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Number Theory and Geometry related to Algebraic Groups

Research Project

Project/Area Number 15340001
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

YUKIE Akihiko  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (20312548)

Co-Investigator(Kenkyū-buntansha) HANAMURA Masaki  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60189587)
ISHIDA Masanori  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (30124548)
NAKAMURA Tetsuo  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90016147)
HARA Nobuo  Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90298167)
OGATA Shouetsu  Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90177113)
梶原 健  東北大学, 大学院・理学研究科, 助手 (00250663)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥9,500,000 (Direct Cost: ¥9,500,000)
Fiscal Year 2005: ¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2004: ¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2003: ¥3,700,000 (Direct Cost: ¥3,700,000)
KeywordsPrehomogeneous vector spaces / Toric Variety / Algebraic cycles / Motif / Frobenius / 対数アーベル多様体 / test ideal / 密度定理 / 密着閉包 / 対数的代数幾何 / 量子群 / 楕円曲線
Research Abstract

(1) Yukie investigated zeta functions associated with prehomogeneous vector spaces and obtained an estimate of the number of quintic fields with discriminant less than or equal to X.
(2) Hanamura investigated mixed motifs and obtained a result on the Kunneth formula for modular varieties.
(3) Ishida established a theory relating ideal theory (of rings) to rational and real fans for toric varieties. Also he found a real fan analogue of blow-ups of algebraic varieties. Especially for a finite number of blow-ups, he confirmed similarities with the case of algebraic varieties. In algebraic geometry, Zariski-Riemann topology can be defined on the set of valuation rings of the function fields. He defined this notion for rational and real fans using the set of all additive orders on free modules and real vector spaces. Using this Zariski-Riemann topology, he proved the existence of the compactification of toric varieties.
(4) Hara generalized the notion of tight closure of ideals I of rings R of … More positive characteristic to "I-tight closure" and proved various properties for the generalized determinantal ideal τ(I), thus made a foundation of the theory. Using this method, he applied to the proof of a special case of the Fujita conjecture on the global generation of adjoint bundles and to a new proof of the Ein-Lazarsfeld-Smith comparison theorem on the symbolic power of ideals of regular local rings.
(5) Nakamura : If a CM elliptic curve is isogenous to all its Galois conjugate, it is called a Q-curve and has important properties. He classified all Q-curves over the absolute class field of a given imaginary quadratic field. It is well-known that the torsion group of an elliptic curve over a number field is finite. He investigated how the torsion changes among isogenous elliptic curves.
(6) Ogata investigated projective normality and the degrees of the generators of the defining ideals of toric varieties by very ample line bundles, and opbtained some criteria of projective normality and some estimates of the degrees of the defining ideals. Less

Report

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

    (32 results)

All 2006 2005 2004 Other

All Journal Article (23 results) Book (3 results) Publications (6 results)

  • [Journal Article] Completion of real fans and Zariski-Riemann spaces.2006

    • Author(s)
      G.Ewald, M.Ishida
    • Journal Title

      Tohoku Math.J. 58-2(To appear)

    • NAID

      110004669750

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the number of quintic fields.2005

    • Author(s)
      A.C.Kable, A.Yukie
    • Journal Title

      Invent. Math. 160

      Pages: 217-259

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Absolute Chow-Kunneth projectors for modular varieties.2005

    • Author(s)
      B.Gordon, M.Hanamura, J.P.Murre
    • Journal Title

      J. Reine Angew. Math. 590

      Pages: 139-155

    • 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 number of quintic fields.2005

    • Author(s)
      A.C.Kable, A.Yukie
    • Journal Title

      Invent.Math. 160

      Pages: 217-259

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Absolute Chow-Kunneth projectors for modular varieties.2005

    • Author(s)
      B.Gordon, M.Hanamura, J.P.Murre
    • Journal Title

      J.Reine Angew.Math. 590

      Pages: 139-155

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Absolute Chow-Kunneth projectors for modular varieties.2005

    • Author(s)
      B.Gordon, M.Hanamura, J.P.Murre
    • Journal Title

      J.Reine Angew.Math. 580

      Pages: 139-155

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A characteristic p analog of multiplier ideals and applications.2005

    • Author(s)
      N.Hara
    • Journal Title

      Comm.Algebra 33

      Pages: 3375-3388

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On quadratic generation of ideals defining projective toric Varieties.2005

