| Co-Investigator(Kenkyū-buntansha) |
MATSUMOTO Keiji Hiroshima Univ., Faculty of Science, Associate Professor, 理学部, 助教授 (30229546)
WATANABE Fumihiko Kyushu Univ., Graduate School of Mathematics, Research Associate, 大学院・数理学研究科, 助手 (20274433)
HANAMURA Masaki Kyushu Univ., Graduate School of Mathematics, Associate Professor, 大学院・数理学研究科, 助教授 (60189587)
KANEKO Masanobu Kyushu Univ., Graduate School of Mathematics, Associate Professor, 大学院・数理学研究科, 助教授 (70202017)
KATO Fumiharu Kyushu Univ., Graduate School of Mathematics, Research Associate, 大学院・数理学研究科, 助手 (50294880)
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| Research Abstract |
Hypergeometric integrals found by Euler was re-formulated in terms of a modern language by many authors : the dualﾟCpairing of twisted homologies and twisted cohomologies. Expected intersection theories were established by M.Kita and Yoshida for homologies, and by K.Cho and Matsumoto for cohomologies. Further developments are in progress, especially those for confluent case by Matsumoto. These can be considered to be twisted versions of Riemann's equality for period integrals. Twisted versions of Riemann's inequality were found, via twisted Hodge theory, by Hanamura and Yoshida.
Modular interpretations of configuration spaces. Let X(k, n) be the configuration space of n-point-sets in the k-1-dimensional projective space. Several configuration spaces can be presented as quotient spaces of symmetric spaces under discontinuous groups ; the original one is X(2, 4) * H/GAMMA(2), where H is the upper half space and GAMMA(2) is an elliptic modular group. Yoshida found, with Matsumoto and T.Sasaki, a modular interpretation of the space X(3, 6) through hepergeometric function of type (3, 6)), which can be summerized as X(3,6) {z * M2(C) I (z -z*)/2i> O}/GAMMA, where GAMMA is an arithmetic subgroup acting on the hermitian symmetric domain of type IV.Yoshida wrote two books about this interpretation.
Kaneko found, with D.Zagier, automorphic forms which connect hypergeometric functions and supersingular elliptic curves. Kaneko found a new arithmetic formulae for the Fourier coefficients of j(gamma).
F.Watanabe established a new very transparent way to find Okamoto transformations for Painlv_ functions by using the Takano's construction of the phase spaces.
F.Kato is ambitiously trying to find examples of algebraic varieties which are p-adically uniformized by Drinfeld symmetric spaces and the uniformizing differential equations ; he already found, with M.Ishida, new fake projective planes, and studied their uniformizations complex anlytically as well as p-adically.
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