# Inverse Galois problem for ordinary curves

## TL;DR

This paper addresses the inverse Galois problem over function fields in positive characteristic by developing a method to construct Galois covers of ordinary curves, leveraging deformation techniques from semi-stable to smooth curves.

## Contribution

It introduces a novel approach to construct Galois covers of ordinary curves in positive characteristic using deformation of semi-stable curves.

## Key findings

- Constructed Galois covers of ordinary semi-stable curves.
- Deformed semi-stable covers into smooth Galois covers.
- Provided new insights into the inverse Galois problem in positive characteristic.

## Abstract

We consider the inverse Galois problem over function fields of positive characteristic p, for example, the inverse Galois problem 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 p-torsion. We do this by constructing Galois covers of ordinary semi-stable curves, and then deforming them into smooth Galois covers.