    • Author(s)
      S.Ogata
    • Journal Title

      Kodai Math.J 28

      Pages: 519-524

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On a construction of quintic rings2004

    • Author(s)
      A.C.Kable, A.Yukie
    • Journal Title

      Nagoya Math. J. 173

      Pages: 163-203

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Mixed motives and algebraic cycles II2004

    • Author(s)
      M.Hanamura
    • Journal Title

      Invent. Math. 158

      Pages: 105-179

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A generalization of test ideals,2004

    • Author(s)
      N.Hara, K.Yoshida
    • Journal Title

      Nagoya Math. J. 175

      Pages: 59-74

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a construction of quintic rings.2004

    • Author(s)
      A.C.Kable, A.Yukie
    • Journal Title

      Nagoya Math.J. 173

      Pages: 163-203

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Mixed motives and algebraic cycles II.2004

    • Author(s)
      M.Hanamura
    • Journal Title

      Invent.Math. 158

      Pages: 105-179

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A generalization of test ideals.2004

    • Author(s)
      N.Hara, K.Yoshida
    • Journal Title

      Nagoya Math.J. 175

      Pages: 59-74

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A construction of quintic rings2004

    • Author(s)
      A.Kable, A.Yukie
    • Journal Title

      Nagoya Math.J. 173

      Pages: 163-203

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the space of quadruples of quinary alternating forms2004

    • Author(s)
      A.Kable, A.Yukie
    • Journal Title

      J.Pure Appl.Algebra 186

      Pages: 277-295

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Mixed motives and algebraic cycles II2004

    • Author(s)
      M.Hanamura
    • Journal Title

      Invent.Math. 158

      Pages: 105-179

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On a generalization of test ideals2004

    • Author(s)
      N.Hara, S.Takagi
    • Journal Title

      Nagoya Math.J. 175

      Pages: 59-74

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Torsion of elliptic curves in isogeny classes,

    • Author(s)
      Y.Fujita, T.Nakamura
    • Journal Title

      Trans. Amer. Math. Soc. (To appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Torsion of elliptic curves in isogeny classes.

    • Author(s)
      Y.Fujita, T.Nakamura
    • Journal Title

      Trans.Amer.Math.Soc. (To appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Torsion of elliptic curves in isogeny classes.

    • Author(s)
      Y.Fujita, T.Nakamura
    • Journal Title

      Trans.Amer.Math.Soc. (To appear)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] k-normality of weighted projective spaces

    • Author(s)
      S.Ogata
    • Journal Title

      Kodai Math.J. To appear

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On the number of quintic fields

    • Author(s)
      A.Kable, A.Yukie
    • Journal Title

      Invent.Math. To appear

    • Related Report
      2004 Annual Research Report
  • [Book] An introduction to algebraic curves-the first course in algebraic geometry.2005

    • Author(s)
      T.Kaijiwara
    • Total Pages
      336
    • Publisher
      Nihon Hyouronsha
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Book] 代数曲線入門-はじめての代数幾何-2004

    • Author(s)
      梶原健
    • Total Pages
      336
    • Publisher
      日本評論社
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Book] 代数曲線入門---はじめての代数幾何2004

    • Author(s)
      梶原健
    • Publisher
      日本評論社
    • Related Report
      2004 Annual Research Report
  • [Publications] A.C.Kable, A.Yukie: "On a construction of quintic rings"Nagoya Math.J.. (To appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] A.C.Kable, A.Yukie: "On the space of quadruples of quinary alternating forms"J.Pure and App.. (To appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] B.Gordon, M.Hanamura, J.P.Murre: "Relative Chow-Kunneth projectors for modular varieties"J.Reine Angew.Math.. 558. 1-14 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] B.Gordon, M.Hanamura, J.P.Murre: "Absolute Chow-Kunneth projectors for modular varieties"J.Reine Angew.Math.. (To appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Nakamura: "A classification of Q-curves with complex multiplication"J.Math.Soc.Japan. (To appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] N.Hara, K.Yoshida: "A generalization of tight closure and multiplier ideals"Trans.Amer.Math.Soc.. 355. 3143-3174 (2003)

    • Related Report
      2003 Annual Research Report

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

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