## Advanced Studies in Pure Mathematics

### Twisted covers and specializations

#### Abstract

The central topic is this question: is a given $k$-étale algebra $\prod_l E_l/k$ the specialization of a given $k$-cover $f : X \to B$ at some unramified point $t_0 \in B(k)$? Our main tool is a twisting lemma that reduces the problem to finding $k$-rational points on a certain $k$-variety. Previous forms of this twisting lemma are generalized and unified. New applications are given: a Grunwald form of Hilbert's irreducibility theorem over number fields, a non-Galois variant of the Tchebotarev theorem for function fields over finite fields, some general specialization properties of covers over PAC or ample fields.

#### Article information

Dates
Revised: 18 September 2011
First available in Project Euclid: 24 October 2018

https://projecteuclid.org/ euclid.aspm/1540417817

Digital Object Identifier
doi:10.2969/aspm/06310141

Mathematical Reviews number (MathSciNet)
MR3051242

Zentralblatt MATH identifier
1321.11114

#### Citation

Dèbes, Pierre; Legrand, François. Twisted covers and specializations. Galois–Teichmüller Theory and Arithmetic Geometry, 141--162, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06310141. https://projecteuclid.org/euclid.aspm/1540417817