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2012 Fiscal Year Annual Research Report

幾何学的モジュライ理論の深化と理論的応用

Research Project

Project/Area Number 23224001
Research InstitutionHokkaido University

Principal Investigator

中村 郁  北海道大学, -, 名誉教授 (50022687)

Co-Investigator(Kenkyū-buntansha) 岩崎 克則  北海道大学, 理学(系)研究科(研究院), 教授 (00176538)
小野 薫  京都大学, 数理解析研究所, 教授 (20204232)
寺尾 宏明  北海道大学, 理学(系)研究科(研究院), 教授 (90119058)
翁 林  九州大学, 数理(科)学研究科(研究院), 教授 (60304002)
Project Period (FY) 2011-05-31 – 2016-03-31
Keywordsアーベル多様体 / モジュライ理論 / レベル構造 / 既約表現 / 安定性 / 整モデル / ゼータ関数 / symplectic 多様体
Research Abstract

研究代表者の目標は
(1)良い素点でのアーベル多様体のモジュライ空間のコンパクト化を記述する.
(2)悪い素点までコンパクト化(以後,整モデルと呼ぶ)を拡張する.そのために,Katz-Mazur理論を高次元に一般化する.
(3)McKay 対応の研究.
今年度は(2)への応用を見込みながら(1)について集中的に研究した.良い素点でのアーベル多様体のモジュライ空間の第二のコンパクト化からAlexeevの完備なモジュライ空間への埋め込み写像を構成し,二つの完備なモジュライ空間の構造を比較した.この過程で,第二のコンパクト化の退化多様体の任意の族にたいして,相対的自己同型群のなかにつねに平坦な半アーベル群概型が存在する,ことが証明できた.これは課題(2)の研究に必要な基本的な結果である.このほか,退化アーベル多様体のコホモロジー群の計算ができたが,これも課題(2)の研究に必要な基本的な結果である.これらの論文は現在投稿中である.これらの結果により,幾何学的不変式論の安定性によって、整モデルを構成できることがほぼ確実になった.したがって,Drinfeldレベル構造の高次元への一般化は,単に「安定なレベル構造」として定義できるであろう.これは課題(2)の整モデルのもっとも望ましい構成方法である.(3)を研究する時間はなかった.Wengはゼータ関数の研究でZagierとともに有限体上の楕円曲線の純非可換Wengゼ-タ関数のリーマン予想を成立することを証明した.小野は2つの symplectic 多様体としての直積の中にHamilton 変形で自分自身から外せない, Lagrangian torus の連続族で,その族の中の異なる2つは Hamilton 変形で移合わないものを構成した.その他は省略する.

Current Status of Research Progress
Current Status of Research Progress

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

Reason

研究はおおむね順調に進展している.ふたつの論文を完成し投稿中であるが,これらの論文で証明できた事実は、ともに整モデルの構成において重要な役割をはたす.その意味で研究は順調に進展している.

Strategy for Future Research Activity

2013年開催予定の3つ国際研究集会での連続講演が内定したので,本研究の遂行と並行して,具体例の詳しい記述も含めた入門,総合報告の論文ないし教科書を執筆することで,進行中の研究を補強することを考えていく.研究は予定通りに進展しており,当初の研究計画に沿って進める.

  • Research Products

    (45 results)

All 2013 2012 Other

All Journal Article (17 results) (of which Peer Reviewed: 16 results) Presentation (28 results) (of which Invited: 28 results)

  • [Journal Article] On G/N-Hilb of N-Hilb2013

    • Author(s)
      A. Ishii, Y. Ito and A. Nolla
    • Journal Title

      Kyoto Journal of Mathematics

      Volume: 53 Pages: 91-130

    • DOI

      10.1215/21562261-1966080

    • Peer Reviewed
  • [Journal Article] RECIPROCITY LAWS THROUGH FORMAL GROUPS2013

    • Author(s)
      O. DEMCHENKO AND A. GUREVICH
    • Journal Title

      PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

      Volume: 141 Pages: 1591-1596

    • DOI

      10.1090/S0002-9939-2012-11632-6

    • Peer Reviewed
  • [Journal Article] Cubic harmonics and Bernoulli numbers2012

    • Author(s)
      K. Iwasaki
    • Journal Title

      Journal of Combinatorial Theory, Ser. A

      Volume: 119 Pages: 1216-1234

    • DOI

      10.1016/j.jcta.2012.02.010

    • Peer Reviewed
  • [Journal Article] Toric degeneration and non-displaceable Lagrangian tori in $S^2 \times S^2$2012

