Monodromy Groups of Coverings of Curves

We consider nite separable coverings of curves f : X → Y over a eld of characteristic p ≥ 0. We are interested in describing the possible monodromy groups of this cover if the genus of X is xed. There has been much progress on this problem over the past decade in characteristic zero. Recently Frohardt and Magaard completed the nal step in resolving the Guralnick Thompson conjecture showing that only nitely many nonabelian simple groups other than alternating groups occur as composition factors for a xed genus. There is an ongoing project to get a complete list of the monodromy groups of indecomposable rational functions with only tame rami cation. In this article, we focus on positive characteristic. There are more possible groups but we show that many simple groups do not occur as composition factors for a xed genus. We also give a reduction theorem reducing the problem to the case of almost simple groups. We also obtain some results on bounding the size of automorphism groups of curves in positive characteristic and discuss the relationship with the rst problem. We note that prior to these results there was not a single example of a nite simple group which could be ruled out as a composition factor of the monodromy group of a rational function in any positive characteristic.