Lifting results for rational points on Hurwitz moduli spaces

Abstract

Abstract. Hurwitz moduli spaces for G-covers of the projective line have two classical variants whether Gcovers are considered modulo the action of PGL 2 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 PGL 2 ) Hurwitz space can be lifted to a rational point p on the non reduced Hurwitz space with [κ(p) : κ(p rd )] ≤ 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.

On the Profinite Regular Inverse Galois Problem

“…We carry out the construction of this obstruction in section 3.1.2. We conclude by giving a moduli formulation of our main result (Corollary 3.6) using variants of the techniques of [3]. §1.…”

“…(2) k is an l-adic field and A has anisotropic reduction 3 . Furthermore, in that case, the constant n A can be taken depending only on the dimension dim(A) 2 Recall that, given a field k, the cyclotomic closure of k in k is the subextension of k obtained from k by adjoining all the nth roots of unity in k; we will denote it by k cyc .…”

“…Furthermore, in that case, the constant n A can be taken depending only on the dimension dim(A) 2 Recall that, given a field k, the cyclotomic closure of k in k is the subextension of k obtained from k by adjoining all the nth roots of unity in k; we will denote it by k cyc . 3 Recall that the identity component of the special fiber A of the Neron model of A is an extension (over the residue field …”

“…To prove this, one can adapt the techniques of [3] as follows. By hypothesis, for each σ ∈ Γ k there exists a weak k-G-isomorphism from f n :…”

Abelian constraints in inverse Galois theory

“…The analog of the Modular Tower Diophantine Conjecture for PGL 2 -reduced modular towers is a priori stronger than the original one; from corollary 3.12 of [Cad07b] the two versions are actually equivalent if the dependence in K of the constants involved is through [K : Q].…”