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

设V是三元二次型对构成的空间，群G=GL(3)XGL(2)作用在V上，(G，V)是12维的非齐次向量空间。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集并未被研究。我们有两个与这个空间相关的数论主题。一个是整数三元二次型对的格L。另一类是3次积分对称矩阵对的格L。根据J.Morales的结果，L的半稳点的积分等价类集与三次域的理想类群的2-挠子群密切相关。另一方面，L的半稳定点的积分等价类集是未知的。作为这一研究的结果，我们证明了它与四次场序的同构类的集合密切相关。现在设n为非零整数，L(N)和L(N)分别表示L中Δ(X)=n的点集x和L的Δ(X)=n中的点集x。我们用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)对应于具有4，2和0实无穷素数的四次域的子集。然后我们猜想，系数为L(i，n)‘S和L’(j，256)‘S的积分等价类数的6个Zeta函数之间存在着一定的联系。这个猜想的一些特例得到了证明。我在2006年6月在莱顿大学举办的“低秩环”研讨会上就这一结果做了一次演讲。

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.