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Integral structures of arithmetic differential equations and geometries behind them

Research Project

Project/Area Number 24654002
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionTohoku University

Principal Investigator

TSUZUKI Nobuo  東北大学, 理学(系)研究科(研究院), 教授 (10253048)

Co-Investigator(Kenkyū-buntansha) YAMAUCHI Takuya  鹿児島大学, 教育学部, 准教授 (90432707)
Co-Investigator(Renkei-kenkyūsha) TAKAHASHI Nobuyoshi  広島大学, 大学院理学研究科, 准教授 (60301298)
Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywords超幾何関数 / 剛性カラビ・ヤウ多様体 / 半安定的退化 / 整係数コホモロジー / ガロア表現 / 保型性 / 2進還元 / 数論幾何学 / カラビ・ヤウ多様体族 / 退化族 / 国際研究者交流(イタリア) / 数論幾何 / 超幾何微分方程式 / カラビ・ヤウ多様体 / 整構造 / 数論幾 / 超幾何微分方程 / 局所系の整構造
Outline of Final Research Achievements

We studied properties of the arithemtic family of Calabi-Yau varieties, constructed by the representative of this research, for which the period integral is a generalized hypergeometric functions. In particular, if the dimension is odd, we found a semistable family around a degenerated fiber such that the number of irreducible components of the special fiber is two among which the one is rational and the other has an interesting natures with respect to arithmetic geometry. In particuler, we proved the modularity of the special fiber in dimension 3.
In the case of dimension 2, we constructed a semistable family over an extension
of Z which is ramified only at 2 and got a K3 surface over an algebraic number field such that it has a good reduction everywhere.

Report

(4 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (27 results)

All 2015 2014 2013 2012 Other

All Journal Article (10 results) (of which Peer Reviewed: 10 results) Presentation (17 results) (of which Invited: 15 results)

  • [Journal Article] Cusp forms for exceptional group of type E_72015

    • Author(s)
      Henry-H. Kim and T. Yamauchi
    • Journal Title

      Compositio Math.

      Volume: 印刷中

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A conditional construction of Artin representations for real analytic Siegel cusp forms of weight (2,1)2015

    • Author(s)
      Henry-H. Kim and T. Yamauchi
    • Journal Title

      Contemporary Mathematics

      Volume: 印刷中

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Clemens-Schmid exact sequence in characteristic p2014

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

      Math. Annalen

      Volume: 358巻 Issue: 3-4 Pages: 971-1004

    • DOI

      10.1007/s00208-013-0980-8

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] An explicit computation of p-stabilized vectors2014

    • Author(s)
      Michitaka Miyauchi, Takuya Yamauch
    • Journal Title

      Journal de Theorie des Nombres de Bordeaux

      Volume: -

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] The closed cone of a rational series is rational polyhedral2014

    • Author(s)
      Kimura, Shunichi; Kuroda, Shigeru; Takahashi, Nobuyoshi
    • Journal Title

      Journal of Algebra

      Volume: 405 Pages: 243-258

    • DOI

      10.1016/j.jalgebra.2014.02.007

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Local newforms and formal exterior square L-functions2013

    • Author(s)
      Michitaka Miyauchi, Takuya Yamauchi
    • Journal Title

      International Journal of Number Theory

      Volume: 9 Issue: 08 Pages: 1995-2010

    • DOI

      10.1142/s179304211350070x

    • Related Report
      2013 Research-status Report
    • Peer Reviewed
  • [Journal Article] Nonstandard Point Counting for Algebraic Varieties2013

    • Author(s)
      Takahashi, Nobuyoshi
    • Journal Title

      Communications in Algebra

      Volume: 41 Issue: 3 Pages: 971-988

    • DOI

      10.1080/00927872.2012.737074

    • Related Report
      2013 Research-status Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] A note on the first rigid cohomology group for geometrically unibranch varieties2012

    • Author(s)
      Nobuo Tsuzuki
    • Journal Title

      Rendiconti del Seminario Matematico della Universita di Padova

      Volume: 128 Pages: 17-53

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Overholonomicity of overconvergent F-isocrystals over smooth varieties2012

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

      Annals of Math

      Volume: 176巻 Issue: 2 Pages: 747-813

    • DOI

      10.4007/annals.2012.176.2.2

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Motivic zeta functions in additive monoidal categories2012

