On the Profinite Regular Inverse Galois Problem

Abstract

Given a field $k$, a $k$-curve $X$ and a $k$-rational divisor $\mathbf{t}\subset X$, we analyze the constraints imposed on $X$ and $\mathbf{t}$ by the existence of abelian $G$-covers $f:Y\to X$ defined over $k$ and unramified outside $\mathbf{t}$. 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 $\ne p$.