Investigation of zeta functions associated with prehomogeneous vector spaces
Project/Area Number  10640014 
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) 
OKAZAKI Masakazu Joetsu University of Education College of Education, Research Assistant, 学校教育学部, 助手 (40303193)
IWASAKI Hiroshi Joetsu University of Education College of Education, Lecturer, 学校教育学部, 講師 (80251867)
溝上 武実 上越教育大学, 学校教育学部, 教授 (90044445)
NUNOKAWA Kazuhiko Joetsu University of Education College of Education, Associate Professor, 学校教育学部, 助教授 (60242468)
高橋 等 上越教育大学, 学校教育学部, 助手 (80293273)

Project Period (FY) 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1999 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 1998 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  prehomogeneous vector space / zeta function / number field 
Research Abstract 
Let L be the lattice of integral binary cubic forms and LィイD4^ィエD4 be the dual lattice of L. The distribution of cubic fields is closely related to the prehomogeneous vector space of binary cubic forms. The zeta functions ξィイD2iィエD2(L, s)(I = 1, 2) associated with this space are expressed as sums of DィイD2KィエD2ィイD1sィエD1ηィイD2KィエD2(2s) over all cubic fields K. Here ηィイD2KィエD2(s) =ζ(2s)ζ(3s  1)ィイD7ζィイD2KィエD2(s)(/)ζィイD2KィエD2(2s)ィエD7. Using this expression and class field theory, I proved Ohno conjecture which statesξィイD21ィエD2(LィイD4^ィエD4, s) = 3ィイD13sィエD1ξィイD22ィエD2(L, s) andξィイD22ィエD2(LィイD4^ィエD4, s) = 3ィイD113sィエD1ξィイD21ィエD2(L, s). As applications of this result, I obtained certain relations among the number of cubic fields of positive and negative discriminants, and a refinement of Sholz's reflection theorem. These results are published in Inventiones mathematicae. I also gave a talk on the results at International Congress of Mathematicians ICM98. I have been studying the prehomogeneous vector spaces of pairs of ternary quadratic forms which is closely related to the distribution of discriminants of quartic fields and 2torsion subgroups of ideal class groups of cubic fields. In particular, I have obtained certain relations between the set of equivalence classes of pairs of integral ternary quadratic forms and 2torsion subgroups of cubic fields. I gave a talk on this result at the symposium on number theory at Tsuda College.

Report
(3results)
Research Output
(4results)