Construction of class fields by Shimura's canonical models

Abstract

In algebraic number theory, it is a classical problem to construct class fields over imaginary quadratic fields. Indeed we have by the theory of complex multiplication that ring or ray class fields can be generated by using the elliptic modular function $\it{j}$ and the Fricke function. And in 1964 by using the Kronecker`s limit formulas Ramachandra obtained one generator which enables us to construct all class fields. Thus theoretically the problem of constructing class fields was fully settled down. But practically the generators are too complicated to compute their minimal polynomials. Since then, some mathematicians tried to find `good` generators in the sense that their minimal polynomials are computable and have small coefficients. In 1993, using the theory of Shimura reciprocity law, Chen and Yui constructed ring class fields generated by singular values of Thompson series and gave the tables of minimal polynomials. And other similar results were obtained by P. Stevenhagen, A. Gee, R. Schertz, J. K. Koo, C. H. Kim and so on. In this thesis we provide certain new methods of constructing ring or ray class fields by using the theory of Shimura`s canonical models and his reciprocity law. More precisely, by constructing Shimura`s canonical models explicitly, we can conclude that ring or ray class fields are generated by singular values of functions fields of Shimura`s canonical models. By our methods we can partially extend several previously known results. For example, if the genus of a Shimura`s canonical model is 0, then we immediately get that the class field corresponding to the model is generated by a singular value of the corresponding Hauptmodul. In the other part of this thesis we consider the modular equations of the Ramanujan- $G\odblac$ llnitz-Gordon continued fraction. Classically we know that the modular equations of the elliptic modular function $\it{j}$ satisfy the so-called Kronecker`s congruences. In 1993 Chen and Yui proved that the mod...