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A local and global study of arithmetic varieties determined by arithmetic differential equations

Research Project

Project/Area Number 21654001
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeSingle-year Grants
Research Field Algebra
Research InstitutionTohoku University

Principal Investigator

TSUZUKI Nobuo  東北大学, 大学院・理学研究科, 教授 (10253048)

Co-Investigator(Renkei-kenkyūsha) YAMAUCHI Takuya  鹿児島大学, 教育学部, 准教授 (90432707)
Project Period (FY) 2009 – 2011
Project Status Completed (Fiscal Year 2011)
Budget Amount *help
¥3,040,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥240,000)
Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2010: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2009: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywords数論幾何、超幾何微分方程式 / カラビ・ヤウ多様体 / 数論的多様体族 / 数論的半安定族 / モノドロミー・重みスペクトル系列 / 保型性 / 超幾何微分方程式 / 超幾何カラビ・ヤウ多様体族 / コホモロジー / ベッチ数 / 超幾何の保型性 / 超幾何局所系 / 数論的コンパクト化 / 高次ガウス超幾何関数 / ルジャンドル族 / ガウス超幾何カラビ・ヤウ族 / 対数的増大度 / フロベニウス傾き / ドウォーク予想
Research Abstract

We constructed an arithmetic family of Calabi-Yau varieties on the projective line whose period integral is a generalized hypergeometric function, and determined the relative cohomology of Betti, de Rham, etale and crystalline realizations. This family is a higher dimensional version of Legenedre's family of elliptic curves. The family of Calabi-Yau is obtained by a desingularization which is given explicitly, and has a semistable degeneration at 0. The cohomologies can be calculated by applying weight spectral sequences. Moreover, we have a result on modularity for rational fibers.

Report

(4 results)
  • 2011 Annual Research Report   Final Research Report ( PDF )
  • 2010 Annual Research Report
  • 2009 Annual Research Report
  • Research Products

    (43 results)

All 2012 2011 2010 2009

All Journal Article (9 results) (of which Peer Reviewed: 9 results) Presentation (34 results)

  • [Journal Article] Overholonomicity of overconvergent F-isocrystals over smooth varieties2012

    • Author(s)
      Daniel Caro, Nobuo Tsuzuki
    • Journal Title

      Annals of Math

      Volume: (印刷中)

    • Related Report
      2011 Annual Research Report 2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] The rank of Jacobian varieties over the maximal abelian extensions of number fields : Toward Frey-Jarden's conjecture2012

    • Author(s)
      Fumio Sairaiji, Takuya Yamauchi
    • Journal Title

      The journal of the Canadian Mathematical Bulletin

      Volume: (印刷中)

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Log-growth filtration and Frobenius slope filtration of F-isocrystals at the generic and special points2011

    • Author(s)
      Bruno Chiarellotto, Nobuo Tsuzuki
    • Journal Title

      Documenta Math

      Volume: 16 Pages: 33-69

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Log-growth filtration and Frobenius slope filtration of $F$-isocrystals at the generic and special points2011

    • Author(s)
      Bruno Chiarellotto, Nobuo Tsuzuki
    • Journal Title

      Documenta Math

      Volume: 16 Pages: 33-69

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A generalization of Sen-Brinon's theory2010

    • Author(s)
      Takuya Yamauchi
    • Journal Title

      Manuscripta Math

      Volume: 133 Pages: 327-346

    • Related Report
      2011 Final Research Report 2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Log-growth filtration and Frobenius slope filtration of F-isocrystals at the generic and special points2010

    • Author(s)
      Bruno Chiarellotto, Nobuo Tsuzuki
    • Journal Title

      Documenta Math. (未定, 掲載確定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The rank of Jacobian varieties over the maximal abelian extensions of number fields : Toward Frey-Jarden's conjecture2010

    • Author(s)
      Takuya Yamauchi
    • Journal Title

      The journal of the Canadian Mathematical Bulletin (未定, 掲載確定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Logarithmic growth and Frobenius filtrations for solutions of p-adic differential equations2009

