Let \(k\) be an algebraically closed field of characteristic \(p\). Fix a smooth base curve \(Y\) over \(k\) and consider Galois covers \(\phi:X \rightarrow Y\) with fixed set of branch points \(B \subset Y\), fixed Galois group \(G\) and fixed inertia subgroups over each point in \(B\). If the cover is tamely ramified, then the Riemann-Hurwitz formula gives a precise formula for the genus of \(X\).
The main purpose of this paper is to show that if wild ramification is present, the genus of \(X\) can be arbitrarily large (in some cases much more precise information is given). The most striking (and interesting) case is an extension of Raynaud’s result about unramified covers of the affine line. If \(G\) is a finite quasi-\(p\) group, then \(G\) occurs as the Galois group of an unramified cover of the affine line by a curve of arbitrarily high genus. More precise results are in fact proved. The methods involve using the patching theory developed by Harbater et al. and by studying local covers. The point is that one can make the higher inertia groups be nontrivial if wild ramification is present and so the genus will increase.