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

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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.