T-adic exponential sums over affinoids

Abstract

We introduce and develop (π1/c,p)-adic Dwork theory for L-functions of exponential sums associated to one-variable rational functions, interpolating pk-order exponential sums over affinoids. Namely, we prove a generalization of the Dwork-Monsky-Reich trace formula and apply it to establish an analytic continuation of the C-function Cf(s,π). We compute the lower (π1/c,p)-adic bound, the Hodge polygon, for this C-function. Along the way, we also show why a strictly π-adic theory will not work in this case.