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
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.
Received: 30 April 2011
Revised: 18 September 2011
First available in Project Euclid: 24 October 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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