On a problem of Hasse and Ramachandra

TL;DR

This paper resolves a question by Hasse and Ramachandra about generating ray class fields of imaginary quadratic fields using Weber function values, specifically for ideals generated by integers greater than one.

Contribution

It provides a complete answer to whether the ray class field modulo (N) can be generated by a single Weber function value for all N > 1.

Findings

Ray class fields modulo (N) are generated by Weber function values for N > 1.

The question of generation by Weber functions is fully answered for the case of ideals generated by integers.

The results extend the understanding of explicit class field theory for imaginary quadratic fields.

Abstract

Let $K$ be an imaginary quadratic field, and let $\mathfrak{f}$ be a nontrivial integral ideal of $K$. Hasse and Ramachandra asked whether the ray class field of $K$ modulo $\mathfrak{f}$ can be generated by a single value of the Weber function. We completely resolve this question when $\mathfrak{f}=(N)$ for an integer $N>1$.