Arithmetic research of hypergeometric differential equation and its Schwarz map

超几何微分方程及其Schwarz图的算术研究

• 批准号: 17540011
• 负责人: SHIGA Hironori
• 金额: $ 2.43万
• 依托单位: Chiba University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540011/
• 关键词: Arithmetic Geometric Mean; Schwarz map; Period Map; Picard Modular form; Hypergeometric Function; Abelian variety; Complex multiplication; アーベル多様体の虚数乗法; 超幾何微分方程式; 保型形式; 周期; テータ函数; 超幾何函数

1) In 1991 Jonathan and Peter Borweins have discovered a nice variant of the GaussAGM (Arithmetic Geometric Mean). It has been an almost unique nice result about GauasAGM. There is no successful study about AGM functions in several variables. In our project we have succeeded to established an AGM function of two variables that is an extension of Borweins' result See the article on J. N. T. 2007.2) We made an investigation on the values of the Schwarz map of the Gauss hypergeometric differential eqations. It can be regarded as a ratio of periods of corresponding hypergeometric curves of the differentioal of the second kind. In our study we fixed the family of hypergeometric curves. And we considered the albebraic values of the Schwarz maps At algebraic points for various differentials of the second kind at a same time. We obtained a general results and some illustrating Examples. See the article Birkhauser Monograph 2007.3) Together with graduate students of the head investigator we made a study of the Schwarz map for the hypergeometric Differential equations of several variables. We obtained some analogous results as 2). The article is under preparation. And We made a investigation about the period map of families of algebraic K3 surfaces. Especially a family with some specific Transcendental lattice of rank 4. That is parametrized by a Hilbert modular surface for the square root of 2. This study is now in Preparation also.

1)1991年，Jonathan和Peter Borweins发现了GaussAGM（算术几何平均）的一个很好的变体。这是GauasAGM几乎独一无二的好结果。关于多元AGM函数的研究还没有成功的文献。在我们的项目中，我们成功地建立了一个二元AGM函数，它是Borweins结果的推广。T. 2007.2）研究了Gauss超几何微分方程的施瓦茨映射的值。它可以看作是第二类微分的相应超几何曲线的周期之比。在我们的研究中，我们固定的家庭超几何曲线。同时考虑了各种第二类微分在代数点上的施瓦茨映射的可解性。我们得到了一般结果和一些例子。见文章Birkhauser专著2007.3）连同研究生的首席调查员，我们做了研究的施瓦茨地图的超几何微分方程的几个变量。我们得到了与2）类似的结果。文章正在编写中。并对代数K3曲面族的周期映射进行了研究。特别是具有某种秩为4的超越格的族。这是参数化的希尔伯特模曲面的平方根2。这项研究目前也在准备中。

项目成果

Complex Function Theory (in Japanese)

复变函数论（日语）

• 发表时间: 2008
• 期刊: Sugakushobou
• 作者: Koike;K. and Shiga;H.;H.shiga
• 通讯作者: H.shiga

A three terms arithmetic-geometric mean

三项算术几何平均数

• 发表时间: 2005
• 期刊: Chiba Technical Report Vol.21-1
• 作者: Shiga;H;K.Koike
• 通讯作者: K.Koike

Extended Gauss AGM corresponding Picard modular forms

扩展高斯 AGM 对应的皮卡德模块化形式

• 发表时间: 2007
• 作者: Shuga;H;K.Koike;Shiga. H.;H. Shiga
• 通讯作者: H. Shiga

Supplement to Extended Gauss AGM, quartic relations among Matsumoto Theta functions

扩展高斯 AGM 的补充，Matsumoto Theta 函数之间的四次关系

• 发表时间: 2006
• 期刊: Technical Report of Mathematical Sciences Chiba Univ. 22
• 作者: K.Koike;H.Shiga;H.Shiga
• 通讯作者: H.Shiga

15週で学ぶ複素関数論

15 周学习复变函数理论

• 发表时间: 2006
• 作者: Sasaki K;Natsugoe S;Ishigami S;Matsumoto M;Okumura H;Setoyama T;Uchikado Y;Kita Y;Tamotsu K;Hanazono K;Owaki T;Aikou T.;豊嶋崇徳;新井仁之;Amagai M;志賀弘典
• 通讯作者: 志賀弘典