Generation of ring class fields by eta-quotients

TL;DR

This paper presents a method to generate ring class fields of imaginary quadratic fields using special values of eta-quotients, leading to minimal polynomials with small coefficients useful for solving quadratic Diophantine equations.

Contribution

It introduces a novel approach linking eta-quotients and Siegel-Ramachandra invariants to construct class fields with practical applications.

Findings

Generated minimal polynomials with small coefficients

Solved quadratic Diophantine equations involving non-convenient numbers

Connected eta-quotients to class field generation

Abstract

We generate ring class fields of imaginary quadratic fields in terms of the special values of certain eta-quotients, which are related to the relative norms of Siegel-Ramachandra invariants. These give us minimal polynomials with relatively small coefficients from which we are able to solve certain quadratic Diophantine equations concerning non-convenient numbers.