2004 Fiscal Year Final Research Report Summary
Research on special values of the standard L-functions of Siegel modular forms and related fields
| Project/Area Number |
15540003
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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 | Muroran Institute of Technology |
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Principal Investigator |
KATSURADA Hidenori Muroran Institute of Technology, Professor, 工学部, 教授 (80133792)
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| Co-Investigator(Kenkyū-buntansha) |
TAKEGAHARA Yugen Muroran Institute of Technology, Associate Professor, 工学部, 教授 (10211351)
HASEGAWA Yuji Muroran Institute of Technology, Associate Professor, 工学部, 助教授 (30287982)
KOHSAKA Yoshihito Muroran Institute of Technology, Associate Professor, 工学部, 助教授 (00360967)
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| Project Period (FY) |
2003 – 2004
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| Keywords | Koecher-Maass series / Saito-Kurokawa lift / Special values / Congruence of modular forms / Standard zeta function / Basis Problem / Doi-Hida-Ishi conjecture / Block-Kato conjecture |
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| Research Abstract |
(1)I considered the relation between special values of the standard zeta functions and the congruence a Siegel cusp form of arbitrary degree. In particular, I proposed a conjecture concerning the congruence of the Saito-Kurokawa lift of a cuspidal Hecke eigenform with respect to SL(2,Z) and the special value of its L-function, and proved it under certain condition. Furthermore we gave some numerical evidence for our conjecture. I gave talks on these topics at the workshop held at Hakuba on October 2004, and at the conference held at Karatsu on March 2005.
(2)Let M and N be square free positive integers, and φ and χ be Dirichlet characters mod M and N, respectively. For a cuspidal Hecke eigenform f of neben type φ I gave an explicit formula for the special values of the standard zeta function of f twisted by χ. I gave numerical examples for these values and checked some results derived from the Block-Kato conjecture.
(3)I proved that the space of Eisenstein series of square free level of quadratic nebentype is spanned by the genus theta series jointly with R. Schulze-Pillot. By using this result, I proved that the space of cusp forms of square free level of quadratic nebentype is spanned by the theta series jointly with S.Boecherer and R.Schulze-Pillot. They gave talks on these topics at the symposium held at Rikkyo University on September 2004.
(4)I gave an explicit form of the Koecher-Maass series for the real analytic Eisenstein series jointly with T.Ibukiyama. I gave a talk on this topic at the symposium held at Rikkyo University on September 2004.
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