Investigation of prehomogeneous vector spaces and ideal class groups of algebraic number fields

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Let V be the vector space of symmetric matrices of degree three. Then the group G = SL(3) x GL(2) acts on V and (G,V) is a prehomogeneous vector space. Let L be the lattice of V consisting of all pairs of matrices with integral coefficients. For any pair x = (x_1,x_2) ∈ L, we define a binary cubic form Φ_x(u,v) by Φ_x(u,v) = det(ux_1 + vs_2). This is an integral binary cubic form. Put Γ = SL(3,Ζ) and consider Γ as a subgroup of G. Then the action of γ∈Γ on x = (x_1,x_2) is given by γx = (γx_1^tx_1γ, γx_2^tx_1γ). It is obvious that Φ_<γx> = Φ_x. So we can consider the following problem: For a given binary form Φ, how many Γ-equivalence classes of pairs x ∈ L with Φ_x = Φ are there? J. Morales generalized this problem and obtained some results under certain assumptions. In this project, we have studied pairs x without his assumptions. We proved that for an integral binary form Φ of degree n, the order associated with Φ is weakly self dual in the meaning of Frohlich if and only if Φ is primitive. Applying this result, we studied the relations between the set of Γ-equivalence classes of pairs in L and the 2-torsion subgroups of ideal class groups of algebraic number fields of degree n. In particular, we obtained some results in the case of n = 2 and n = 3 when Φ is not primitive. These results are to be published in Acta Arithemetica. I also gave a talk on the results at Journees Arithmetiques 2001.