Difference equations related with higher genus algebraic curves

2014 Fiscal Year Final Research Report

• 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
• 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

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.

Free Research Field

可積分系とテータ関数