$S_n$-extensions with prescribed norms

TL;DR

This paper investigates the distribution of $S_n$-extensions of a number field with prescribed norm conditions, deriving explicit formulas for their densities using local mass calculations, especially for prime and small degrees.

Contribution

It provides explicit formulas for the density of $S_n$-extensions with prescribed norms, including Euler product expressions and computational algorithms for local masses.

Findings

Density expressed as an explicit Euler product for prime $n$

Formulas for local masses at various places of $k$

Efficient algorithm for computing local masses when $n=4$

Abstract

Given a number field $k$, a finitely generated subgroup $\mathcal{A}\subseteq k^\times$, and an integer $n\geq 3$, we study the distribution of $S_n$-extensions of $k$ such that the elements of $\mathcal{A}$ are norms. For $n\leq 5$, and conjecturally for $n \geq 6$, we show that the density of such extensions is the product of so-called ``local masses'' at the places of $k$. When $n$ is an odd prime, we give formulas for these local masses, allowing us to express the aforementioned density as an explicit Euler product. For $n=4$, we determine almost all of these masses exactly and give an efficient algorithm for computing the rest, again yielding an explicit Euler product.