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Study on interrelations among quadratic forms, automorphic forms and related various zeta-functions

Research Project

Project/Area Number 15540049
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

AOKI Noboru  Rikkyo University, Department of Math, Professor, 理学部, 教授 (30183130)

Co-Investigator(Kenkyū-buntansha) SATO Fumihiro  Rikkyo University, Department of Math, Professor, 理学部, 教授 (20120884)
FUJII Akio  Rikkyo University, Department of Math, Professor, 理学部, 教授 (50097226)
OHSUGI Hidefumi  Rikkyo University, Dept. of Math, Associate Professor, 理学部, 助教授 (80350289)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2005: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2003: ¥1,600,000 (Direct Cost: ¥1,600,000)
KeywordsGross conjecture / elliptic curve / Tate-Shafarevich group / Fermat curve / Jacobi sum / prehomogenous vector space / Riemann zeta function / Groebner basis / アーベル多様体 / テイト・シャファレヴィッチ群 / 二次形式 / 局所密度 / Eisenstein級数 / トーリック多様体 / Tate-Safarevich群 / ゼータ関数 / トーリック環 / 保型形式 / トーリックイデアル / L関数
Research Abstract

I studied, with Ki-Seng Tan (Taiwan) and Lee Joongul (Korea), a generalization of Tate's refinement of the Gross conjecture on the Stickelberger element obtained from special values of L-functions of number fields and function fields of one-variable over finite fields, and formulated a new conjecture. As a result, I succeeded in proving the generalized conjecture for elementary abelian extensions under certain conditions. Moreover, I proved Tate's original refinement in the case of number fields. I also studied some arithmetic properties of elliptic curves and abelian varieties. In particular, I proved unboundedness of the 3-part of the Tate-Shafarevich group of elliptic curves over the rational number field. Although this has been already known by Cassels, I proved it for a new family of elliptic curves. Furthermore, I proved the Hodge conjecture for Catalan curves, and this is one of my results in the study of arithmetic properties of jacobian varieties of algebraic curves. As a related topic, expecting application for congruent zeta functions of Fermat curves, I studied purity problem of Jacobi sums and solved negatively a problem posed by Evans.
Sato studied the local density of symmetric matrices over p-adic integers and extended a result of Schulze-Pillot to higher level cases. Fujii studied the behavior of the argument of the Riemann zeta function on the critical line. Ohsugi studied Gorensteinness of toric rings.

Report

(4 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (30 results)

All 2006 2005 2004 Other

All Journal Article (22 results) Book (2 results) Publications (6 results)

  • [Journal Article] Multiplicity One Property and the Decomposition of b-Functions2006

    • Author(s)
      F.Sato, K.Sugiyama
    • Journal Title

      Intern. J. Math. 17

      Pages: 195-229

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Multiplicity One Property and the Decomposition of b-Functions2006

    • Author(s)
      F.Sato, K.Sugiyama
    • Journal Title

      Intern.J.Math.

      Pages: 195-229

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Multiplicity one property and the decomposition of b-functions2006

    • Author(s)
      Fumihiro Sato, Kazunari Sugiyama
    • Journal Title

      International Journal of Mathematics 17

      Pages: 195-229

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the Farey series and the Riemann Hypothesis2005

    • Author(s)
      A.Fujii
    • Journal Title

      Commentarii Mat. Univ. Soncty Pauli 54

      Pages: 211-235

    • NAID

      110007689256

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the Farey series and the Riemann Hypothesis,2005

    • Author(s)
      Akio Fujii
    • Journal Title

      Commentarii Mat.Univ.Sancti Pauli, 2005 54

      Pages: 211-235

    • NAID

      110007689256

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The Siegel series and spherical functions on O(2n)/(O(n)×O(n))2005

    • Author(s)
      Fumihiro Sato, Yumiko Hironaka
    • Journal Title

      Automorphic Forms and Zeta Functions, Proceedings of the conference in memory of Tsuneo Arakawa 論文集

      Pages: 150-169

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The $h$-vector of a Gorenstein toric ring of a compressed polytope2005

    • Author(s)
      H.Ohsugi, T.Hibi
    • Journal Title

      Electron.J.Combin., 11(2004/06), no.2, Note 4, 5pp.(electronic) 11

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Fourier coefficients of Eisenstein series of GL(n), local densities of square matrices and subgroups of finite abelian groups2005

    • Author(s)
      Fumihiro Sato
    • Journal Title

      Commentarii Mathematici Universitatis Sancti Pauli 54(印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On Tate's refinement for a conjecture of Gross and its generalization2004

