Arithmetic research of hypergeometric differential equation and its Schwarz map

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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.