    • Author(s)
      K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono
    • Journal Title

      International Mathematical Research Notices 2012

      Volume: 13 Pages: 2942-2993

    • DOI

      10.1093/imrn/rnr128

    • Peer Reviewed
  • [Journal Article] Symplectic fillings of links of quotient surface singularities2012

    • Author(s)
      M. Bhupal and K. Ono
    • Journal Title

      Nagoya Mathematical Journal

      Volume: 207 Pages: 1-45

    • DOI

      10.1215/00277630-1630014

    • Peer Reviewed
  • [Journal Article] Lagrangian Floer theory on compact toric manifolds: survey2012

    • Author(s)
      K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono
    • Journal Title

      Surveys in Differential Geometry XVII In Memory of C. C. Hsiung edited by Huai-Dong Cao and Shing-Tung Yau

      Volume: 17 Pages: 229-298

  • [Journal Article] The Shi arrangement of the type $D_\ell$2012

    • Author(s)
      R. Gao, D. Pei, H. Terao
    • Journal Title

      Proceedings of the Japan Academy, Ser. A Mathematical Sciences

      Volume: 88 Pages: 41-45

    • DOI

      10.3792/pjaa.88.41

    • Peer Reviewed
  • [Journal Article] The Shi arrangements and the Bernoulli polynomials2012

    • Author(s)
      D. Suyama, H. Terao
    • Journal Title

      Bulletin London Mathematical Society

      Volume: 44 Pages: 563-570

    • DOI

      10.1112/blms/bdr118

    • Peer Reviewed
  • [Journal Article] Arrangements stable under the Coxeter groups2012

    • Author(s)
      H. Kamiya, A. Takemura, H. Terao
    • Journal Title

      ``Configuration Spaces: Geometry, Topology and Combinatorics,'' Proceedings of a special period atthe De Giorgi Center, Scuola Normale in Pisa, May - June, 2010

      Volume: なし Pages: 327-354

    • Peer Reviewed
  • [Journal Article] Equivariant multiplicities of Coxeter arrangements and invariant bases2012

    • Author(s)
      T. Abe, H. Terao, A. Wakamiko
    • Journal Title

      Advances in Mathematics

      Volume: 230 Pages: 2364-2377

    • DOI

      10.1016/j.aim.2012.04.015

    • Peer Reviewed
  • [Journal Article] A note on Chern-Schwartz-MacPherson class2012

    • Author(s)
      T. Ohmoto
    • Journal Title

      IRMA Lectures in Mathematics and Theoretical Physics, European Mathematical Society

      Volume: 20 Pages: 117-131

    • DOI

      10.4171/118-1/7

    • Peer Reviewed
  • [Journal Article] Linear dependence of certain L-values of half-integral weight modular forms2012

    • Author(s)
      H. Katsurada and Y. Mizuno
    • Journal Title

      Journal of the London Mathematical Society

      Volume: 85 Pages: 455-471

    • DOI

      10.1112/jlms/jdr057

    • Peer Reviewed
  • [Journal Article] Foliations on the moduli space of rank two connections on the projective line minus four points2012

    • Author(s)
      F. Loray, M-H. Saito, C. Simpson
    • Journal Title

      Seminaires et Congres

      Volume: 27 Pages: 115-168

    • Peer Reviewed
  • [Journal Article] Gauss sums on finite groups2012

    • Author(s)
      K. Shinoda (with Y. Gomi and T. Maeda)
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: 35 Pages: 165-179

    • DOI

      10.3836/tjm/1342701349

    • Peer Reviewed
  • [Journal Article] Arrangements, multiderivations, and adjoint quotient map of type ADE2012

    • Author(s)
      M. Yoshinaga
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: 62 Pages: 523-552

    • Peer Reviewed
  • [Journal Article] Quintic surface over p-adic local fields with infinite p-primarytorsion in the Chow group of 0-cycles2012

