2011/06/30 by Pierre Dèbes, Dèbes, Pierre, François Legrand +1
Mathematics · #12E25 #12E30 #14G05 #14Gxx #14H10 #14H30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Primary 11R58 #Secondary 12Fxx
paper · pdf · doi:10.48550/arxiv.1106.6151
openalex publication_date 2011/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper has three main applications. The first one is this Hilbert-Grunwald statement. If f:X→ \Pp1 is a degree n \Qq-cover with monodromy group Sn over \Qq, and finitely many suitably big primes p are given with partitions \dp,1,..., dp,sp\ of n, there exist infinitely many specializations of f at points t0∈ \Qq that are degree n field extensions with residue degrees dp,1,..., dp,sp at each prescribed prime p. The second one provides a description of the se-pa-ra-ble closure of a PAC field k of characteristic p\not=2: it is generated by all elements y such that ym-y∈ k for some m≥ 2. The third one involves Hurwitz moduli spaces and concerns fields of definition of covers. A common tool is a criterion for an étale algebra ∏lEl/k over a field k to be the specialization of a k-cover f:X→ B at some point t0∈ B(k). The question is reduced to finding k-rational points on a certain k-variety, and then studied over the various fields k of our applications.