Arithmetic research of hypergeometric differential equation and its Schwarz map

1) 1991年，乔纳森和彼得·鲍温斯发现了高斯均值（算术几何均值）的一个很好的变体。这是关于gaasagm的一个几乎独一无二的好结果。关于多变量的AGM函数，目前还没有成功的研究。在我们的项目中，我们成功地建立了一个两个变量的AGM函数，这是对Borweins结果的扩展（见文章j.n.t. 2007.2）。我们对高斯超几何微分方程的Schwarz映射的值进行了研究。它可以看作是对应的第二类微分超几何曲线的周期之比。在我们的研究中，我们固定了超几何曲线族。我们同时考虑了Schwarz映射在代数点上的代数值对于各种二阶微分。我们得到了一般的结果和一些举例说明。（见文章Birkhauser Monograph 2007.3）与首席研究员的研究生一起，我们对几个变量的超几何微分方程的Schwarz映射进行了研究。我们得到了一些类似的结果如2)。文章正在准备中。并对代数K3曲面族的周期图进行了研究。特别是具有特定的4阶超越格的一族。它被根2的希尔伯特模曲面参数化了。这项研究现在也在准备阶段。

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.