2016/05/30 by Pierre Dèbes, Dèbes, Pierre
Mathematics · #11Gxx #11R58 #12E30 #14E20 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Primary 12F12 #Secondary 14E22
paper · pdf · doi:10.48550/arxiv.1605.09363
openalex publication_date 2016/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We disprove a strong form of the Regular Inverse Galois Problem: there exist finite groups G which do not have a realization F/\Qq(T) that induces all Galois extensions L/\Qq(U) of group G by specializing T to f(U) ∈ \Qq(U). For these groups, we produce two extensions L/\Qq(U) that cannot be simultaneously induced, thus even disproving a weaker Lifting Property. Our examples of such groups G include symmetric groups Sn, n≥ 7, infinitely many PSL2(\Ffp), the Monster. Two variants of the question with \Qq(U) replaced by \Cc(U) and \Qq are answered similarly, the second one under a diophantine "working hypothesis" going back to a problem of Schinzel. We introduce two new tools: a comparizon theorem between the invariants of an extension F/\Cc(T) and those obtained by specializing T to f(U) ∈ \Cc(U), and, given two regular Galois extensions of k(T), a finite set of polynomials P(U,T,Y) that say whether these extensions have a common specialization E/k.