2005 Fiscal Year Final Research Report Summary
Study on interrelations among quadratic forms, automorphic forms and related various zeta-functions
| Project/Area Number |
15540049
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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 | Rikkyo University |
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Principal Investigator |
AOKI Noboru Rikkyo University, Department of Math, Professor, 理学部, 教授 (30183130)
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| Co-Investigator(Kenkyū-buntansha) |
SATO Fumihiro Rikkyo University, Department of Math, Professor, 理学部, 教授 (20120884)
FUJII Akio Rikkyo University, Department of Math, Professor, 理学部, 教授 (50097226)
OHSUGI Hidefumi Rikkyo University, Dept. of Math, Associate Professor, 理学部, 助教授 (80350289)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Gross conjecture / elliptic curve / Tate-Shafarevich group / Fermat curve / Jacobi sum / prehomogenous vector space / Riemann zeta function / Groebner basis |
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| Research Abstract |
I studied, with Ki-Seng Tan (Taiwan) and Lee Joongul (Korea), a generalization of Tate's refinement of the Gross conjecture on the Stickelberger element obtained from special values of L-functions of number fields and function fields of one-variable over finite fields, and formulated a new conjecture. As a result, I succeeded in proving the generalized conjecture for elementary abelian extensions under certain conditions. Moreover, I proved Tate's original refinement in the case of number fields. I also studied some arithmetic properties of elliptic curves and abelian varieties. In particular, I proved unboundedness of the 3-part of the Tate-Shafarevich group of elliptic curves over the rational number field. Although this has been already known by Cassels, I proved it for a new family of elliptic curves. Furthermore, I proved the Hodge conjecture for Catalan curves, and this is one of my results in the study of arithmetic properties of jacobian varieties of algebraic curves. As a related topic, expecting application for congruent zeta functions of Fermat curves, I studied purity problem of Jacobi sums and solved negatively a problem posed by Evans.
Sato studied the local density of symmetric matrices over p-adic integers and extended a result of Schulze-Pillot to higher level cases. Fujii studied the behavior of the argument of the Riemann zeta function on the critical line. Ohsugi studied Gorensteinness of toric rings.
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Research Products
(12 results)
