Investigation of prehomogeneous vector spaces and ideal class groups of algebraic number fields
Project/Area Number 
12640018

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Joetsu University of Education 
Principal Investigator 
NAKAGAWA Jin Joetsu University of Education, College of Education, Associate Professor, 学校教育学部, 助教授 (30183883)

CoInvestigator(Kenkyūbuntansha) 
NUNOKAWA Kazuhiko Joetsu University of Education, College of Education, Associate Professor, 学校教育学部, 助教授 (60242468)
MIZOKAMI Takemi Joetsu University of Education, College of Education, Professor, 学校教育学部, 教授 (90044445)

Project Period (FY) 
2000 – 2001

Project Status 
Completed (Fiscal Year 2001)

Budget Amount *help 
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2001: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2000: ¥1,300,000 (Direct Cost: ¥1,300,000)

Keywords  prehomogeneous vector space / ideal class group / number field / 概均的ベクトル空間 / ゼータ関数 
Research Abstract 
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 Γequivalence 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 Γequivalence classes of pairs in L and the 2torsion 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.

Report
(3 results)
Research Products
(3 results)