Construction of class fields over certain CM fields

Abstract

In this thesis we mainly focus on the generation of ray class fields over cyclotomic fields and imaginary biquadratic fields. We first construct Siegel modular functions $\Phi_{(r,s)}$ for rational vectors $r$, $s$ by the quotient of two theta constants and investigate transformation fomulas of $\Phi_{(r,s)}$. And by using Shimura`s reciprocity law we also construct primitive generators of the ray class fields of cyclotomic fields in terms of singular values of $\Phi_{(r,s)}$ at the CM-point. Over imaginary biquadratic fields $K$, we present certain class fields of $K$ which are generated by ray class invariants of imaginary quadratic subfields of $K$ and provide a necessary and sufficient condition for these class fields to be the ray class fields of $K$. We also generate a primitive generator of ray class fields over the Hilbert class field of the real quadratic subfield of $K$ by using norms of the above ray class invariants, and further find its normal basis.