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Study on the K3 modular function and its arithmetic aspects

Research Project

Project/Area Number 12640010
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionCHIBA UNIVERSITY

Principal Investigator

SHIGA Hironori  Graduate School of Science and Technology at Chiba University, Professor, 大学院・自然科学研究科, 教授 (90009605)

Co-Investigator(Kenkyū-buntansha) SUGIYAMA Ken-ichi  Faculty of Science at Chiba University, Associate Professor, 理学部, 助教授 (90206441)
KITAZUME Masaaki  Faculty of Science at Chiba University, Professor, 理学部, 教授 (60204898)
MATSUDA Shigeki  Faculty of Science at Chiba University, Assistant, 理学部, 助手 (90272301)
TSUTSUI Toru  Faculty of Science at Chiba University, Assistant, 理学部, 助手 (00197732)
ISHIMURA Ryu-ichi  Faculty of Science at Chiba University, Professor, 理学部, 教授 (10127970)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2001: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2000: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsModular function of several variables / K3 surface / period map / Abelian variety / Holonomic system / Hypergeometric Differrential Equation / Bounded symmetric domain / ホロノミックシステム / ミラ対称性 / 超幾何微分方程式 / モノドロミー / 格子 / 志村多様体
Research Abstract

(a) We made a study on the 3 dimensional congruent number problem and rational cuboid problems. We reformed the problem to the study of rational points on a certain type of K3 surfaces and showed the way of construction for some type of rational cuboid.
(b) We studied the quotient of an orthogonal group by the level 2 principal congruence subgroup with respect to a certain typical indefinite quadratic form with 2 positive eigen values. We determined the structure of such quotients.
c We made studies from various aspects on the family of K3 surfaces with a certain fixed structure of the Picard lattice of rank 14. In fact we obtained the following facts :
(i) the explicit model for the member of such a family with a defining equation,
(ii) the explicit moduli space for the periods, that is a bounded symmetric domain of type II,
(iii) description of the differential equation of the periods, that is a holonomic system of rank 8,
(iv) the moduli space can be embedded in the Siegel upper space of degree 8, it is characterized as a certain type of Shimura variety,
(v) the above Shimura variety is induced from the starting Hodge structure of the K3 surface via the construction of Kuga-Satake.
(d) We made an investigation on the Fuchsian differential equation coming from the family of punctured 1 dimensional tori. We obtained several observational results.

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (26 results)

All Other

All Publications (26 results)

  • [Publications] (1) N.Narumiya, H.Shiga: "On certain rational cuboid problems"Nihonkai Mathematical Journal. 12. 75-88 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] (2) M.Kitazume, K.Koike, H.Shiga: "Notes on a certain integral orthogonal group and its quotient by the principal congruence subgroup of level 2"Kyushu Journal of Mathematics. 54. 257-271 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] (3) K.Koike, H.Shiga, N.Takayama, T.Tsutsui: "Study on the family of K3 surfaces induced from the lattice (D_4)^3+<-2>+<2>"International Journal of Mathematics. 12. 1049-1085 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] (4) M.Kitazume, C.H.Lam, M.Yamada: "A class of vertex operator algebra constructed from Z_8 codes"Journal of Algebra. 242. 338-359 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] (5) M.Kitazume, C.H.Lam, M.Yamada: "Decomposition of the Moonshine vertex operator algebra as Virasoro modules"Journal of Algebra. 226. 893-919 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] (6) S.Matsuda: "Katz correspndence for quasi-unipotent overconvergent isocrystals"Compositio Mathematics. (掲載予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] N. Narumiya, H. Shiga: "On certain rational cuboid problems"Nihonkai Math. J.. vol 12. 75-88 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Kitazume, K. Koike, H. Shiga: "Notes on a certain integral orthogonal group and its quotient by the principal congruence subgroup of level 2"Kyushu J. Math.. vol. 54. 257-271 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] K. Koike, H. Shiga, N. Takayama, T. Tsutsui: "Study on the family of K3 surfaces induced from the lattice (D_4)^3+<-2>+<2>"Int. J. Math.. vol. 12. 1049-1085 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Kitazume, C.H. Lam, M. Yamada: "A class of vertex operator algebra constructed from Z_8 codes"J. Algebra. vol. 242. 338-359 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M. Kitazume, C.H. Lam, M. Yamada: "Decomposition of the Moonshine vertex operator algebra as Virasoro modules"J. Algebra. vol. 226. 893-919 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Matsuda: "Katz correspondence for quasi-unipotent overconvergent isocrystals"Compositio Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] N. Narumiya, H. Shiga: "The mirror map for a family of K3 surfaces induced from the simplest 3-dimensional reflexive polytope"CMR Proceeding and Lecturenote. (to appear). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H. Shiga, T. Tsutsui, J. Wolfart: "Summary note on the moduli of punctured tori and related Fuchsian differential equations"R.I.M.S. Kokyuroku. 1212. 43-49 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Narumiya, N., Shiga, H.: "On Certain Rational Cuboid Problems"Nihonkai Mathematical Journal. 12. 75-88 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kitazume, M., Koike, K., Shiga, H.: "Notes on a certain integral orthogonal group and its guotient by the principal congruence subgroup of level 2"Kyushu Journal of Mathematics. 54. 257-271 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Koike, K., Shiga, H., Takayama, N., Tsutsui, T.: "Study on the family of K3 surfaces induced from the latltce(D_4)^3【symmetry】<-2>【symmetry】<2>"International Journal of Mathematics. 12. 1049-1085 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kitazume, M., Lam, C.H., Yamada, H.: "A class of vertex operator algebra constructed from ZZ_8 codes"Journal of Algebra. 242. 338-359 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kitazume, M., Lam, C.H., Yamada, H.: "Decomposition of the Moonshine vertex operator algebra as Virasoro modules"Journal of Algebra. 226. 893-919 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Matsuda, S.: "Katz correspondence for guasi-unipotent overconvergent idiecrystals"Compositio Mathematica. (掲載予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Koike,M.Kitazume,H.Shiga: "Notes on a certain integral orthogonal group and its guotient by the principal congruence subgroup of level 2"Kyushu Journal of Mathematics. 54. 257-271 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] N.Narumiya,H.Shiga: "On centain national euboid problems"Nihonkai Mathematical Journal. (未定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] N.Narumiya,H.Shiga: "The mirror map for a family of K3 surfaces"CMR Proceeding and Lecture note. (未定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Koike,H.Shiga,T.Tsursui,T.Takayama: "Study on the family of k3 surfaces niduced from the lattice 〓<-2>〓<2>"International Journal of Mathematics. (未定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Kitazume: "Some Non-Split Extenstions of the Qshogonal Group Ω(7,3)"Journal of Algebra. 224. 59-76 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Kitazume: "Bromein Identity and Vertex Operator Algebras"Journal of Number Theory. 82. 100-108 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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