One visualization of Shimura's complex multiplication theorem via hypergeometric modular functions (Algebraic Number Theory and Related Topics 2015)

"Algebraic Number Theory and Related Topics 2015". November 30 - December 4, 2015. edited by Hiroki Takahashi, Yasuo Ohno and Takahiro Tsushima. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.This article is a sketch of our research work on the subject given by the above title. In the work "Construction of class fields and zeta functions of algebraic curves" (1967) by Goro Shimura (cf. [SmrA], [SmrB] ), it was proved that there exists a modular function (that is called canonical model) that enables to obtain a certain class field (the Shimura class field) of some kind of CM field. In this article we show that for the case of the CM field embedded into the quaternion albebra coming from a co - compact arithmetic triangle group we can determine the canonical model as a hypergeometric modular function in an explicit way. Moreover we give several examples of Hilbert class fields of such kind of CM fields coming from the triangle group Delta(3, 3, 5) . For our work, we use Shimura'sreciplocity law and the existense of the canonical model together with the result by K. Takeuchi (1977) (see [Tku1], [Tku2]). To construct explicit examples we use the modular function for genus 4 pentagonal curves discovered by K. Koike ([Kik]). The first author has written the same subject in the book [Shg] (chapter 8). There he made a detailed explanation of the modular function for Delta(3, 3, 5) . In contract, here we tried to explain the back ground of our research work. By both of two explanations he expects that the readers will have a nicer perspective of the story. For the full argument with exact proofs refer the paper [N - S]