T-adic exponential sums over affinoids

We introduce and develop (pi 1/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, pi). We compute the lower (pi 1/c, p)-adic bound, the Hodge polygon, for this C -function. Along the way, we also show why a strictly pi-adic theory will not work in this case.(c) 2022 Elsevier Inc. All rights reserved.