Difference equations related with higher genus algebraic curves

• Project/Area Number: 23540245
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Global analysis
• Research Institution: Tsuda College
• Principal Investigator: NAKAYASHIKI Atsushi 津田塾大学, 学芸学部, 教授 (10237456)
• Co-Investigator(Renkei-kenkyūsha): ONISHI Yoshihiro 名城大学, 理工学部数学科, 教授 (60250643) ; CHO Koji 九州大学, 大学院数理学研究院, 准教授 (10197634) ; KONNO Hitoshi 東京海洋大学, 大学院海洋科学技術研究科, 教授 (00291477)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 高種数シグマ関数 / モジュラー不変性 / リーマンの特異点定理 / シューア関数 / 可積分階層のタウ関数 / 普遍グラスマン多様体 / テータ関数 / ギャップ列 / テータ関数の展開 / モジュラー不変 / 国際研究者交流（ロシア、イギリス、カナダ） / 国際研究者交流（ロシア） / KP階層の佐藤理論 / タウ関数 / リーマン面 / 可換微分・差分作用素 / (n,s)曲線 / 多変数シグマ関数 / KP階層 / 可換偏差分作用素 / テレスコピック曲線 / 国際研究者交流、ロシア / 交際情報交流、イギリス / 可積分系 / 加法定理 / 代数曲線 / 差分方程式 / 相関関数

Outline of Final Research Achievements

Using the structure of integrable systems we have studied two kinds of expansions of the theta function associated with a Riemann surface of an arbitrary genus. One is the Taylor expansion of the theta function at an arbitrary point on the theta divisor. We have determined the initial term as the Schur function . The other is the expansion of the theta function considered as a function on the direct product of the Riemann surface. We have described the initial term of the expansion in one of the variables of the product by the explicitly given derivative of the theta function. As an application of the these results we have generalized the celebrated Riemann's singularity theorem. As other applications we have derived certain addition formulae of the multivariate sigma function and proved the modular invariance of it.

Research Products

• [Journal Article] On Addition formulae for sigma functions of telescopic curves (2013)
  • Author(s): Takanori Ayano and Atsushi Nakayashiki
  • Journal Title: Symmetry, Integrability and Geometry: Methods and Applications Volume: 9 Pages: 46-59
  • Peer Reviewed
• [Journal Article] Derivatives of Schur, tau and sigma functions on Abel-Jacobi images (2013)
  • Author(s): Atsushi Nakayashiki and Keijiro Yori
  • Journal Title: Springer Proceedings in Mathematics & Statistics Volume: 40 Pages: 429-462
  • Peer Reviewed
• [Presentation] On series expansions of higher genus sigma functions (2015)
  • Author(s): Atsushi Nakayashiki
  • Organizer: workshop on Curves, Moduli and Integrable Systems
  • Place of Presentation: 津田塾大学
  • Year and Date: 2015-02-17 – 2015-02-19
  • Invited
• [Presentation] リーマンの特異点定理の精密化について (2014)
  • Author(s): 中屋敷 厚
  • Organizer: 研究会「リーマン面に関 係する位相幾何学」
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2014-08-25 – 2014-08-29
  • Invited
• [Presentation] An application of the theory of tau function to Riemann's theta function (2014)
  • Author(s): Atsushi Nakayashiki
  • Organizer: Tsuda College mini-workshop on Calabi Yau varieties: Arithmetic Geometry and Physics
  • Place of Presentation: 津田塾大学
  • Year and Date: 2014-08-07 – 2014-08-08
• [Presentation] An application of the theory of tau function to Riemann's theta function (2014)
  • Author(s): Atsushi Nakayashiki
  • Organizer: workshop on Algebraic Curves with Symmetries, Their Jacobians and Integrable Dynamical Systems
  • Place of Presentation: National University of Kyiv-Mohyla Academy, Kiev, Ukraine
  • Year and Date: 2014-07-30 – 2014-08-01
  • Invited
• [Presentation] A Refined Riemann's singularity theorem for sigma functions (2013)
  • Author(s): Atsushi Nakayashiki
  • Organizer: Algebraic topology and Abelian functions
  • Place of Presentation: Steklov Institute, Moscow, Russia
  • Invited
• [Presentation] Riemann surfaces and Schur functions (2013)
  • Author(s): Atsushi Nakayashiki
  • Organizer: String theory, Integrable Systems and Representation theory
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] Riemann surfaces and Schur functions (2013)
  • Author(s): Atsushi Nakayashiki
  • Organizer: Geometry Days in Novosibirsk
  • Place of Presentation: Sobolev Institute of Mathematics、Novosibirsk,Russia
  • Invited
• [Presentation] リーマン面のシューア関数 (2013)
  • Author(s): 中屋敷 厚
  • Organizer: 玉原特殊多様体研究集会
  • Place of Presentation: 東京大学玉原セミナーハウス
• [Presentation] The prime form as a derivative of sigma function (2012)
  • Author(s): 中屋敷 厚
  • Organizer: International Conference on Jacobian varieties, Abelian functions, and Kummer surfaces(招待講演)
  • Place of Presentation: 山梨大学
• [Presentation] Abel-Jacobi map and Schur functions (2011)
  • Author(s): 中屋敷 厚
  • Organizer: Symmetries, Integrable Systems and Representations(招待講演)
  • Place of Presentation: リヨン大学、フランス