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2012 Fiscal Year Annual Research Report

数学・物理学の様々な局面に現れるモジュラー・準モジュラー形式と多重ゼータ値の研究

Research Project

Project/Area Number 23340010
Research InstitutionKyushu University

Principal Investigator

金子 昌信  九州大学, 数理(科)学研究科(研究院), 教授 (70202017)

Project Period (FY) 2011-04-01 – 2016-03-31
Keywordsモジュラー形式 / 準モジュラー形式 / 多重ゼータ値 / 共形場理論 / 位相不変量
Research Abstract

代表者は永友と共同して,二次元共形場理論の指標の分類を行った Mathurらの結果を頂点作用素代数の立場,および代表者が以前 Zagier と研究した保型微分方程式を道具として構成し直し,解に対数項が現れる場合に一般化した. それを基に,ある種の有理頂点作用素代数の非存在に関する結果を得た.またこの保型微分方程式のモジュラーでない解について,ハワイ大の Guerzhoy との議論から,mixed modular form となっている解が発見された.これは近年の同関数の研究の飛躍的進展に鑑みても非常に興味深いもので,引き続き共同研究を行いたい.鍬田(中央大)より専門的知識の提供を受けて始まった,古典的なヤコビのテータ関係式をレベル5や7の場合に考察し,楕円曲面の数論へ応用する試みは,レベル6の場合に予想とは異なる現象に行き当たり,現在この場合を含む理論を模索中である.いずれにせよ今後の発展が大いに期待される.有限多重ゼータ値について Zagier と直接議論をする機会を得,実数上の場合での極めて興味深い類似物を発見,その性質を共同で研究中である.村上による体積公式に関連する研究,細野によるミラー対称性に関する研究,樋上によるK3曲面の楕円種数の研究やムーンシャイン現象に関する研究,田口による今井の定理の一般化とその岩澤理論への応用に関する研究にも一定の進展があった.

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

申請書に記した五つの観点のいずれにおいても,一定の研究の進展があったため.

Strategy for Future Research Activity

これまで通り,連携研究者や共同研究者との連絡をとりながら,申請書の研究目的に沿った研究を推進する.今年度は特に海外共同研究者の Zagier 教授のほか,Brown 教授や Pellarin教授らとの議論の機会を設け,多重ゼータ値および extremal modular form に関する議論を行い今後の更なる発展につなげる.

  • Research Products

    (39 results)

All 2013 2012 Other

All Journal Article (11 results) (of which Peer Reviewed: 11 results) Presentation (25 results) (of which Invited: 23 results) Book (1 results) Remarks (2 results)

  • [Journal Article] Volume of a doubly truncated hyperbolic tetrahedron2013

    • Author(s)
      Alexander Kolpakov, Jun Murakami
    • Journal Title

      Aequationes Math.

      Volume: 85 Pages: 449-463

    • DOI

      10.1007/s00010-012-0153-y

    • Peer Reviewed
  • [Journal Article] Modular Forms and Second Order Ordinary Differential Equations: : Applications to Vertex Operator Algebras2013

    • Author(s)
      Masanobu Kaneko, Kiyokazu Nagatomo, Yuichi Sakai
    • Journal Title

      Letters in Mathematical Physics

      Volume: 103(4) Pages: 439-453

    • DOI

      10.1007/s11005-012-0602-5

    • Peer Reviewed
  • [Journal Article] Introduction to Arithmetic Mapping Class Groups2013

    • Author(s)
      Makoto Matsumoto
    • Journal Title

      IAS-Park City Mathematical Insutute Lecture Note Series

      Volume: - Pages: 317-351

    • Peer Reviewed
  • [Journal Article] Mirror Symmetry and Projective Geometry of Reye Congruences I2013

    • Author(s)
      Shinobu Hosono, Hiromich Takagi
    • Journal Title

      Journal Algebraic Geometry

      Volume: - Pages: -

    • Peer Reviewed
  • [Journal Article] On SL(2, C) quantum 6j-symbol and its relation to the hyperbolic volume2013

    • Author(s)
      Francesco Costantino, Jun Murakami
    • Journal Title

      Quantum Topology

      Volume: - Pages: -

    • Peer Reviewed
  • [Journal Article] A generalization of a theorem of Imai and its applications to Iwasawa theory2013

    • Author(s)
      Yusuke Kubo, Yuichiro Taguchi
    • Journal Title

      Mathematische Zeitschrift

      Volume: - Pages: -

    • Peer Reviewed
  • [Journal Article] On Fourier coefficients of some meromorphic modular forms2012

    • Author(s)
      Yutaro Honda, Masanobu Kaneko
    • Journal Title

      Bulletin of the Korean Mathematical Society

      Volume: 49(6) Pages: 1349-1357

    • DOI

      10.4134/BKMS.2012.49.6.1349

    • Peer Reviewed
  • [Journal Article] Volume formulas for a spherical tetrahedron2012

    • Author(s)
      Jun Murakami
    • Journal Title

      Proc. Amer. Math. Soc.