    • Author(s)
      Bruno Chiarellotto, Nobuo Tsuzuki
    • Journal Title

      J. Inst. Math. Jussieu

      Volume: 8 Pages: 465-505

    • Related Report
      2011 Final Research Report
    • Peer Reviewed
  • [Journal Article] Logarithmic growth and Frobenius filtrations for solutions of p-adic differential equations2009

    • Author(s)
      Bruno Chiarellotto, Nobuo Tsuzuki
    • Journal Title

      J.Inst.Math.Jussieu 8

      Pages: 465-505

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Presentation] L-function of some Siegel threefold of low level : non-neat case2012

    • Author(s)
      山内卓也
    • Organizer
      Workshop on p-adic arithmetic geometry and motives
    • Place of Presentation
      東北大学
    • Year and Date
      2012-01-25
    • Related Report
      2011 Annual Research Report 2011 Final Research Report
  • [Presentation] Arithmetic families of Calabi-Yau varieties having a generalized hypergeometric function as a period integral2011

    • Author(s)
      都築暢夫
    • Organizer
      代数的整数論とその周辺
    • Place of Presentation
      京都大学
    • Year and Date
      2011-11-28
    • Related Report
      2011 Final Research Report
  • [Presentation] Arithmetic families of Calabi-Yau varieties having a generalized hypergeometric function as a period integral2011

    • Author(s)
      都築暢夫
    • Organizer
      代数的整数論とその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2011-11-28
    • Related Report
      2011 Annual Research Report
  • [Presentation] ある一般超幾何関数を周期積分に持つ数論的カラビ・ヤウ多様体族について2011

    • Author(s)
      都築暢夫
    • Organizer
      代数学セミナー
    • Place of Presentation
      九州大学
    • Year and Date
      2011-11-04
    • Related Report
      2011 Annual Research Report 2011 Final Research Report
  • [Presentation] 超幾何層に付随するCalabi-Yau多様体とその応用2011

    • Author(s)
      山内卓也
    • Organizer
      第147回数理情報科学談話会
    • Place of Presentation
      鹿児島大学
    • Year and Date
      2011-04-22
    • Related Report
      2011 Annual Research Report 2011 Final Research Report
  • [Presentation] 超幾何層に付随するカラビーヤウ多様体族について2011

    • Author(s)
      山内卓也
    • Organizer
      近畿大学数学講演会
    • Place of Presentation
      近畿大学
    • Year and Date
      2011-02-14
    • Related Report
      2011 Final Research Report
  • [Presentation] 超幾何層に付随するカラビーヤウ多様体族について2011

    • Author(s)
      山内卓也
    • Organizer
      近畿大学数学講演会
    • Place of Presentation
      大阪大学(豊中市)
    • Year and Date
      2011-02-14
    • Related Report
      2010 Annual Research Report
  • [Presentation] A family of Calabi-Yau related to certain generalized hypergeometric local systems2011

    • Author(s)
      都築暢夫
    • Organizer
      rom p-adic differential equations to arithmetic algebraic geometry
    • Place of Presentation
      パドバ大学(イタリア)
    • Year and Date
      2011-02-05
    • Related Report
      2011 Final Research Report
  • [Presentation] A family of Calabi-Yau related to certain generalized hypergeometric local systems2011

    • Author(s)
      都築暢夫
    • Organizer
      From p-adic differential equations to arichmetic algebraic geometry
    • Place of Presentation
      パドバ大学(イタリア)
    • Year and Date
      2011-02-05
    • Related Report
      2010 Annual Research Report
  • [Presentation] Rigid cohomology2010

    • Author(s)
      都築暢夫
    • Organizer
      2010年台湾数学会年会
    • Place of Presentation
      国立彰化師範大学
    • Year and Date
      2010-12-11
    • Related Report
      2011 Final Research Report
  • [Presentation] Rigid cohomology2010