    • Author(s)
      N.Aoki
    • Journal Title

      J. Theorie des Nombres de Bordeaux 16

      Pages: 457-486

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The Hodge conjecture for the jacobian varieties of generalized Catalan curves2004

    • Author(s)
      N.Aoki
    • Journal Title

      Tokyo J. Math. 27

      Pages: 313-335

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A finiteness theorem on pure Gauss sums2004

    • Author(s)
      N.Aoki
    • Journal Title

      Comment. Math. Univ. Sancti Pauli 53

      Pages: 145-168

    • NAID

      110007689265

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the Tate-Shafarevich groups of semistable elliptic curves with a rational 3-torsion2004

    • Author(s)
      N.Aoki
    • Journal Title

      Acta Arithmetica 112

      Pages: 209-227

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On Tate's refinement for a conjecture of Gross and its generalization2004

    • Author(s)
      N.Aoki
    • Journal Title

      J.Theorie des Nombres de Bordeaux 16

      Pages: 457-486

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The Hodge conjecture for the jacobian varieties of generalized Catalan curves2004

    • Author(s)
      N.Aoki
    • Journal Title

      Tokyo J.Math. 27

      Pages: 313-335

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A finiteness theorem on pure Gauss sums2004

    • Author(s)
      N.Aoki
    • Journal Title

      Comment.Math.Univ.Sancti Pauli 53

      Pages: 145-168

    • NAID

      110007689265

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the Tate-Shafarevich groups of semistable elliptic curves with a rational 3-torsion2004

    • Author(s)
      Noboru Aoki
    • Journal Title

      Acta Arithmetica 112

      Pages: 209-227

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The Hodge conjecture for the jacobian varieties of generalized Catalan curves2004

    • Author(s)
      Noboru Aoki
    • Journal Title

      Tokyo Journal of Mathematics 27

      Pages: 313-335

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A finiteness theorem on pure Gauss sums2004

    • Author(s)
      Noboru Aoki
    • Journal Title

      Commentarii Mathematici Universitatis Sancti Pauli 53

      Pages: 145-168

    • NAID

      110007689265

    • Related Report
      2004 Annual Research Report
  • [Journal Article] An explicit estimate in the theory of Distribution of the zeros of the Riemann zeta function2004

    • Author(s)
      Akio Fujii
    • Journal Title

      Commentarii Mathematici Universitatis Sancti Pauli 53

      Pages: 85-114

    • NAID

      110007689261

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the Farey series and the Riemann Hypothesis

    • Author(s)
      A.Fujii
    • Journal Title

      Commentarii Mat.Univ.Sancti Pauli 54

      Pages: 211-235

    • NAID

      110007689256

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Special simplices and Gorenstein toric rings

    • Author(s)
      H.Ohsugi, T.Hibi
    • Journal Title

      J.Combinatorial Theory, Series A (in press)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Prestable ideals and Sagbi bases

    • Author(s)
      Hidehumi Ohsugi, T.Hibi
    • Journal Title

      Mathematica Scandinavica (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Book] Automorphic Forms and Zeta Functions2005

    • Author(s)
      Fumihiro Sato 他3名 編
    • Total Pages
      400
    • Publisher
      World Scientific
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Book] Automorphic Forms and Zeta Functions, Proceedings of the conference in memory of Tsuneo Arakawa2005

    • Author(s)
      Fumihiro Sato他3名編
    • Total Pages
      400
    • Publisher
      World Scientific
    • Related Report
      2005 Annual Research Report
  • [Publications] N.Aoki: "On the generalized Gross-Tate conjecture for elementary abelian 2-extensions"Commentarii Mathematici Universitatis Sancti Pauli. 52・2. 197-206 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] F.Sato: "関数等式の様々な一般化"第10回整数論サマースクール「概均質ベクトル空間」報告集. 91-113 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.Fujii: "On the zeros of the Riemann zeta function II"Commentarii Mathematici Universitatis Sancti Pauli. 52・2. 165-190 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Ohsugi, T.Hibi: "Normalized volumes of configurations related with root systems and complete bipartite graphs"Discrete Mathematics. 268. 217-242 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Arakawa, S.Boecherer: "Vanishing of certain spaces of elliptic modular forms and some applications"Journal fuer die reine und angewandte Mathematik. 559. 25-51 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Arakawa: "Shimura correspondence for Maass wave forms and Selberg zeta functions"京都大学数理解析研究所講究録. 1338. 1-14 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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