Lifting results for rational points on Hurwitz moduli spaces
Résumé
Hurwitz moduli spaces for G-covers of the pro jective line have two classical variants whether G- covers are considered modulo the action of PGL2 on the base or not. A central result of this paper is that, given an integer r ≥ 3 there exists a bound d(r) ≥ 1 depending only on r such that any rational point p rd of a reduced (i.e. modulo PGL2 ) Hurwitz space can be lifted to a rational point p on the non reduced Hurwitz space with [κ(p) : κ(prd )] ≤ d(r). This result can also be generalized to infinite towers of Hurwitz spaces. Introducing a new Galois invariant for G-covers, which we call the base invariant, we improve this result for G-covers with a non trivial base invariant. For the sublocus corresponding to such G-covers the bound d(r) can be chosen depending only on the base invariant (no longer on r) and ≤ 6. When r = 4, our method can still be refined to provide effective criteria to lift k-rational points from reduced to non reduced Hurwitz spaces. This, in particular, leads to a rigidity criterion, a genus 0 method and, what we call an expansion method to realize finite groups as regular Galois groups over Q. Some specific examples are given.