2007 Fiscal Year Final Research Report Summary
Stadies on Andytic Priepeitied of autoinoiptic L-functions
| Project/Area Number |
16340002
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Nagoya University |
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Principal Investigator |
MATSUMOTO Koji Nagoya University, Grad. Sch. Math, Prof (60192754)
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| Co-Investigator(Kenkyū-buntansha) |
KONDO Shigeyuki Nagaya Univ, Grad. Sch. Math, Prof (50186847)
KANEMITU Shigeru Kinki Univ, Dept. Eng, Prof. (60117091)
KANEKO Masahiko Kyoetsu Univ, Grad. Sch. Math, Prof (70202017)
EGAMI Shigeki Tayama Univ, Fac. Eng, Assit. Prof (60168771)
KOJIMA Hisashi Saitama Univ, Fac. Sci, Prof (90146118)
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| Project Period (FY) |
2004 – 2007
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| Keywords | automcrphic L-function / automacrphic form / automcrphic L-fiuiction / Ikeda lifting / mean-valat fheoreul / multiple zeta-fanction / multiple L-function |
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| Research Abstract |
First, we have developed the theory of modular relations and have given a unified treatment of functional equations, approximate functional equations and asymptotic expansions in view of modular relations. From this viewpoint, we gave a modular-theoretic proof of asymptotic expansion formulas of Katsurada and Matsumoto on Dirichlet L-functions and Hurwitz zeta-functions. Secondly, on Ikeda liftings of Siegel modular forms, we have shown-a part of Kohnen's conjecture on the image space of Ikeda liftings. We have proved rather sharp asymptotic formula for the mean square of standard L-functions attached to modular forms which are Ikeda lifts. In particular, applying a large value lemma, we almost determined the true order of the mean square in some cases. As for the theory of multiple zeta-functions, we have developed the study of double shuffle relations of multiple zeta values, and have clarified their arithmetic structure. Also we discovered that the functional relations among multiple zeta-functions can be explained by the symmetry of Weyl groups of underlying Lie algebras. Moreover we have found that the recursive structure of the family of multiple zeta-functions can be described in terms of Dynkin diagrams.
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