L-functions of Exponential Sums over One-dimensional Affinoids: Newton over Hodge

This paper proves a sharp lower bound for Newton polygons of L-functions of exponential sums of one-variable rational functions. Let p be a prime and let Fp be the algebraic closure of the finite field of p elements. Let f(x) be any one-variable rational function over Fp with l poles of orders d1, . . . , dl. Suppose p is coprime to d1 · · · dl. We prove that there exists a tight lower bound which we call Hodge polygon, depending only on the dj ’s, to the Newton polygon of L-function of exponential sums of f(x). Moreover, we show that for any f(x) these two polygons coincide if and only if p ≡ 1 mod dj for every 1 ≤ j ≤ l. As a corollary, we obtain a tight lower bound for the p-adic Newton polygon of zeta-function of an Artin-Schreier curve given by affine equations y − y = f(x).