Advanced Studies in Pure Mathematics

Twisted covers and specializations

Pierre Dèbes and François Legrand

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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

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

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


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