      Volume: 140 Pages: 3289-3295

    • DOI

      10.1090/S0002-9939-2012-11182-7

    • Peer Reviewed
  • [Journal Article] Twisted elliptic genus for K3 and Borcherds product2012

    • Author(s)
      Tohru Eguchi, Kazuhiro Hikami
    • Journal Title

      Letters in Mathematical Physics

      Volume: 102 Pages: 203-222

    • DOI

      10.1007/s11005-012-0569-2

    • Peer Reviewed
  • [Journal Article] N=2 Moonshine2012

    • Author(s)
      Tohru Eguchi, Kazuhiro Hikami
    • Journal Title

      Physics Letters B

      Volume: 717 Pages: 266-273

    • DOI

      10.1016/j.physletb.2012.09.037

    • Peer Reviewed
  • [Journal Article] Some Remarks on Pseudo-Trace Functions for Orbifold Models Associated with Symplectic Fermions2012

    • Author(s)
      Yusuke Arike, Kiyokazu Nagatomo
    • Journal Title

      International Journal of Mathematics

      Volume: 24 Pages: -

    • DOI

      10.1142/S0129167X13500080

    • Peer Reviewed
  • [Presentation] On the logarithmic knot invariants and the hyperbolic volume2013

    • Author(s)
      村上 順
    • Organizer
      Low dimensional topology and number theory V
    • Place of Presentation
      ソフトリサーチパーク(福岡)
    • Year and Date
      20130314-20130314
    • Invited
  • [Presentation] On the Complex Volume of Knots2013

    • Author(s)
      樋上 和弘
    • Organizer
      Bethe Ansatz, quantum groups and beyond
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      20130309-20130309
    • Invited
  • [Presentation] On congruences of Galois representations of global fields2013

    • Author(s)
      Yuichiro Taguchi
    • Organizer
      The Arithmetic of Function Fields and Related Topics
    • Place of Presentation
      Haeundae Riviera Hotel (釜山)
    • Year and Date
      20130218-20130218
    • Invited
  • [Presentation] Fourier-Mukai partners, mirror symmetry, and BPS numbers2013

    • Author(s)
      細野 忍
    • Organizer
      Seminar weeks on Calabi-Yau manifolds, mirror symmetry, and derived categories
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      20130214-20130214
    • Invited
  • [Presentation] Cluster algebra and complex volume of 2-bridge knots2013

    • Author(s)
      Kazuhiro Hikami
    • Organizer
      Facets of integrability: random patterns, stochastic processes, hydrodynamics, gauge theories and condensed matter systems
    • Place of Presentation
      Simons Center for Geometry and Physics, USA
    • Year and Date
      20130125-20130125
    • Invited
  • [Presentation] Logarithmic invariants of knots in three manifolds2013

    • Author(s)
      村上 順
    • Organizer
      Exact results in SUSY gauge theories and integrable systems
    • Place of Presentation
      立教大学
    • Year and Date
      20130112-20130112
    • Invited
  • [Presentation] mod p 多重ゼータ値について2012

    • Author(s)
      金子 昌信
    • Organizer
      関西多重ゼータ研究会
    • Place of Presentation
      大阪工業大学
    • Year and Date
      20121117-20121117
    • Invited
  • [Presentation] Calabi-Yau threefolds of Reye congruences2012

    • Author(s)
      Shinobu Hosono
    • Organizer
      Everytopic seminar
    • Place of Presentation
      Brandeis University, USA
    • Year and Date
      20121116-20121116
    • Invited
  • [Presentation] Mirror symmetry of determinantal quintics in P42012

    • Author(s)
      Shinobu Hosono
    • Organizer
      Physics seminar
    • Place of Presentation
      Brandeis University, USA
    • Year and Date
      20121113-20121113
    • Invited
  • [Presentation] セール微分と楕円曲線のモジュラー一意化2012

    • Author(s)
      金子 昌信
    • Organizer
      香川セミナー
    • Place of Presentation
      香川大学
    • Year and Date
      20121111-20121111
    • Invited
  • [Presentation] Double zeta values and modular forms2012

    • Author(s)
      Masanobu Kaneko
    • Organizer
      Modular form seminar
    • Place of Presentation
      National Center for Theoretical Sciences, Taiwan
    • Year and Date
      20121107-20121107
    • Invited
  • [Presentation] The Ramanujan-Serre differential operators and certain elliptic curves2012

