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A study of Siegel modular forms of half integral weight by a method of algebraic geometry

Research Project

Project/Area Number 10640044
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionMeiji University

Principal Investigator

INATOMI Akira (1999-2000)  Meiji Univ., Faculty of Science and Technology, Prof., 理工学部, 教授 (20061872)

対馬 龍司 (1998)  明治大学, 理工学部, 助教授 (20118764)

Co-Investigator(Kenkyū-buntansha) NAKAMURA Yukio  Meiji Univ., Faculty of Science and Technology, Associated Prof., 理工学部, 講師 (00308066)
SATO Atsushi  Meiji Univ., Faculty of Science and Technology, Associated Prof., 理工学部, 助教授 (70178705)
GOTO Shiro  Meiji Univ., Faculty of Science and Technology, Prof., 理工学部, 教授 (50060091)
TSUSHIMA Ryuji  Meiji Univ., Faculty of Science and Technology, Associated Prof., 理工学部, 助教授 (20118764)
稲富 彬  明治大学, 理工学部, 教授 (20061872)
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2000: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1999: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1998: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsSiegel modular form / Jacobi form / Algebraic Geometry / Riemann-Roch formula / Satake compactification / リーマン・ロッホの定理
Research Abstract

Siegel modular forms of half integral weight are identified with holomorphic sections of a certain holomorphic line bundle over a quotient space of Siegel upper half plane by a discrete group. We computed the dimension of the spaces of Siegel modular forms of degree two and half integral weight by applying the formula of Riemann-Roch (holomorphic Lefschetz fixed point theorem) and Kodaira vanishing theorem to this line bundie. We classified fixed points by using computer.
The space of Siegel modular forms of half integral weight has a subspace called plus space. This subspace is a very important subspace concerning the lifting theory of modular forms. There exists an isomorphism between this plus space and the space of Jacobi forms of index one. We computed the dimension of the spaces of Jacobi forms of degree two to know the dimension of the plus space by this isomorphism. In this way we knew the dimension of the plus space and its structure was determined (Ibukiyama and Hayashida).
Jac … More obi forms are holomorphic functions on the product space of Siegel upper half plane and complex vector space which behave like modular forms with respect to the variables of Siegel upper half plane and behave like theta functions with respect to the variables of complex vector space. Since Jacobi forms of index m behave like theta functions of degree 2m with respect to the variables of complex vector space, they are represented by a linear combination of theta series which consist of a basis of theta functions of degree 2m. The coefficients of this combination are holomorphic functions on Siegel upper half plane. The vector consisting of these coefficients becomes a vector valued modular form with respect to a certain automorphic factor on Siegel upper half plane. Therefore Jacobi forms are identified with holomorphic sections of a certain holomorphic vector bundle on a quotient space of Siegel upper half plane by a discrete group. We computed the dimension of the space of holomorphic sections which is the dimension of the space of Jacobi forms by applying the formula of Riemann-Roch and the vanishing theorem of Kodaira-Nakano. Less

Report

(4 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (33 results)

All Other

All Publications (33 results)

