Non-parametricity of rational translates of regular Galois extensions

TL;DR

This paper proves that for regular Galois extensions over number fields, most rational translates are non-parametric, meaning they do not generate all Galois extensions with the same group through specialization.

Contribution

It generalizes Legrand's result by showing that almost all rational translates of a given regular Galois extension are non-parametric over number fields.

Findings

Most rational translates are non-$G$-parametric.

Non-parametricity holds for almost all rational functions of a given degree.

The result applies to Galois extensions over number fields with group $G$.

Abstract

We generalize a result of F.\ Legrand about the existence of non-parametric Galois extensions for a given group $G$. More precisely, for a $K$-regular Galois extension $F|K(t)$, we consider the translates $F(s)|K(s)$ by an extension $K(s)|K(t)$ of rational function fields (in other words, $s$ is a root of $g(X)-t$ for some rational function $g\in K(X)$). We then show that if $F|K(t)$ is a $K$-regular Galois extension with group $G$ over a number field $K$, then for any degree $k\ge 2$ and almost all (in a density sense) rational functions $g$ of degree $k$, the translate of $F$ by a root field of $g(X)-t$ over $K(t)$ is non-$G$-parametric, i.e.\ not all Galois extensions of $K$ with group $G$ arise as specializations of $F(s)|K(s)$.