2000 Fiscal Year Final Research Report Summary
A study of Siegel modular forms of half integral weight by a method of algebraic geometry
| Project/Area Number |
10640044
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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 | Meiji University |
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Principal Investigator |
INATOMI Akira Meiji Univ., Faculty of Science and Technology, Prof., 理工学部, 教授 (20061872)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAMURA Yukio Meiji Univ., Faculty of Science and Technology, Associated Prof., 理工学部, 講師 (00308066)
SATO Atsushi Meiji Univ., Faculty of Science and Technology, Associated Prof., 理工学部, 助教授 (70178705)
GOTO Shiro Meiji Univ., Faculty of Science and Technology, Prof., 理工学部, 教授 (50060091)
TSUSHIMA Ryuji Meiji Univ., Faculty of Science and Technology, Associated Prof., 理工学部, 助教授 (20118764)
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| Project Period (FY) |
1998 – 2000
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| Keywords | Siegel modular form / Jacobi form / Algebraic Geometry / Riemann-Roch formula / Satake compactification |
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| Research Abstract |
Siegel modular forms of half integral weight are identified with holomorphic sections of a certain holomorphic line bundle over a quotient space of Siegel upper half plane by a discrete group. We computed the dimension of the spaces of Siegel modular forms of degree two and half integral weight by applying the formula of Riemann-Roch (holomorphic Lefschetz fixed point theorem) and Kodaira vanishing theorem to this line bundie. We classified fixed points by using computer.
The space of Siegel modular forms of half integral weight has a subspace called plus space. This subspace is a very important subspace concerning the lifting theory of modular forms. There exists an isomorphism between this plus space and the space of Jacobi forms of index one. We computed the dimension of the spaces of Jacobi forms of degree two to know the dimension of the plus space by this isomorphism. In this way we knew the dimension of the plus space and its structure was determined (Ibukiyama and Hayashida).
Jac
… More
obi forms are holomorphic functions on the product space of Siegel upper half plane and complex vector space which behave like modular forms with respect to the variables of Siegel upper half plane and behave like theta functions with respect to the variables of complex vector space. Since Jacobi forms of index m behave like theta functions of degree 2m with respect to the variables of complex vector space, they are represented by a linear combination of theta series which consist of a basis of theta functions of degree 2m. The coefficients of this combination are holomorphic functions on Siegel upper half plane. The vector consisting of these coefficients becomes a vector valued modular form with respect to a certain automorphic factor on Siegel upper half plane. Therefore Jacobi forms are identified with holomorphic sections of a certain holomorphic vector bundle on a quotient space of Siegel upper half plane by a discrete group. We computed the dimension of the space of holomorphic sections which is the dimension of the space of Jacobi forms by applying the formula of Riemann-Roch and the vanishing theorem of Kodaira-Nakano. Less
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Research Products
(21 results)
