1999 Fiscal Year Final Research Report Summary
Investigation of zeta functions associated with prehomogeneous vector spaces
| Project/Area Number |
10640014
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Joetsu University of Education |
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Principal Investigator |
NAKAGAWA Jin Joetsu University of Education College of Education, Associate Professor, 学校教育学部, 助教授 (30183883)
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| Co-Investigator(Kenkyū-buntansha) |
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)
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| Project Period (FY) |
1998 – 1999
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| Keywords | prehomogeneous vector space / zeta function / number field |
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| 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D1-3s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D11-3s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 2-torsion 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 2-torsion subgroups of cubic fields. I gave a talk on this result at the symposium on number theory at Tsuda College.
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Research Products
(2 results)