    • Author(s)
      Masanobu Kaneko
    • Organizer
      Number theory seminar
    • Place of Presentation
      National Center for Theoretical Sciences, Taiwan
    • Year and Date
      20121107-20121107
    • Invited
  • [Presentation] セール微分と楕円曲線のモジュラー一意化2012

    • Author(s)
      金子 昌信
    • Organizer
      愛知数論セミナー
    • Place of Presentation
      名古屋工業大学
    • Year and Date
      20121027-20121027
    • Invited
  • [Presentation] On congruences of Galois representations of function fields2012

    • Author(s)
      Yuichiro Taguchi
    • Organizer
      L-functions and Arithmetic
    • Place of Presentation
      延世大学校(ソウル)
    • Year and Date
      20121023-20121026
    • Invited
  • [Presentation] Rational torsions points of abelian varieties over a large extension of a local field2012

    • Author(s)
      Yuichiro Taguchi
    • Organizer
      Rational points on curves: A p-adic and computational computational perspective
    • Place of Presentation
      Mathematical Institute, University of Oxford
    • Year and Date
      20120925-20120925
    • Invited
  • [Presentation] Quantum 6j-symbols for non-integral highest weight representations of U_q(sl_2) at root of unity2012

    • Author(s)
      村上 順
    • Organizer
      日本数学会秋期総合分科会
    • Place of Presentation
      九州大学
    • Year and Date
      20120919-20120919
  • [Presentation] 結び目の量子不変量とその応用2012

    • Author(s)
      村上 順
    • Organizer
      Summer School 数理物理 2012 結び目の数理と物理
    • Place of Presentation
      東京大学
    • Year and Date
      20120907-20120909
    • Invited
  • [Presentation] Logarithmic invariants for knots in three manifolds2012

    • Author(s)
      村上 順
    • Organizer
      研究集会 ホップ代数と量子群--応用の可能性
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      20120903-20120903
    • Invited
  • [Presentation] Decomposition of elliptic genera in terms of superconformal characters2012

    • Author(s)
      Kazuhiro Hikami
    • Organizer
      Low-Dimensional Topology and Number Theory
    • Place of Presentation
      MFO, Germany
    • Year and Date
      20120831-20120831
    • Invited
  • [Presentation] Quantum invariants of knots and the hyperbolic volume2012

    • Author(s)
      Jun Murakami
    • Organizer
      The 29th international colloquium on group-theoretical methods in physic
    • Place of Presentation
      Chern Institute of Mathematics, 天津, 中国
    • Year and Date
      20120824-20120824
    • Invited
  • [Presentation] Mirror symmetry of some determinantal quintics2012

    • Author(s)
      Shinobu Hosono
    • Organizer
      Mirror Symmetry and Related Topics
    • Place of Presentation
      Kunming University of Science and Technology,China
    • Year and Date
      20120822-20120822
    • Invited
  • [Presentation] Superconformal algebra, mock theta functions, and moonshine2012

    • Author(s)
      Kazuhiro Hikami
    • Organizer
      Hypergeometric Series and Their Generalizations in Algebra, Geometry, Number Theory and Physics
    • Place of Presentation
      Institut Henri Poincare, France
    • Year and Date
      20120529-20120529
    • Invited
  • [Presentation] The Ramanujan-Serre differential operators and certain elliptic curves2012

    • Author(s)
      金子 昌信
    • Organizer
      Workshop on Modular Forms
    • Place of Presentation
      九州大学伊都キャンパス
    • Year and Date
      20120507-20120507
  • [Presentation] Moonshine from Character Decompositions of Elliptic Genera2012

    • Author(s)
      樋上 和弘
    • Organizer
      Workshop on Modular Forms
    • Place of Presentation
      九州大学
    • Year and Date
      20120507-20120507
    • Invited
  • [Presentation] Markoff数について2012

    • Author(s)
      金子 昌信
    • Organizer
      整数論・力学系セミナー
    • Place of Presentation
      弘前大学理工学部
    • Year and Date
      20120427-20120427
    • Invited
  • [Book] 線形代数とMathematica2012

    • Author(s)
      谷口 義治,永友 清和
    • Total Pages
      286
    • Publisher
      牧野書店
  • [Remarks] 金子昌信 論文・著書等

    • URL

      http://www2.math.kyushu-u.ac.jp/~mkaneko/paper.html

  • [Remarks] 金子昌信 講演等

    • URL

      http://www2.math.kyushu-u.ac.jp/~mkaneko/lecture.html

URL: 

Published: 2014-07-24  

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