Arithmetic research of hypergeometric differential equation and its Schwarz map

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。对于GauasAGM来说，这几乎是一个独一无二的好结果。目前还没有关于AGM函数在多个变量中的成功研究。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，见J.N.T.，2007.2)我们研究了Gauss超几何微分方程解的Schwarz映射值。它可以看作是相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。同时考虑了第二类微分在代数点上的Schwarz映射的反照值。我们得到了一般性的结果和一些例证。见文章Birkhauser专著2007.3)与首席研究员的研究生一起研究了多元超几何微分方程解的Schwarz映射。我们得到了一些类似于2)的结果。这篇文章正在准备中。并对代数K3曲面族的周期映射进行了研究。特别是具有一些特定的4阶超验格的族。这是由2的平方根的Hilbert模曲面参数化的。这项研究现在也在准备中。

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.