    • Author(s)
      都築暢夫
    • Organizer
      台湾数学会年会
    • Place of Presentation
      国立彰化師範大学(台湾)
    • Year and Date
      2010-12-11
    • Related Report
      2010 Annual Research Report
  • [Presentation] Log-growth and Frobenius for p-adic differential equations2010

    • Author(s)
      都築暢夫
    • Organizer
      Berkovich Spaces and p-adic Differential Equations
    • Place of Presentation
      ストラスブルク大学(フランス)
    • Year and Date
      2010-11-08
    • Related Report
      2011 Final Research Report
  • [Presentation] Log-growth and Frobenius for p-adic differential equations2010

    • Author(s)
      都築暢夫
    • Organizer
      Berkovich Spaces and P-adic Differential Equations
    • Place of Presentation
      ストラスブルク大学(フランス)
    • Year and Date
      2010-11-08
    • Related Report
      2010 Annual Research Report
  • [Presentation] GSp(4)の肥田理論の紹介2010

    • Author(s)
      山内卓也
    • Organizer
      GSp(4)の数論を中心とした基礎的ワークショップ
    • Place of Presentation
      大阪大学
    • Year and Date
      2010-08-06
    • Related Report
      2011 Final Research Report
  • [Presentation] GSp(4)の肥田理論の紹介2010

    • Author(s)
      山内卓也
    • Organizer
      GSp(4)の数論を中心とした基礎的ワークショップ
    • Place of Presentation
      大阪大学(豊中市)
    • Year and Date
      2010-08-06
    • Related Report
      2010 Annual Research Report
  • [Presentation] Generalized hypergeometric functions and arithmetic families of Calabi-Yau varieties2010

    • Author(s)
      都築暢夫
    • Organizer
      玉原数論幾何研究集会2010
    • Place of Presentation
      東京大学玉原国際セミナーハウス
    • Year and Date
      2010-05-26
    • Related Report
      2011 Final Research Report
  • [Presentation] 超幾何層に付随するカラビヤウ多様体とその応用について2010

    • Author(s)
      山内卓也
    • Organizer
      大数理セミナー-保型形式の整数論月例セミナー
    • Place of Presentation
      東京大学
    • Year and Date
      2010-05-15
    • Related Report
      2011 Final Research Report
  • [Presentation] 超幾何層に付随するカラビヤウ多様体とその応川について2010

    • Author(s)
      山内卓也
    • Organizer
      東大数理セミナー-保型形式の整数論月例セミナー
    • Place of Presentation
      東京大学(東京都)
    • Year and Date
      2010-05-15
    • Related Report
      2010 Annual Research Report
  • [Presentation] リジッド・コホモロジー論の最近の進展2010

    • Author(s)
      都築暢夫
    • Organizer
      日本数学会年会
    • Place of Presentation
      慶応大学
    • Year and Date
      2010-03-26
    • Related Report
      2011 Final Research Report 2009 Annual Research Report
  • [Presentation] Logarithmic growth and Frobenius slopes2010

    • Author(s)
      都築暢夫
    • Organizer
      Effective methods in p-adic cohomology
    • Place of Presentation
      オックスフォード、イギリス
    • Year and Date
      2010-03-16
    • Related Report
      2009 Annual Research Report
  • [Presentation] H^1 of rigid cohomology for geometrically unibranch varieties2010

    • Author(s)
      都築暢夫
    • Organizer
      Seminair de Geometrie Arithmetique et Motivique
    • Place of Presentation
      パリ北大学(フランス)
    • Year and Date
      2010-01-29
    • Related Report
      2011 Final Research Report
  • [Presentation] H^1 of rigid cohomology for geometrically unibranch varieties2010

    • Author(s)
      都築暢夫
    • Organizer
      Seminair de Geometrie Arithmetique et Motivique
    • Place of Presentation
      パリ、フランス
    • Year and Date
      2010-01-29
    • Related Report
      2009 Annual Research Report
  • [Presentation] Endoscopic lift to the Siegel paramodular threefold related to Klein's cubic threefold2010

