## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Galois–Teichmüller Theory and Arithmetic Geometry, H. Nakamura, F. Pop, L. Schneps and A. Tamagawa, eds. (Tokyo: Mathematical Society of Japan, 2012), 141 - 162

### Twisted covers and specializations

Pierre Dèbes and François Legrand

#### 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**

Received: 30 April 2011

Revised: 18 September 2011

First available in Project Euclid:
24 October 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/aspm/06310141

**Mathematical Reviews number (MathSciNet)**

MR3051242

**Zentralblatt MATH identifier**

1321.11114

**Subjects**

Primary: 11R58: Arithmetic theory of algebraic function fields [See also 14-XX] 12E30: Field arithmetic 12E25: Hilbertian fields; Hilbert's irreducibility theorem 14G05: Rational points 14H30: Coverings, fundamental group [See also 14E20, 14F35]

Secondary: 12Fxx: Field extensions 14Gxx: Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx]

**Keywords**

Specialization algebraic covers twisting lemma Hilbert's irreducibility theorem PAC fields finite fields local fields global fields

#### 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