Journal of the
Korean Mathematical Society JKMS

We show in many cases that the Siegel-Ramachandra invariants generate the ray class fields over imaginary quadratic fields. As its application we revisit the class number one problem done by Heegner and Stark, and present a new proof by making use of inequality argument together with Shimura's reciprocity law.

Keywords: class field theory, complex multiplication, class numbers, elliptic and modular units, modular and automorphic functions

MSC numbers: Primary 11G16; Secondary 11F03, 11G15, 11R29, 11R37