    • Author(s)
      山内卓也
    • Organizer
      「保型形式・保型表現およびそれに伴うL函数と周期の研究」RIMS研究集会
    • Place of Presentation
      東京
    • Year and Date
      2010-01-18
    • Related Report
      2009 Annual Research Report
  • [Presentation] A Calabi-Yau family associated to some hypergeometric sheaf and its applications2010

    • Author(s)
      山内卓也
    • Organizer
      Kobe Number Theory Workshop
    • Place of Presentation
      神戸大学
    • Year and Date
      2010-01-15
    • Related Report
      2011 Final Research Report
  • [Presentation] A Calabi-Yau family associated to some hypergeometric sheaf and its applications2010

    • Author(s)
      山内卓也
    • Organizer
      Kobe Number Theory Workshop
    • Place of Presentation
      神戸
    • Year and Date
      2010-01-15
    • Related Report
      2009 Annual Research Report
  • [Presentation] Overconvergent F-isocrystals and rigid cohomology on curves2010

    • Author(s)
      都築暢夫
    • Organizer
      Summer school p-adic cohomologies and arithmetic applications
    • Place of Presentation
      セビリア大学(スペイン)
    • Related Report
      2011 Final Research Report
  • [Presentation] Overconvergent F-isocrystals and rigid cohomology on curves2010

    • Author(s)
      都築暢夫
    • Organizer
      Summer school "p-adic cohomologies and arithmetic applications"
    • Place of Presentation
      セビリア大学(スペイン)
    • Related Report
      2010 Annual Research Report
  • [Presentation] 超幾何層に付随するCalabi-Yau多様体族とその応用2009

    • Author(s)
      山内卓也
    • Organizer
      p-進特殊関数と数論幾何
    • Place of Presentation
      蔵王町
    • Year and Date
      2009-10-28
    • Related Report
      2011 Final Research Report
  • [Presentation] 超幾何層に付随するCalabi-Yau多様体族について2009

    • Author(s)
      都築暢夫
    • Organizer
      p-進特殊関数と数論幾何
    • Place of Presentation
      蔵王町
    • Year and Date
      2009-10-28
    • Related Report
      2011 Final Research Report
  • [Presentation] 超幾何層に付随するCalabi-Yau多様体族について2009

    • Author(s)
      都築暢夫
    • Organizer
      p-進特殊関数と数論幾何
    • Place of Presentation
      宮城県蔵王町
    • Year and Date
      2009-10-28
    • Related Report
      2009 Annual Research Report
  • [Presentation] 超幾何層に付随するCalabi-Yau多様体族とその応用2009

    • Author(s)
      山内卓也
    • Organizer
      p-進特殊関数と数論幾何
    • Place of Presentation
      宮城県蔵王町
    • Year and Date
      2009-10-28
    • Related Report
      2009 Annual Research Report
  • [Presentation] On rigid cohomology with partially compact supports2009

    • Author(s)
      都築暢夫
    • Organizer
      Mini-workshop on isocrystals
    • Place of Presentation
      東北大学
    • Year and Date
      2009-10-16
    • Related Report
      2011 Final Research Report 2009 Annual Research Report
  • [Presentation] GSp4に対するセール予想、同伴形式、テータ作用素2009

    • Author(s)
      山内卓也
    • Organizer
      東北大学代数学セミナー
    • Place of Presentation
      東北大学
    • Year and Date
      2009-07-09
    • Related Report
      2011 Final Research Report
  • [Presentation] GSp4に対するセール予想、同伴形式、テータ作用素2009

    • Author(s)
      山内卓也
    • Organizer
      東北大学代数学セミナー
    • Place of Presentation
      仙台
    • Year and Date
      2009-07-09
    • Related Report
      2009 Annual Research Report

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Published: 2009-04-01   Modified: 2016-04-21  

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