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Elaborate construction of a concrete theoryof Abelian functions

Research Project

Project/Area Number 22540006
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionUniversity of Yamanashi

Principal Investigator

ŌNISHI Yoshihiro  山梨大学, 教育学研究科, 教授 (60250643)

Project Period (FY) 2010 – 2012
Project Status Completed (Fiscal Year 2012)
Budget Amount *help
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2012: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords数論 / 数論幾何 / Jacobi 多様体 / Abel 函数 / 加法公式 / sigma 函数 / Jacobian varieties / Abelian functions / sigma functions / 代数曲線 / アーベル函数 / ヤコビ多様体 / 代数的加法公式
Research Abstract

The first year, I participate in an international work shop on this subject,and gave a talk and had discussions with its participants. Then I planed the direction of this threeyear research supported by JSPS.
I invited foreign researcers working on the subject. They and Igot a quite new addition formula for general genus one curve.
With some foreign researchers, I gota explicit and simple system of equations of the Kummer variety coming from a general genus threetrigonal curve. This also gave a nice realization of A. Coble's unique results.
We investigated on theexpansions of some sigma function at the origin.

Report

(4 results)
  • 2012 Annual Research Report   Final Research Report ( PDF )
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • Research Products

    (26 results)

All 2013 2012 2011 2010 Other

All Journal Article (12 results) (of which Peer Reviewed: 8 results) Presentation (6 results) (of which Invited: 1 results) Book (1 results) Remarks (7 results)

  • [Journal Article] Explicit realization of Coble's hypersurfaces in terms of multivari-ate }-functions2013

    • Author(s)
      大西良博
    • Journal Title

      第17回早稲田大学整数論研究集会報告集

      Volume: 16

    • Related Report
      2012 Final Research Report
  • [Journal Article] Re-lationship between the prime form and thesigma function for some cyclic (r; s) curves,Journal of Physics A2013

    • Author(s)
      J. Gibbons, S. Matsutani and Y. ^ Onishi
    • Journal Title

      Mathematical and The-oretical

      Volume: 46 Pages: 17-17

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Prime form and sigma function2013

    • Author(s)
      John Gibbons, Shigeki Matsutani, Yoshihiro Onishi
    • Journal Title

      Journal of Physics A (Mathematical and Theoretical)

      Volume: 未定

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Some new addition formulae for Weierstrasselliptic functions2012

    • Author(s)
      M. England, J.C. Eilbeck and Y. ^ Onishi
    • Related Report
      2012 Final Research Report
  • [Journal Article] Some new addition formulae for Weierstrass elliptic functions2012

    • Author(s)
      Matthew England, J. Chris Eilbeck and Yoshihiro Onishi
    • Journal Title

      http://arxiv.org/abs/1207.6274

      Volume: --- Pages: 1-16

    • Related Report
      2012 Annual Research Report
  • [Journal Article] Some new addition formulae for Weierstrass elliptic functions.2012

    • Author(s)
      J. C. Eilbeck, M. England, Y. Onishi
    • Journal Title

      http://arxiv.org/abs/1207.6274

      Volume: --- Pages: 1-16

    • Related Report
      2011 Annual Research Report
  • [Journal Article] Generalized Bernoulli-Hurwitznumbers and the universal Bernoulli num-bers2011

    • Author(s)
      Y. ^ Onishi
    • Journal Title

      Russian Mathematical Surveys

      Volume: 66:5 Pages: 871-932

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Abelian Functions associated with genusthree algebraic curves2011

    • Author(s)
      M. England, J.C. Eilbeck and Y. ^ Onishi
    • Journal Title

      London Math. Soc. Jour. of Computationand Math.

