Galois specialization to symmetric points and the inverse Galois problem up to $S_n$

TL;DR

This paper explores a version of Hilbert's irreducibility theorem involving Galois covers and symmetric points, providing new insights into the inverse Galois problem and its solutions over number fields.

Contribution

It introduces a Galois specialization method to symmetric points, advancing the understanding of the inverse Galois problem for arbitrary finite groups.

Findings

Existence of degree n, S_n-extensions over Q for large n

Solutions to the inverse Galois problem for any finite group G

Corollaries for moduli spaces of various types

Abstract

The paper is concerned with the following version of Hilbert's irreducibility theorem: if $\pi: X \to Y$ is a Galois $G$-covering of varieties over a number field $k$ and $H \subset G$ is a subgroup, then for all sufficiently large and sufficiently divisible $n$ there exist a degree $n$ closed point $y \in |Y|$ and $x \in \pi^{-1}(y)$ for which $k(x)/k(y)$ is a Galois $H$-extension, and $k(y)/k$ is an $S_n$-extension. The result has interesting corollaries when applied to moduli spaces of various kinds. For instance, for every finite group $G$ there is a constant $N$ such that for all $n>N$ there is a degree $n$, $S_n$-extension $F/\mathbb{Q}$ such that over $F$ the inverse Galois problem for $G$ has a solution.