Specialization results in Galois theory

Abstract:

The central topic of this paper is this question: is a given $k$-étale algebra $\prod _lE_l/k$ the specialization of a given $k$-cover $f:X\rightarrow B$ of the same degree at some unramified point $t_0\in B(k)$? We reduce it to finding $k$-rational points on a certain $k$-variety, which we then study over various fields $k$ of diophantine interest: finite fields, local fields, number fields, etc. We have three main applications. The first one is the following Hilbert-Grunwald statement. If $f:X\rightarrow \mathbb {P}^1$ is a degree $n$ $\mathbb {Q}$-cover with monodromy group $S_n$ over $\overline {\mathbb {Q}}$, and finitely many suitably large primes $p$ are given with partitions $\{d_{p,1}, \ldots , d_{p,s_p}\}$ of $n$, there exist infinitely many specializations of $f$ at points $t_0\in \mathbb {Q}$ that are degree $n$ field extensions with residue degrees $d_{p,1}, \ldots , d_{p,s_p}$ at each prescribed prime $p$. The second one provides a description of the separable closure of a PAC field $k$ of characteristic $p\not =2$: it is generated by all elements $y$ such that $y^m-y\in k$ for some $m\geq 2$. The third one involves Hurwitz moduli spaces and concerns fields of definition of covers.