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Research on determinantal formulae and Bernoulli-Hurwitz numbers in the theory of Abelian functions

Research Project

Project/Area Number 16540002
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

ONISHI Yoshihiro  Iwate University, Faculty of Humanities and Social Sciences, Assistant Professor, 人文社会科学部, 助教授 (60250643)

Co-Investigator(Kenkyū-buntansha) ODAI Yoshitaka  Iwate University, Faculty of Humanities and Social Sciences, Assistant Professor, 人文社会科学部, 助教授 (10204215)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2006: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2005: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2004: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsAbelian function / Algebraic function / Algebraic curve / hyperelliptic curve / division polynomial / Frobenius-Stickelberger / Frobenins-Stickelberger / Bernoulli数 / Frobenius-Stickelbergerの公式 / Bernoulli-Hurwitz数 / 普遍Bernoulli数
Research Abstract

Thanks to the Grant-in-Aid for Scientific Research (C), academic years 2000-2002, with theme "Research on product formula for special values of Abelian functions", the head investigator (Y.O) made two progresses :
(1)Nicely generalized Bernoulli-Hurwitz numbers to algebraic functions of cyclotomic type, and
(2)Gave determinantal expressions of addition formulae for Abelian functions associated to higher genus algebraic curves,
Hence, we aimed to investigate these results deeply as follows :
(1)should be improved by connecting the result to application on L-series as in the case of Bernoulli, and Hurwitz numbers;
(2)should be generalized to non-hyperelliptic curves.
The results :
The problem (1)is so difficult. We only preparated a manuscript to submit a academic journal. We, however, made a big progress on (2), and published a lot of papers as included in this report.
(1)is needed to investigate further ;
(2)is only proved by a dedious way, and should be proved more directly.
We will continue to research on these problem.
Finally, we hope our results be good hints for researchers who are interesting on theory of Abelian functions.

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (27 results)

All 2007 2006 2005 2004 Other

All Journal Article (27 results)

  • [Journal Article] 構円Gauss和とHurwity数を結ぶ合同式及びTate Shafarevich群2007

    • Author(s)
      大西良博
    • Journal Title

      2006年 北陸数論小研究集会報告集

      Pages: 21-21

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Determinant Expressions for purely pentagonal curves of degree six2007

    • Author(s)
      Yoshihiro Onishi
    • Journal Title

      http://arxiv.org/org/abs/math.NT/

      Pages: 1-75

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Congruence of elliptic Gauss sums with Hurwitge numbers and Tate-Shafarevich groups2007

    • Author(s)
      Yoshihiro Onishi
    • Journal Title

      http://arxiv.org/abs/math/NT/

      Pages: 1-21

    • Related Report
      2006 Annual Research Report
  • [Journal Article] 楕円Gauss和とHurwitz数を結ぶ合同式及びTate-Shafarevich群2007

    • Author(s)
      大西 良博
    • Journal Title

      2006年北陸数論小研究集会報告集

      Pages: 1-21

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Theory of generaliyed Bernoulli-Hurwity numbers and the universal Bernoulli numbers2006

    • Author(s)
      Yoshihiro Onishi
    • Journal Title

      http://arxiv. Org/abs/math. NT/0406096

      Pages: 91-91

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Abelian functions for purely frignal curres of genus three2006

    • Author(s)
      J.C.Eilbeck, V.Z.Enoyskii, S.Matsutani, Y.Onishi, E.Previato
    • Journal Title

      hppt : //arxiv.org/abs/math/AG/0610019

      Pages: 1-26

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Addition formulae over the Jacobian pre-image of hyperelliptic Wirtinger varieties2006

    • Author(s)
      J.C.Eilbeck, V.Z.Enoyskii, S.Matsutani, Y.Onishi, E.Previato
    • Journal Title

      http://web.cc.iwate-u.ac.jp//reseach/pub/

      Pages: 1-11

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Abelian functions for cyclic trigonal curves of genus four2006

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

      http://arxiv.org/abs/math/0612654

      Pages: 1-23

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Determinant Expression for purely pentagonal curves of degree six2006

    • Author(s)
      Yoshihiro Onishi
    • Journal Title

      http://arxiv.org/abs/math.NT/

      Pages: 1-75

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Explicit addition formulae for purely trigonal curves of degree four2006

