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

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.