    • Author(s)
      M. Asakura
    • Journal Title

      Regulators, Contemporary Mathematics

      Volume: 571 Pages: 1-17

    • DOI

      10.1090/conm/571

    • Peer Reviewed
  • [Journal Article] The category O for a general Coxeter system2012

    • Author(s)
      N. Abe
    • Journal Title

      Journal of Algebra

      Volume: 367 Pages: 1-25

    • DOI

      10.1016/j.jalgebra.2012.06.002

    • Peer Reviewed
  • [Presentation] Non-displaceable Lagrangian submanifolds2013

    • Author(s)
      Kaoru Ono
    • Organizer
      代数・解析・幾何セミナー
    • Place of Presentation
      鹿児島大学(鹿児島市)
    • Year and Date
      20130220-20130220
    • Invited
  • [Presentation] Lagrangian Floer theory (two lectures)2013

    • Author(s)
      Kaoru Ono
    • Organizer
      UK-Japan Mathematical Forum on Algebraic and Symplectic Geometry
    • Place of Presentation
      Keio University(横浜市)
    • Year and Date
      20130125-20130126
    • Invited
  • [Presentation] Primitive derivations, Shi arrangements and Bernoulli polynomials2012

    • Author(s)
      Hiroaki Terao
    • Organizer
      2012 Shanghai Conference on Algebraic Combinatorics
    • Place of Presentation
      Shanghai Jiao Tong University(上海,中国)
    • Year and Date
      20120820-20120820
    • Invited
  • [Presentation] Non-Abelian zeta functions2012

    • Author(s)
      Lin Weng
    • Organizer
      p-adic modular forms and Arithmetic
    • Place of Presentation
      UCLA(ロサンゼルス, 米国)
    • Year and Date
      20120623-20120623
    • Invited
  • [Presentation] Lagrangian Floer theory and mirror symmetry on compact toric manifolds2012

    • Author(s)
      Kaoru Ono
    • Organizer
      Toric Geometry
    • Place of Presentation
      Mathematisches Forschungsinstitut(オーバーボールバッハ,ドイツ)
    • Year and Date
      20120415-20120421
    • Invited
  • [Presentation] Compactification of the moduli of abelian varieties

    • Author(s)
      Iku Nakamura
    • Organizer
      Seminar on arithmetic geometry
    • Place of Presentation
      McGill University(モントリオール,カナダ)
    • Invited
  • [Presentation] Compactification of the moduli of abelian varieties and Closed immersions of $SQ_{g,K}$ into Alexeev's moduli

    • Author(s)
      Iku Nakamura
    • Organizer
      KIAS colloquium
    • Place of Presentation
      Korea Institute of Advanced Study (ソウル,韓国)
    • Invited
  • [Presentation] Compactification of the moduli of abelian varieties and Closed immersions of $SQ_{g,K}$ into Alexeev's moduli

    • Author(s)
      Iku Nakamura
    • Organizer
      Workshop on arithmetic geometry
    • Place of Presentation
      Kyushu University(福岡市)
    • Invited
  • [Presentation] Compactification of the moduli of abelian varieties

    • Author(s)
      Iku Nakamura
    • Organizer
      第11回岡シンポジウム
    • Place of Presentation
      奈良女子大学(奈良市)
    • Invited
  • [Presentation] Closed immersions of $SQ_{g,K}^{\rm{toric}}$ into Alexeev's moduli

    • Author(s)
      Iku Nakamura
    • Organizer
      Arithmetic and Algebraic Geometry 2013
    • Place of Presentation
      University of Tokyo(東京都)
    • Invited
  • [Presentation] 多面体と関数方程式

    • Author(s)
      岩崎克則, 蓮井美希
    • Organizer
      函数方程式サマーセミナー 2012
    • Place of Presentation
      KKR 蔵王白銀荘(山形市)
    • Invited
  • [Presentation] 超幾何和の超幾何性

    • Author(s)
      岩崎克則
    • Organizer
      琉球超幾何セミナー
    • Place of Presentation
      琉球大学(中頭郡西原町)
    • Invited
  • [Presentation] 超幾何和とサイン・サイン

    • Author(s)
      岩崎克則
    • Organizer
      超幾何方程式研究会 2013
    • Place of Presentation
      神戸大学(神戸市)
    • Invited
  • [Presentation] Floer theory for Lagrangian submanifolds

