集合与三次域的理想类群的2-扭转子群密切相关。另一方面，L中半稳定点的积分等价类的集合是未知的。研究结果证明了它与四次域的阶同构类集密切相关。就在将结果提交给期刊之前，我们知道布阿尔加瓦在2004年底发表了同样的结果。现在设n为非零整数，用L(n)和L‘(n)分别表示L中点x的集合Δ (x)=n，和L’中点x的集合Δ (x)=n。我们分别用L（1,n）、L（2,n）、L（3,n）、L‘（1,n）、L’（2,n）和L‘（3,n）表示L(n)或L’(n)对应于4、2和0实数无穷素数的四次域的子集。然后我们推测了系数为L(i, n)'s和L'(j, 256)'s的积分等价类数的6个ζ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，我在莱顿大学举办的研讨会“低秩环”上做了一个关于这个结果的演讲。

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.