Inverse Galois problem for ordinary curves

Abstract

We consider the inverse Galois problem over function fields of positive characteristic [Formula: see text], for example, over the projective line. We describe a method to construct certain Galois covers of the projective line and other curves, which are ordinary in the sense that their Jacobian has maximal [Formula: see text]-torsion. We do this by constructing Galois covers of ordinary semi-stable curves, and then deforming them into smooth Galois covers.