| Co-Investigator(Kenkyū-buntansha) |
ARAKAWA Tsumeo Faculty of Engnerring, Muroran Institute of Technology Prof., 理学部, 教授 (60097219)
KATSUNADA Hidenori Faculty of Engineering, Muroran Institute of Technology Prof., 工学部, 教授 (80133792)
SATO Fumihara Faculty of Science, Rikkyo Univ. Professor, 理学部, 教授 (20120884)
ODA Takayuki Graduate School of Mathematical Science, The University of Tokyo Professor, 大学院・数理科学研究科, 教授 (10109415)
SATO Hiroshi Graduate school of Human and Environmental Studies, Kyoto University Professor, 大学院・人間・環境学研究科, 教授 (20025464)
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| Research Abstract |
In these three years, the principal researcher of this subject organized a conference at Oberwolfach in Germany, three autumn workshops and two mini conferences in Japan, hence 6 conferences in total, and studied effectively with many foreign researchers and cooperative researchers in Japan. In particular in three autumn workshops, we took three themes, Koecher Maass series, Eisenstein series, dimension formulae of automorphic forms, as initially planned. There we studied the foundation and development of arithmetic zeta functions, and obtained satisfactory results, and besides we published three volumes of proceedings of 640 total pages which can be regarded as fundamental references of the research of this direction. More concretely, the principal researcher defined Koecher-Maass series for automorphic forms of any tube domains, proved their analytic continuation and gave functional equations. Explicit forms of Koecher-Maass series for modular forms such as Eisenstein series, or thos
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e closely related to liftings were also obtained (jointly with Katsurada) and established the meaning of Koecher-Maass series as "arithmetic" zeta functions. Also, he obtained a lifting conjecture on Siegel modular forms of half integral weights (joint with Hayashida), developed theories on modular forms of rational weights and modular varieties, and applied differential operators to the study of structures of vector valued Siegel modular forms. As you can see from the list of their papers, the other researchers of our project also actively studied on spherical functions and zeta functions of prehomogeneous vector spaces, their explicit forms and convergence, Jacobi forms, liftings, explicit density formulae, adele geometry, real analysis on automorphic forms, Siegel modular forms mod p, fundamental lemma, the Fourier expansion of modular forms on bounded symmetric domains of non-tube type and other various researches. These are all related to the development of our research project and we think our project was very successful. Now we think we are in the stage to go from fundamental research to various applications, and expecting and planning a project of next stage such as explicit theories of graded rings of automorphic forms and vertex operator algegras. Less
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