On the Profinite Regular Inverse Galois Problem

Abstract

Given a field k, a k-curve X and a k-rational divisor t ⊂ X, we analyze the constraints imposed on X and t by the existence of abelian G-covers f : Y → X defined over k and unramified outside t. We show that these constraints produce an obstruction to the weak regular inverse Galois problem for a whole class of profinite groups - we call p-obstructed - when k is a finitely generated field of characteristic ≠ p.