    • Author(s)
      J.C.Eilbeck, V.Z.Enolskii, S.Matsutani, Y.Onishi, E.Peviato
    • Journal Title

      http://arxiv.org/abs/math.AG/

      Pages: 1-14

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Determinant expressions for hyperelliptic functions (with an Appendix by S. Matsutani)2005

    • Author(s)
      Yoshihiro Onishi
    • Journal Title

      Proc. of Edinburgh Math. Soc 48

      Pages: 705-742

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Determinant expressions for hyperelliptic functions (with an Appendix by Shigeki Matsutani : Connection of The formula of Cantor and of Brioschi-Kiepert type)2005

    • Author(s)
      Y.Onishi
    • Journal Title

      Proc.Edinburgh Math.Soc. 48

      Pages: 705-742

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Determinant Expressions for hyperelliptic functions2005

    • Author(s)
      Yoshihiro Onishi
    • Journal Title

      Proc. Edinburgh Math.Soc. 48

      Pages: 705-742

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Determinant Expressions for purely trigonal curves of degree four2005

    • Author(s)
      Yoshihiro Onishi
    • Journal Title

      http://arxiv.org/abs/math.NT/0503696

      Pages: 1-14

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The addition law attached to a stratification for a hyperelliptic Jacobian variety2005

    • Author(s)
      V.Z.Enolskii, S.Matsutani, Y.Onishi
    • Journal Title

      http://arxiv.org/abs/math.AG/0508366

      Pages: 1-13

    • Related Report
      2005 Annual Research Report
  • [Journal Article] 円分型台数函数版Bernoulli-Hurwitz数と普遍Bernoulli数の理論2005

    • Author(s)
      大西 良博
    • Journal Title

      2004数論セミナー静岡報告集

      Pages: 1-91

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Determinant expressions for hyperelliptic functions in genus three2004

    • Author(s)
      Yoshihiro Onishi
    • Journal Title

      Tokyo Journal of Mathematics 27

      Pages: 299-312

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Determinantal expressions for hyperelliptic functions in genus three2004

    • Author(s)
      Y.Onishi
    • Journal Title

      Tokyo J.Math. 27

      Pages: 299-312

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Determinantal Expressions in hyperelliptic functions in genus three2004

    • Author(s)
      Yoshihiro Onishi
    • Journal Title

      Tokyo Journal of Mathematics 27・2

      Pages: 299-312

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Addition formulae over the Jacobian pre-image of hyperelliptic Wirtinger varieties

    • Author(s)
      J.C.Eilbeck, V.Z.Enolskii, S.Matsutani, Y.Onishi, E.Previato
    • Journal Title

      J. fur reine und angew. Math. (未定)(掲載受理)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The addition law attached to a stratification for hyperelliptic Jacobian variety

    • Author(s)
      V.Z.Enolskii, S.Matsutani, Y.Onishi
    • Journal Title

      Tokyo Journal of Math (未定)(掲載受理)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Addition formulae over the Jacobian pre-image of hyperelliptic Wirtinger varieties.

    • Author(s)
      J.C.Eilbeck, V.Z.Enol'skii, S.Matustani, Y.Onishi, E.Previato
    • Journal Title

      J.fur reine und angew.Math. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The addition law attached to a stratification for a hyperelliptic Jacobian variety.

    • Author(s)
      V.Z.Enolskii, S.Matsutani, Y.Onishi
    • Journal Title

      Tokyo J.Math. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Theory of generalized Bernoulli-Hurwitz numbers for algebraic functions of cyclotomic type and universal Bernoulli numbers

    • Author(s)
      Y.Onishi
    • Journal Title

      http://arxiv.org/abs/math.NT/0406096

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Theory of generalized Bernoulli-Hurwitz numbers in the algebraic functions of cyclotomic type,

    • Author(s)
      Y.Onishi
    • Journal Title

      http://arxiv.org/abs/math.NT/0304377

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Determinant expressions in Abelian functions for purely pentagonal curves of degree six.

    • Author(s)
      Y.Onishi
    • Journal Title

      http://web.cc.hss.iwate-u.ac.jp/~onishi/publications.html

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Determinantal Expressions in hyperelliptic functions(with an Appendix by S.Matsutani)

    • Author(s)
      Yoshihiro Onishi
    • Journal Title

      Proceedings of Edinburgh Math.Soc. 未定

    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2019-10-16  

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