2004 Fiscal Year Final Research Report Summary
Study on automorphic forms on algebraic groups and associated zeta functions
| Project/Area Number |
13440016
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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 Sangyo University |
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Principal Investigator |
MURASE Atsushi Kyoto Sangyo University, Faculty of Science, Professor, 理学部, 教授 (40157772)
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| Co-Investigator(Kenkyū-buntansha) |
SUGANO Takashi Kanazawa University, Faculty of Science, Professor, 理学部, 教授 (30183841)
ITO Masami Kyoto Sangyo University, Faculty of Science, Professor, 理学部, 教授 (50065843)
NARITA Hiroaki Kyoto Sangyo University, Faculty of Science, Guest researcher (Japan Society for the Promotion of Science, Postdoctoral Fellows (PD)), 理学部, 客員研究員(日本学術振興会特別研究員PD)
HIRANO Miki Ehime University, Faculty of Science, Associated Professor, 理学部, 助教授 (80314946)
OHNO Yasuo Kinki University, Faculty of Science, Lecturer, 理工学部, 講師 (70330230)
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| Project Period (FY) |
2001 – 2004
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| Keywords | Kudla lift / Automorphic forms on unitary groups / Siegel Weil formula / Periods of automorphic forms / Automorphic L-functions |
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| Research Abstract |
1. Metaplectic representations of unitary groups :
We studied metaplectic representations of unitary groups over local fields and gave their "universal" splitting, which are in particular useful in the study of theta lifting. As an application of this result, we gave an explicit character formula for metaplectic representations.
2. Fourier-Jacobi expansion of automorphic form on unitary groups of degree three :
We reformulated Shintani' s theory on Fourier-Jacobi expansion of automorphic forms on unitary groups of degree three in adelic language, and calculated explicit form for Fourier-Jacobi expansion of Eisenstein series and Kudla lifts, theta lifts from elliptic modular forms. As an application, we gave a criterion for the non-vanishing of Kudla lifts.
3. Siegel-Weil formula :
We studied a non-regularized Siegel-Weil formula in the case of the dual reductive pair (U(2,2), U(2, 1)).
4. Inner product formula for Kudla lifts :
Using the formula stated in 3, we gave an explicit formula for the Petersson norms of Kudla lifts in term of special values of automorphic L-functions. As an application, we gave a criterion for the non-vanishing of Kudla lifts different from the one given in 2. (The studies 2- 4 are joint works with Takashi Sugano).
5. Support for the Summer School of Number Theory :
We supported financially the Summer School of Number Theory held annually.
The themes were as follows : "Zeta functions" in 2001, "Prehomogeneous vector spaces" in 2002, "Iwasawa theory" in 2003 and "Fundamental groups and Galois representations" in 2004.
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