Generation of class fields by Siegel-Ramachandra invariants

TL;DR

This paper demonstrates that Siegel-Ramachandra invariants can generate ray class fields of imaginary quadratic fields, providing a modern perspective on classical solutions to the class number one problem through Shimura's reciprocity law.

Contribution

It establishes that Siegel-Ramachandra invariants serve as primitive generators for ray class fields, linking classical and modern approaches in algebraic number theory.

Findings

Siegel-Ramachandra invariants generate ray class fields

Provides a modern explanation for class number one problem

Connects classical solutions with Shimura's reciprocity law

Abstract

We show that the Siegel-Ramachandra invariants could be primitive generators of the ray class fields of imaginary quadratic fields. By using Shimura's reciprocity law we give a modern explanation of the solution of class number one problem given by Heegner and Stark.