Wildly ramified covers with large genus

Abstract

We study wildly ramified G-Galois covers $\ddot{I}•:Y\to X$ branched at B (defined over an algebraically closed field of characteristic p). We show that curves Y of arbitrarily high genus occur for such covers even when G, X, B and the inertia groups are fixed. The proof relies on a Galois action on covers of germs of curves and formal patching. As a corollary, we prove that for any nontrivial quasi-p group G and for any sufficiently large integer $\ddot{I}ƒ$ with $p\nmid \ddot{I}ƒ$, there exists a G-Galois étale cover of the affine line with conductor $\ddot{I}ƒ$ above the point $\infty$.