2003 Fiscal Year Final Research Report Summary
Research on special values and zeros of L-funcions and on automorphic forms
| Project/Area Number |
13440007
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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 | Kyoto University |
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Principal Investigator |
YOSHIDA Hiroyuki Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (40108973)
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| Co-Investigator(Kenkyū-buntansha) |
UMEDA Toru Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (00176728)
HIRAGA Kaoru Kyoto. Univ., Graduate School of Science, Assistant, 大学院・理学研究科, 助手 (10260605)
IKEDA Tamotsu Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (20211716)
FUJII Akio Rikkyo Univ., Fuculty of Science, Professor, 理学部, 教授 (50097226)
FUJIWARA Kazuhiro Nagoya Univ., Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (00229064)
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| Project Period (FY) |
2001 – 2003
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| Keywords | CM-period / Automorphic form / L-function / zeros of L-functions |
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| Research Abstract |
Yoshida studied various problems centered around the absolute CM-period. Also he found a direct method to obtain cohomology classes from automorphic forms of a wide class, by decomposing the integral of Eichler-Shimura type. He organized his research results in a book, which was published by American Mathematical Society. He studied a p-adic analogue of the absolute CM-period, in collaboration with Ph. D student Tomokazu Kashio. There are strong evidences that a p-adic analogue holds in a perfect manner. A few years ago, Ikeda constructed a lifing from elliptic modular forms to Siegel modular forms of several variables. Taking as the kernel function the restriction to the diagonal of this lifting, he constructed a new lifing, which contains Miyawaki's lifting as a apecial case.. He formulated a conjecture which relates the nonvanishing of this lifting to special values of certain L-functions. Hiraga formulated a conjecture which relates the Zelevinskii involution and A-packet conjectured by Arthur. As an evidence, he proved the commutativity of endoscopic lifts and the Zelevinskii involution. He further studied Arthur's conjecture 'on automorphic representations. Umeda studied three problems on the center of the univeral enveloping algebra of a Lie algebra of classical type, namely concrete description of generating system, relations to the other generating system and explicit representations of a generation system. These problems originated from the Capelli identity which is the identity of invariant differential operators. Fujiwara studied Leopoldt's conjecture using the Taylor-Wiles-Fujiwara theory and arrived at the new point of view that number. fields and hyperbolic manifolds are analogous. Fujii studied on the Montgomery conjecture on the pair correlations of zeros, the Montgomery sum and higher 'moments of the argument of the Riemann zeta function.
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