Moments, Exponential Sums, and Monodromy Groups

We determine the geometric monodromy groups attached to various families, both one-parameter and multi-parameter, of exponential sums over finite fields, or, more precisely, the geometric monodromy groups of the ℓ-adic local systems on affine spaces in characteristic p> 0 whose trace functions are these exponential sums. The exponential sums here are much more general than we previously were able to consider. As a byproduct, we determine the number of irreducible components of maximal dimension in certain intersections of Fermat surfaces. We also show that in any family of such local systems, say parameterized by an affine space S, there is a dense open set of S over which the geometric monodromy group of the corresponding local system is a fixed known group.