      Volume: 14 Pages: 291-326

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Determinant formulae in Abelianfunctions for a general trigonal curve of de-gree five2011

    • Author(s)
      Y. ^ Onishi
    • Journal Title

      Computational Methods and Function The-ory

      Volume: 11:2 Pages: 547-574

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Addition formulae for Abelian functions as-sociated with specialized curves. Phil2011

    • Author(s)
      J.C. Eilbeck, S. Matsutani and Y. ^ Onishi
    • Journal Title

      Phil.Trans.Royal Society A

      Volume: 369 Pages: 1245-1263

    • Related Report
      2012 Final Research Report
    • Peer Reviewed
  • [Journal Article] Addition formula for Abelian functions associated with specialized curves.2011

    • Author(s)
      J.C.Eilbeck, S.Matustani, Y.Onishi
    • Journal Title

      PhiL.Trans.Royal Society A

      Volume: 369 Pages: 1245-1263

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Determinant formulae in Abelian functions for a general trigonal curve of degree five2011

    • Author(s)
      Y.Onishi
    • Journal Title

      Computational Methods and Function Theory

      Volume: 掲載確定(未定)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Presentation] Explicit realization of Coble'sHypersurfaces in Terms of Multivariate }-Functions2013

    • Author(s)
      大西良博
    • Organizer
      第17回整数論研究集会
    • Place of Presentation
      早稲田大学
    • Year and Date
      2013-03-17
    • Related Report
      2012 Final Research Report
  • [Presentation] Explicit realization of Coble's hypersurfaces in terms of multivariate P-functions2013

    • Author(s)
      Yoshihiro Onishi
    • Organizer
      2013早稲田整数論研究集会
    • Place of Presentation
      早稲田大学西早稲田キャンパス
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] 高い種数の代数関数体に対する一般化されたBernoulli-Hurwitz 数と, 関連する諸問題2011

    • Author(s)
      大西良博
    • Organizer
      保型形式の整数論月例セミナー
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2011-11-19
    • Related Report
      2012 Final Research Report
  • [Presentation] Frobenius-Stickelberger-type for-mulae for general curves2010

    • Author(s)
      Y. ^ Onishi
    • Organizer
      Work shop on "The higher-genus sigma func-tion and applications"
    • URL

      http://icms.org.uk/workshops/sigma

    • Place of Presentation
      ICMS at Edinburgh, U.K
    • Year and Date
      2010-10-11
    • Related Report
      2012 Final Research Report
  • [Presentation] Frobenius-Stickelberger-type formulae for general curves2010

    • Author(s)
      Y.Onishi
    • Organizer
      The higher-genus sigma function and applications
    • Place of Presentation
      International Center of Math.Science(連合王国) 招待講演
    • Year and Date
      2010-10-11
    • Related Report
      2010 Annual Research Report
  • [Presentation] Hurwitz integrality of power se-ries expansion of the sigma function in genustwo2010

    • Author(s)
      大西良博
    • Organizer
      早稲田整数論セミナー
    • Place of Presentation
      早稲田大学
    • Year and Date
      2010-07-02
    • Related Report
      2012 Final Research Report
  • [Book] Abel 函数論

    • Author(s)
      大西良博
    • Total Pages
      139
    • Publisher
      中央大学数学教室
    • Related Report
      2012 Final Research Report
  • [Remarks] 成果の一部を公表してゐるWeb page

    • URL

      http://www.ccn.yamanashi.ac.jp/~yonishi/#publications

    • Related Report
      2012 Final Research Report
  • [Remarks] International Conference on Jacobian va-rieties, Abelian functions, and Kummer sur-faces, University of Yamanashi

    • Related Report
      2012 Final Research Report
  • [Remarks] Y. ^ Onishi : "The main congruences on gen-eralized Bernoulli-Hurwitz numbers for thecurves of cyclotomic type", 26 Aug., 2011

    • Related Report
      2012 Final Research Report
  • [Remarks] Y. ^ Onishi : "The sigma function of a gen-eral curve of genus two", 7, June 2012

    • Related Report
      2012 Final Research Report
  • [Remarks] Publications

    • URL

      http://www.ccn.yamanashi.ac.jp/~yonishi/#publications

    • Related Report
      2012 Annual Research Report
  • [Remarks] ---

    • URL

      http://www.ccn.yamanashi.ac.jp/~yonishi/#publications

    • Related Report
      2011 Annual Research Report
  • [Remarks]

    • URL

      http://www.ccn.yamanashi.ac.jp/~yonishi/#publications

    • Related Report
      2010 Annual Research Report

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Published: 2010-08-23   Modified: 2019-07-29  

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