2004 Fiscal Year Final Research Report Summary
A Study on Jacobi Forms by a Method of Algebraic Geometry
| Project/Area Number |
14540047
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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 |
TSUSHIMA Ryuji Meiji Univ., School of Science and Technology, Prof., 理工学部, 教授 (20118764)
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| Co-Investigator(Kenkyū-buntansha) |
GOTO Shiro Meiji Univ., School of Science and Technology, Prof., 理工学部, 教授 (50060091)
NAKAMURA Yukio Meiji Univ., School of Science and Technology, Assoc.Prof., 理工学部, 助教授 (00308066)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Algebraic Geometry / Automorhic Function / Siegel Modular Form / Jacobi Form / Satake Compactification |
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| Research Abstract |
We computed the dimension of the spaces of skew-holomorphic Jacobi forms of weight k and degree two. This is an extension of the former result in which we computed the dimension of the spaces of holomorphic Jacobi forms of degree two. Moreover we computed the formula of Riemann-Roch which is needed to compute the dimension of the spaces of skew-holomorphic Jacobi forms also in the vector valued case. This is not a dimension formula yet, since we have not proved the vanishing theorem. But this is needed to compute the dimension formula and sufficient to estimate the dimension formula.
If we apply the result of T. Ibukiyama to our result, we obtain the dimension formula for the subspaces called plus spaces in the spaces of Siegel modular forms of half integral weight (this is a conjecture in the vector valued case). The spaces of Siegel modular forms of half integral weight were explicitly determined by the former work of the dimension formula for these spaces. The plus spaces were also e
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xplicitly determined (due to a computer calculation by S. Hayashida). The lifting theory of Siegel modular forms of Saito-Kurokawa is constructed by an isomorphism of the plus space and the space of Jacobi forms of upper degree. By this result the lifting theory from degree two to degree three is completed.
In the 1970's G.Shimura found a correspondence from modular forms of integral weight to modular forms of half integral weight in the case of degree one which is called Shimura correspondence today. It is a natural idea to try an extension this correspondence to the case of higher degree. But the clue have not been found after 30 years. Since the Shimura correspondence is not a surjection to the space of modular forms of half integral weight, it is important to know the dimension of the plus space which is expected to be the image of the correspondence. Moreover it is absolutely important to study the general case of vector valued modular forms because the Shimura correspondence of higher degree is a correspondence between vector valued modular forms. If we compare the dimension of the space of Siegel modular forms of degree two and integral weight and the dimension of the space of Jacobi forms assuming the vanishing theorem, we can determine the dimension of the plus pace which is expected to be isomorphic to the space of Siegel modular forms of integral weight. And it is known that the dimensions of these spaces are equal to each other (due to T.Ibukiyama). From this the existence of the Shimura correspondence in the case of degree two is conjectured. Less
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