2002 Fiscal Year Final Research Report Summary
Study of zeta functions related to Jacobi forms and Siegel modular forms
| Project/Area Number |
12640045
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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 |
ARAKAWA Tsuneo Rikkyo University, Faculty of Science, Professor, 理学部, 教授 (60097219)
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| Co-Investigator(Kenkyū-buntansha) |
ENDO Mikihiko Rikkyo University, Faculty of Science, Professor, 理学部, 教授 (40062616)
AOKI Noboru Rikkyo University, Faculty of Science, Associate Professor, 理学部, 助教授 (30183130)
SATO Fumihiro Rikkyo University, Faculty of Science, Professor, 理学部, 教授 (20120884)
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| Project Period (FY) |
2000 – 2002
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| Keywords | modular forms / Jacobi forms / Selberg zeta functions / Maass wave forms / prime geodesic theorem / multiple L-values / prehomogeneous vector space / b-function |
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| Research Abstract |
1. On Siegel modular forms : We formulated the converse theorem of Imai in such a manner that it can be applica* to not necessarily cuspidal Siegel modular forms of degree two. As an application we reconstructed the Saito-Kurokawa lifting for Siegel modular forms of degree two by using Duke-Imamogle's method (joint work with the coworker F.Sato and I.Makino).
2. Solution of the Hashimoto conjecture : We solved the conjecture presented by K.Hashimoto concerning the that series associated to definite quaternion algebras over Q by using Jacobi forms, the theorem of Deligne-Serre on modular forms of weight one (joint work with S.Boecherer).
3. The study of Selberg zeta functions : The head investigator studied the Shimura correspondence for Maass wave forms via Selberg trace formulas and Selberg zeta functions and reduced the problem of bijectivity of the correspondence to a simple relation of the associated two Selberg zeta functions. On the other hand he with the co-operation of S.Koyama and M.Nakasuji obtained explicit forms of the arithmetic Selberg zeta functions attached to indefinite quaternion aogebras over Q and the prime geodesic theorem of Brun-Titchmarsh type.
4. On multiple L-values : The head investigator and M.Kaneko formulated the (regularized) double shuffle relations for multiple L-values. We also determined the principal part at the pole s=0 of the associated multiple L-function by using the zeta regularization.
5. The study of prehomogeneous vector spaces : The coworker F.Sato and K.Sugiyama showed that the b-function also has the decomposition corresponding to that of the associated representations of the vector space.
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Research Products
(14 results)
