p-adic estimates of exponential sums on curves

The purpose of this article is to prove a "Newton over Hodge" result for exponential sums on curves. Let X be a smooth proper curve over a finite field F-q of characteristic p >= 3 and let V subset of X be an affine curve. For a regular function (f) over bar on V, we may form the L-function L((f) over bar V, s) associated to the exponential sums of (f) over bar. In this article, we prove a lower estimate on the Newton polygon of L((f) over bar, V, s). The estimate depends on the local monodromy of f around each point x is an element of X - V. This confirms a hope of Deligne that the irregular Hodge filtration forces bounds on p-adic valuations of Frobenius eigenvalues. As a corollary, we obtain a lower estimate on the Newton polygon of a curve with an action of Z/pZ in terms of local monodromy invariants.