Nonrigid constructions in Galois theory

Harbater-Mumford subvarieties of moduli spaces of covers

“…We first recall succintly the description of G-covers defined over R with prescribed invariants given in [DF94]. We will use it in the next paragraph.…”

“…In this context, two main methods can be distinguished: on the one hand, genus 0 methods [M89] which may provide Q or Q ab -rational points on usually low-dimensional Hurwitz spaces and, on the other hand, large field methods [DF94], [D95], [Des95] 1 , which combine irreducibility Conway and Parker type results [FV91], realization results over local fields [H03], [DF94] and the local-global principle for varieties [Mo89], [P96] to provide Q Σ2 -rational points. Our main theorem (theorem 2.2) conjoins these two aspects: it is, as Conway and Parker's theorem, a global structure result about high-dimensional Hurwitz spaces but, as genus 0 methods, it deals with low-dimensional closed subvarieties (of those high-dimensional Hurwitz spaces) obtained by specializing most of the branch points.…”

Counting real Galois covers of the projective line

“…One of the main results of [Dèbes and Fried 1994] is that, given t ∈ ᐁ r ordered according to the Convention above, there exists an identification ( r,G ) −1 (t ) sn(C, G), as recalled above, such that sn mod,‫ޒ‬ (C; r 1 , r 2 ) is exactly the set of those G-covers in sn(C, G) whose field of moduli is contained in ‫,ޒ‬ and sn ‫ޒ‬ (C; r 1 , r 2 ) is the set of those G-covers in sn(C, G) that are defined over ‫.ޒ‬ A complete proof of this can be found in [Dèbes and Fried 1994]. We only recall the main ideas.…”

“…We also consider the related problem of how many G-covers have their field of moduli contained in ‫.ޒ‬ For these two questions P. Dèbes and M. Fried [1994] showed that there is also a group theoretic characterization: the r -tuples (g 1 , . .…”

Introduction to modular towers: generalizing dihedral group–modular curve connections

“…Here G is the dihedral group D p of order 2p. Also, r = 4 a n d C is four repetitions of the involution conjugacy class in D p DFr,x5.1{5.2]. Part I (especially, xI.D) discusses example problems with ready applications for modular towers.…”