L-functions of exponential sums over one-dimensional affinoids: Newton over Hodge

Abstract

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 F¯p be the algebraic closure of the finite field of p elements. Let f¯(x) be any one-variable rational function over F¯p 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 yp−y=f¯(x)⁠.