1996 Fiscal Year Final Research Report Summary
Arithmetic research of automorphic functions
| Project/Area Number |
06452003
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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 UNIVERSITY |
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Principal Investigator |
SAITO Hiroshi Kyoto Univ, Human and Environmental studo.P., 大学院・人間・環境学研究科, 教授 (20025464)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHINO Yuji Kyoto, Univ, Integrated Human Studies A.P., 総合人間学部, 助教授 (00135302)
MORIMOTO Yoshinori Kyoto Univ, Human and Environmented studies A.P., 総合人間学部, 助教授 (30115646)
YAMAUTI Masatoshi Kyoto Univ, Integrated Huyman studies P., 総合人間学部, 教授 (30022651)
GYOJA Akihiko Kyoto Univ, Integrated Huyman Studies A.P., 総合人間学部, 助教授 (50116026)
KATO Shinichi Kyoto Univ, Integrated Huyman Studies A.P., 総合人間学部, 助教授 (90114438)
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| Project Period (FY) |
1994 – 1996
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| Keywords | Siegel modular form / prehomogeneous vector space / zeta function / dimension formula / speherical homogeneous space / sphrical function / Hecke ring / Gauss sum |
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| Research Abstract |
In this research project, we mainly seek the application of the theory of prehomogeneous spaces to the theory of Siegel modular forms. It has been known long before that such zeta functions and their special vulues play important roles in the theory of Siegel modular forms. But It was generally thought that these special values were difficult to determine in general. But in our research, it was shown that those zeta function, which are zeta functions associated to the prehomogeneous vector space of the spaces of symmetric matrices, can be expressed by Rimeann's zeta functions and Dirichlet series associated to Eisenstein series of half integral weight. By this results the special values of zeta functions mentioned above can be easily obtained and are shown to be expressed by Bernoulli numbers. From this we formulated a conjecture on the explicit dimension formula of Siegel modular forms. The method used in this calculation of zeta functions associated to the space of symmetric matrices can be applied to the calculation of zeta functions of the other types of prehomogeneous vector spaces. By this method, we can reduce the calculation of global zeta functions of prehomogeneous vector spaces to that of local orbital zeta functions, and we hope this method allow us to compute those global zeta functions in many cases. Gyoja's results on Gauss sums of prehomogeneous vector spaces over finite fields and Hecke ring are important not only in the thoery of representations of reductive groups over finite fields but also in that of Siegel modular form. Kato's results on spehrical homogeneous spaces and spherical functions have many applications in the analytic theory of zeta functions of automorphic forms, especially in their integral representation and analytic continuation.
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