Construction of ray class fields by smaller generators and applications

TL;DR

This paper extends the generation of ray class fields over imaginary quadratic fields using Siegel-Ramachandra invariants, producing smaller generators and applying them to solve quadratic Diophantine equations.

Contribution

It introduces a method to generate ray class fields with smaller generators via Siegel-Ramachandra invariants and applies these to solve specific quadratic Diophantine equations.

Findings

Generated ray class fields using Siegel-Ramachandra invariants.

Constructed ray class invariants with minimal polynomials having small coefficients.

Solved certain quadratic Diophantine equations using these invariants.

Abstract

We first generate ray class fields over imaginary quadratic fields in terms of Siegel-Ramachandra invariants, which would be an extension of Schertz's result. And, by making use of quotients of Siegel-Ramachandra invariants we also construct ray class invariants over imaginary quadratic fields whose minimal polynomials have relatively small coefficients, from which we are able to solve certain quadratic Diophantine equations.