  • [Publications] 対馬龍司: "On the dimension formula for the spaces of Jacobi forms of degree two"数理解析研究所講究録. 1103. 96-110 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 対馬龍司: "On the dimension formula for the spaces of Siegel cusp forms of half integral weight and degree two"数理解析研究所講究録. 1052. 42-57 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎,原井川聡,居相真一郎: "Complete intersection in overrings of a certain one-dimensional Gorenstein graded local ring"Journal of Algebra. 233. 772-790 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎: "Cohen-Macaulayness and negativity of A-invariants in Rees algebras associated to m-primary ideals of minimal multiplicity"Journal of Pure and Applied Algebra. 152. 93-107 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎,居相真一郎: "Embeddings of certain graded rings into their canonical modules"Jouranl of Algebra. 228. 377-396 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎,中村幸男,西田康二: "On the Gorensteiness in graded rings associated to certain ideals of analytic deviation one"Japanese Jouranl of Mathematics. 23. 377-396 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎: "Buchsbaumness in Rees algebras associated to ideals of minimal multiplicity"Jouranl of Algebra. 213. 604-661 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 原井川聡,後藤四郎: "イデアル化によって得られたArtin Gorenstein局所環内のgood idealsの構造と分布について"明治大学科学技術研究所紀要. 38. 9-24 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎,西田康二: "Catenarity in module finite algebras"Proceedings of American Mathematical Society. 127. 3495-3502 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎,西田康二: "Minimal injective resolutions of Cohen-Macaulay isolated singularities"Archiv der Mathematik. 73. 249-255 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 中村幸男: "On the Buchsbaum property of associated graded rings"Jouranl of Algebra. 209. 345-366 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Ryuji Tsushima: "On the dimension formula for the spaces of Jacobi forms of degree two"Surikaisekikenkyusho Kokyuroku. 1103. 96-110 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Ryuji Tsushima: "On the dimension formula for the spaces of Siegel cusp forms of half integral weight and degree two"Surikaisekikenkyusho Kokyuroku. 1052. 42-57 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Shiro Goto: "Satoshi Haraikawa, Shin-ichiro Iai, Complete intersection in overrings of a certain one-dimensional Gorenstein graded local ring."J.of Algebra. 233. 772-790 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Shiro Goto: "Cohen-Macaulayness and negativity of A-invariants in Rees algebras associated to m-primary ideals of minimal multiplicity"J.Pure and Appl.Algebra. 152. 93-107 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Shiro Goto and Shin-ichiro Iai: "Embeddings of certain graded rings into their canonical modules."J.of Algebra. 228. 377-396 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Shiro Goto, Yukio Nakamura, Koji Nishida: "On the Gorensteiness in graded rings associated to certain ideals of analytic deviation one, Japan."J.of Math.. 23. 377-396 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Shiro Goto: "Buchsbaumness in Rees algebras associated to ideals of minimal multiplicity"J.of Algebra. 213. 604-661 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Shiro Goto, Koji Nishida: "Gatanarity in module finite algebras"Proc. of Amer, Math.Soc.. 127. 3495-3502 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Shiro Goto, Koji Nishida: "Minimal injective resolution of Cohen-Macaulay isolated singularities"Archiv der Math.. 73. 249-255 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Yukio Nakamura: "On the Buchsbaum property of associated graded rings"J.of Algebra. 209. 345-366 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 対馬龍司: "On the dimension formula for the spaces of Siegel cusp forms of half integral weight and degree two"数理解析研究所講究録. 1052. 42-57 (1998)

    • Related Report
      2000 Annual Research Report
  • [Publications] 対馬龍司: "On the dimension formula for the spaces of Jacobi forms of degree two"数理解析研究所講究録. 1103. 96-110 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Goto: "Cohen-Macaulayness versus negativity of a-invariants in Rees algebras associated to ideals odminimal muliplicity"J.Pure and Applied Algebra. 152. 93-107 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Goto,S.Iai: "Embeddings of certain graded rings into their canonical modules"Journal of Algebra. 228. 377-396 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Y.Nakamura: "Multiplicity and tight closures of Parameter ideals"明治大学理工学部研究報告. to appear.

    • Related Report
      2000 Annual Research Report
  • [Publications] Shiro Goto: "Buchsbaumness in Rees algebras associated to ideals of minimal multiplicity"Journal of Algebra. 213. 604-661 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Shiro Goto: "Cohen-Macaulayness versus negativity of a-invariants in Rees algebras associated to ideals of minimal multiplicity"Jourunal of Pure and Applied algebra. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] Yukio Nakamura: "On the Buchsbaum property of associated graded rings"Journal of Algebra. 209. 345-346 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] 対馬龍司: "On the dimension formula for the spaces of Siegel cusp forms of half integral weight and degree two" 京大数理研講究録. 1052. 42-57 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 対馬龍司: "On the dimension for the spaces of Jacobi fores of degree two" 京大数理研講究録. (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 後藤四郎: "Certenarity in module-finite algebra" Proc.Ameri.Math.Soc.(to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] Thomas Korb・中村幸男: "On the Cohen-Macaulayness of multi-Rees algebras and Rees algebras of powers of ideals" L.Math.Soc.Japan. 50. 451-467 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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