    • Author(s)
      Kimura, Kimura, Takahashi
    • Journal Title

      Journal of K-theory

      Volume: (印刷中) Issue: 3 Pages: 459-473

    • DOI

      10.1017/is011011006jkt174

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] On a Lefschetz type problem and variation of slopes for F-isocrystals2015

    • Author(s)
      Nobuo Tsuzuki
    • Organizer
      Workshop on recent trends in $p$-adic cohomology
    • Place of Presentation
      mperial College London(イギリス)
    • Year and Date
      2015-03-25
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] The weight in Serre's conjecture for GSp42015

    • Author(s)
      山内卓也
    • Organizer
      東北大学整数論セミナー
    • Place of Presentation
      東北大学
    • Year and Date
      2015-03-18
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Cusp forms on E_72014

    • Author(s)
      山内卓也
    • Organizer
      金沢大学組合せ数学セミナー
    • Place of Presentation
      金沢大学
    • Year and Date
      2014-12-01
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On constancy of Newton polygons of F-isocrystals2014

    • Author(s)
      Nobuo Tsuzuki
    • Organizer
      Workshop of arithmetic geometry in Tohoku
    • Place of Presentation
      東北大学
    • Year and Date
      2014-10-31
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] The Clemens Schmidt sequence2014

    • Author(s)
      Bruno Chiarellotto, Nobuo Tsuzuki
    • Organizer
      ALGANT summer school on MONODROMY
    • Place of Presentation
      Dobbiacco(イタリア)
    • Year and Date
      2014-06-23 – 2014-06-27
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Clemens-Schmid exact sequence in characteristic p2014

    • Author(s)
      Nobuo Tsuzuki
    • Organizer
      Mini-Workshop on Arithmetic
    • Place of Presentation
      台湾大学(台湾)
    • Year and Date
      2014-05-08
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Singular fibers of the arithmetic family of hypergeometric Calabi-Yau varieties

    • Author(s)
      都築暢夫
    • Organizer
      Arithmetic, differentials, and geometry
    • Place of Presentation
      東北大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] On integral cohomologies for an arithmetic family of hypergeometric Calabi-Yau

    • Author(s)
      都築暢夫
    • Organizer
      Seminaire de Mathematique
    • Place of Presentation
      フランス・IHES
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] On the singular fiber of the arithmetic family of hypergeometric Calabi-Yau

    • Author(s)
      都築暢夫
    • Organizer
      Seminaire de Geometrie Arithmetique et Motivique
    • Place of Presentation
      フランス・パリ13大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] The L-function of some Siegel modular 3-folds and endoscopic lifts

    • Author(s)
      山内卓也
    • Organizer
      Seminar at Jussieu
    • Place of Presentation
      フランス・パリ6大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Artin representations for GSp(4) attached to real analytic Siegel modular forms

    • Author(s)
      山内卓也
    • Organizer
      Number Theory / Representation Theory Seminar Academic year 2013-2014
    • Place of Presentation
      カナダ・トロント大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] p-adic Clemens-Schmid exact sequence

    • Author(s)
      都築暢夫
    • Organizer
      Symposium on ARITHMETIC & GEOMETRY
    • Place of Presentation
      九州大学
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] The first rigid cohomology groups for mildly singular varieties

    • Author(s)
      都築暢夫
    • Organizer
      p-adic cohomology and its applications to arithmetic geometry
    • Place of Presentation
      東北大学
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] On the first rigid cohomology groups for mildly singular varieties

    • Author(s)
      都築暢夫
    • Organizer
      Mini-workshop on Arithmetic
    • Place of Presentation
      台湾大学(台湾)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] On the first rigid cohomology groups for mildly singular varieties

    • Author(s)
      都築暢夫
    • Organizer
      Seminaire de geometrie algebrique
    • Place of Presentation
      Rennes大学(フランス)
    • Related Report
      2012 Research-status Report
  • [Presentation] Endoscopic lift to the Siegel paramodular threefold related to Klein cubic threefold

    • Author(s)
      山内卓也
    • Organizer
      Number Theory / Representation Theory Seminar
    • Place of Presentation
      Toronto大学(カナダ)
    • Related Report
      2012 Research-status Report
  • [Presentation] Arithmetic Calabi-Yau families associated to generalized hypergeometric local systems and its applications

    • Author(s)
      山内卓也
    • Organizer
      第8回代数・解析・幾何学セミナー
    • Place of Presentation
      鹿児島大学
    • Related Report
      2012 Research-status Report
    • Invited

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Published: 2013-05-31   Modified: 2019-07-29  

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