    • Author(s)
      Kaoru Ono
    • Organizer
      The First Symposium of the Institute for Basic Science in 2012
    • Place of Presentation
      Lotte Hotel Seoul(ソウル,韓国)
    • Invited
  • [Presentation] Spectral invariants and Lagrangian Floer theory

    • Author(s)
      Kaoru Ono
    • Organizer
      Symplectic Geometry and Mathematical Physics
    • Place of Presentation
      Beijing International Center for Mathematical Research(北京,中国)
    • Invited
  • [Presentation] Non-displaceable Lagrangian tori

    • Author(s)
      Kaoru Ono
    • Organizer
      Interactions between Algebra and Dynamics in Sympletic Topology
    • Place of Presentation
      Technion(ハイファ,イスラエル)
    • Invited
  • [Presentation] Lagrange 部分多様体の displaceability 及び displacement energy について

    • Author(s)
      小野 薫
    • Organizer
      第59回トポロジーシンポジウム
    • Place of Presentation
      佐賀大学(佐賀市)
    • Invited
  • [Presentation] Spectral invariant and Lagrangian Floer theory

    • Author(s)
      Kaoru Ono
    • Organizer
      The conference of Mirror Symmetry and Related Topics
    • Place of Presentation
      Kunming University of Science and Technology(昆明,中国)
    • Invited
  • [Presentation] Hamilton 微分同相写像群のスペクトル不変量とその応用

    • Author(s)
      小野 薫
    • Organizer
      日本数学会秋季総合分科会 企画特別講演
    • Place of Presentation
      九州大学(福岡市)
    • Invited
  • [Presentation] Height basis of Weyl arrangements

    • Author(s)
      Hiroaki Terao
    • Organizer
      "Arrangements in Pyrenees" School on hyperplane arrangements and related topics
    • Place of Presentation
      Universite de Pau (ポー,フランス)
    • Invited
  • [Presentation] Primitive derivations, Shi arrangements, Bernoulli polynomials and the height-free conjecture

    • Author(s)
      Hiroaki Terao
    • Organizer
      FPSAC '12 (The 24th International Conference on Formal Power Series and Algebraic Combinatorics)
    • Place of Presentation
      Nagoya University(名古屋市)
    • Invited
  • [Presentation] Primitive derivations, Shi arrangements and Bernoulli polynomials

    • Author(s)
      Hiroaki Terao
    • Organizer
      colloquium
    • Place of Presentation
      Northeastern Normal University(長春,中国)
    • Invited
  • [Presentation] Logarithmic derivation modules of Shi arrangements associated with root systems

    • Author(s)
      Hiroaki Terao
    • Organizer
      Combinatorial Representation Theory and Related Topics
    • Place of Presentation
      RIMS, Kyoto University(京都市)
    • Invited
  • [Presentation] Logarithmic derivation modules of Shi arrangements associated with root systems

    • Author(s)
      Hiroaki Terao
    • Organizer
      Workshop on "Free Divisors and Differential Equations"
    • Place of Presentation
      東京農工大学(小金井市)
    • Invited
  • [Presentation] Logarithmic derivation modules of Shi arrangements associated with root systems

    • Author(s)
      Hiroaki Terao
    • Organizer
      Conference on "Hyperplane Arrangements: Combinatorial and Geometric Aspects"
    • Place of Presentation
      Ruhr-Universitat Bochum(ボーフム,ドイツ)
    • Invited
  • [Presentation] Tamagawa number conjecture and new zeta functions for Riemann surfaces

    • Author(s)
      Lin Weng
    • Organizer
      Symposium on arithmetic and geometry
    • Place of Presentation
      Kyushu University(福岡市)
    • Invited
  • [Presentation] Non-abelian zeta functions for elliptic curves and their zeros

    • Author(s)
      Lin Weng
    • Organizer
      Conference on L-Functions
    • Place of Presentation
      Shineville Resort Jeju (済州, 韓国)
    • Invited
  • [Presentation] Higher rank zeta function and Riemann Hypothesis for elliptic curves

    • Author(s)
      Lin Weng
    • Organizer
      Arithmetic and Algebraic Geometry2013
    • Place of Presentation
      University of Tokyo(東京都)
    • Invited

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Published: 2015-05-28  

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