To the Hilbert class field from the hypergeometric modular function

Abstract

In this article we make an explicit approach to the problem: “For a given CM field M, construct its maximal unramified abelian extension C(M) by the adjunction of special values of certain modular functions” in some restricted cases with [M:Q]≥4. We make our argument based on Shimura's main result on the complex multiplication theory of his article in 1967. His main result treats CM fields embedded in a quaternion algebra B over a totally real number field F. We determine the modular function which gives the canonical model for all B's coming from arithmetic triangle groups. That is our main theorem. As its application, we make an explicit case-study for B corresponding to the arithmetic triangle group Δ(3,3,5). By using the modular function of K. Koike obtained in 2003, we show several examples of the Hilbert class fields as an application of our theorem to this triangle group.