Twisted covers and specializations

2011/06/30 by Dèbes, Pierre, Legrand, François
#12E25 #12E30 #14G05 #14Gxx #14H10 #14H30 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary 11R58 #Secondary 12Fxx

paper · doi:10.48550/arxiv.1106.6154

Abstract

The central topic is this question: is a given k-étale algebra ∏_lE_l/k the specialization of a given k-cover f:X→ B at some point t₀∈ B(k)? Our main tool is a \it 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.

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