To the Hilbert class field from the hypergeometric modular function

TL;DR

This paper explicitly constructs the Hilbert class field of certain CM fields using special values of hypergeometric modular functions, based on Shimura's complex multiplication theory and explicit case studies.

Contribution

It determines the modular functions providing canonical models for quaternion algebras from arithmetic triangle groups, extending complex multiplication theory to explicit higher degree cases.

Findings

Explicit construction of Hilbert class fields via hypergeometric modular functions.

Identification of canonical models for quaternion algebras from arithmetic triangle groups.

Examples of Hilbert class fields embedded in specific quaternion algebra cases.

Abstract

In this article we make an explicit approach to the higher degree case of the problem: " For a given $CM$ field $M$, construct its maximal abelian extension $C(M)$ (i.e. the Hilbert class field) by the adjunction of special values of certain modular functions" in a restricted case. We make our argument based on Shimura's main result on the complex multiplication theory of his article in 1967. His main result is constructed for a quaternion algebra $B$ over a totally real number field $F$. We determine the modular function which gives the canonical model for the case $B$ is coming from an arithmetic triangle group. That is our main theorem. And we make an explicit case-study for $B$ corresponding to the triangle group $\Delta (3,3,5)$. The corresponding canonical model appears as a restriction of the Appell's hypergeometric modular function on a 2-dimensional hyperball to a hyperplane section. That is a modular function for the family of the Koike pentagonal curves $w^5=z(z-1)(z-\lambda_1)(z-\lambda_2)$ with two parameters $\lambda_1,\lambda_2$. We use the result of K. Koike in 2003 to get an theta representation of the canonical model function. By using this expression, we show several examples of the Hilbert class fields of the $CM$ fields those are embedded in the above $B$.