• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to project page

1998 Fiscal Year Final Research Report Summary

Various aspects of hypergeometric functions

Research Project

Project/Area Number 08404004
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionKYUSHU UNIVERSITY

Principal Investigator

YOSHIDA Masaaki  Kyushu Univ., Graduate School of Mathematics, Professor, 大学院・数理学研究科, 教授 (30030787)

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)
Project Period (FY) 1996 – 1998
Keywordshypergeometric function / hypergeometric equation / period mapping / projective structure / conformal structure / arithmetic subgroup / reflection group / twisted (co) homology
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.

  • Research Products

    (40 results)

All Other

All Publications (40 results)

  • [Publications] N.Takayama and M.Yoshida: "CR-geometry on the configuration space of 5 points on the projective line" Funkcial.Ekvac.39. 165-181 (1996)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Yoshida: "The democratic compactifications of the configuration spaces of point-sets on the real projective line" Kyushu J.Math.50. 493-512 (1996)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Sasaki, K.Yamaguchi and M.Yoshida: "On the rigidity of differential systems modelled on hermitian symmetric spaces and disproofs of a conjecture concerning modular interpretations of configuration spaces" CR-Geometry and Overdetermind Systems(Advanced Studies Pure Math.). 25. 318-354 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] J.Sekiguchi and M.Yoshida: "W(E_6)-orbits of the configuration space of 6 lines on the real projective plane" Kyushu J.Math. 51. 1-58 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] F.Apery and M.Yoshida: "Pentagonal structure of the configuration space of five points in the real projective" Kyushu J.Math. 2. 1-14 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Yoshida: "The real loci of the configuration space of six points on the projective line and a Picard modular 3-fold"

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kumamoto J.Math. 11. 43-67 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Hanamura and M.Yoshida: "Hodge structure of twisted cohomologies and twisted Riemann inequalities I" Nagoya Math J. to appear.

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Matsumoto and M.Yoshida: "Recent progress of interesection theory for twisted (co)homologies" Advanced studies in pure Math. to appear.

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Kaneko: "poly-Bernoulli numbers" J.de Theorie des Nombres. 9. 199-206 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Kaneko and D.Zagier: "Supersingular j-invariants hypergeometric series, and Atkin's orthogonal polynomials," Studies in Advanced Mathematics. vol.7. 97-126 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Matsumoto and T.Sasaki: "On the system of differetial equations associated with a quadric and hyperplanes" Kyushu J.of Math.vol.50. 93-131 (1996)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Matsumoto and M.Kita: "Duality for Hypergeometric Functions and Invariant Gauss-Manin Systems" Compositio Math.108. 77-106 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Matsumoto: "Intersection numbers for 1-forms associated with confluent hypergeometric functions" Funkcal. Ekvac. 41. 291-308 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Matsumoto: "Intersection numbers for logarithmic k-forms" Osaka J.Math. 35. 873-893 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] F.Kato: "Log smooth deformation theory" Tohoku Math.Journal. 48. 317-354 (1996)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] F.Kato and Masanori Ishida: "The strong rigidity theorem for non-archimedean uniformization" Tohoku Math.Journal. 50. 537-555 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] F.Kato: "The relative log Poincare lemma and relative log de Rham theory" Duke Math.Jounal. 93. (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] F.Kato: "Functors of log Artin rings" Manuscripta Mathematica. 96. 97-112 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Yoshida: ":Hypergeometric Functions, My Love : Modular Interpretations of Confuguration Spaces." :Vieweg Verlag, Wiesbaden, 292 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 吉田正章: "私説超幾何関数" 共立出版, 258 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] N.Takayama and M.Yoshida: "CR-geometry on the configuration space of 5 point on the projective line" Funkcial.Ekvac.39. 165-181 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Yoshida: "The democratic compactifications of the configuration spaces of point-sets on the real projective line" Kyushu J.Math.50. 493-512 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Sasaki, K.Yamaguchi and M.Yoshida: "On the rigidity of differen-tial systems modelled on hermitian symmetric spaces and disproofs of a conjecture concerning modular interpretations of configuration spaces" Advanced Studies in Pure Math.CR-Geometry and Overdeter-mined Systems. 25. 318-354 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] J.Sekiguchi and M.Yoshida: "W (E_6)-orbits of the configuration space of 6 lines on the real projective plane" Kyushu J.Math.51. 1-58 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] F.Apery and M.Yoshida: "Pentagonal structure of the configuration space of five points in the real projective line" Kyushu J.Math.52. 1-14 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Yoshida: "The real loci of the configuration space of six points on the projective line and a Picard modular 3-fold" Kumamoto J.Math.11. 43-67 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Hanamura and M.Yoshida: "Hodge structure of twisted coho-mologies and twisted Riemann in qualities I" Nagoya Math J. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Matsumoto and M.Yoshida: "Recent progress of interesection theory for twisted (co) homologies" Advanced studies in pure Math. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Kaneko: "Poly-Bernoulli numbers" J.de Theorie des Nombres. 9. 199-206 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Kaneko and D.Zagier: "Supersingular j-invariants, hypergeomet-ric series, and Atkin's orthogonal polynomials" Studies in Advanced Mathematics. vol.7. 97-126 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Matsumoto and T.Sasaki: "On the system of differential equa-tions associated with a quadric and hyperplanes" Kyushu J.of Math.vol.50. 93-131 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Matsumoto and M.Kita: "Duality for Hypergeometric Functions and Invariant Gauss-Manin Systems" Compositio Math.108. 77-106 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Matsumoto: "Intersection numbers for 1-forms associated with confluent hypergeometric functions" Funkcial.Ekvac.41. 291-308 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Matsumoto: "Intersection numbers for logarithmic k-forms" Osaka J.Math.35. 873-893 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] F.Kato: "Log smooth deformation theory" Tohoku Math.Journal. 48. 317-354 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] F.Kato and Masanori Ishida: "The strong rigidity theorem for non-archimedean uniformization" Tohoku Math.Journal. 50. 537-555 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] F.Kato: "The relative log Poincare lemma and relative log de Rham theory" Duke Math.Journal. 93. (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] F.Kato: "Functors of log Artin rings" Manuscripta Mathematica. 96. 97-112 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Yoshida: Hypergeometric Functions, My Love : Modular Inter-pretations of confuguration Spaces.Vieweg Verlag, Wiesbaden, 292 (1997)

    • Description
      「研究成果報告書概要(欧文)」より

URL: 

Published: 1999-12-08  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi