Representing ideal classes of ray class groups by product of prime ideals of small size

TL;DR

This paper proves that every class in the narrow ray class group of a number field can be represented by a product of three small unramified prime ideals, using explicit bounds derived from advanced analytic number theory techniques.

Contribution

It introduces explicit bounds for representing classes in ray class groups by small prime ideals, improving understanding of their structure and distribution.

Findings

Every class contains a product of three unramified prime ideals of bounded norm.

Derived explicit bounds for the least prime ideal in quadratic subgroups.

Established improved Brun-Titchmarsh theorems for ray class groups.

Abstract

We prove that, for every modulus $\mathfrak{q}$, every class of the narrow ray class group $H_{\mathfrak{q}}(\mathbf{K})$ of an arbitrary number field $\mathbf{K}$ contains a product of three unramified prime ideals $\mathfrak{p}$ of degree one with $\mathfrak{N}\mathfrak{p}\le (t(\mathbf{K})\mathfrak{N}\mathfrak{q})^3$, where $t(\mathbf{K})$ is an explicit function of $\mathbf{K}$ described in the paper. To achieve this result, we first obtain a sharp explicit Brun-Titchmarsh Theorem for ray classes and then an equally explicit improved Brun-Titchmarsh Theorem for large subgroups of narrow ray class groups. En route, we deduce an explicit upper bound for the least prime ideal in a quadratic subgroup of a narrow ray class group and also for the size of the least ideal that is a product of degree one primes in any given class of $H_\mathfrak{q}(\mathbf{K})$.