L-functions of Exponential sums over one-dimensional affinoid: Newton over Hodge

2003/02/28 by Hui June Zhu
Mathematics · #math.NT #math.AG #msc:11 #msc:14

paper · pdf

published as Inter. Math. Research Notices, 2004, no. 30, (2004), 1529--1550 · 17 pages, LaTEX

arxiv created 2005/02/01 · arxiv updated 2009/11/30

Abstract

Let p be a prime and let F_pbar be the algebraic closure of the finite field of p elements. Let f(x) be any one variable rational function over F_pbar with n poles of orders d₁, ...,d_n. Suppose p is coprime to d_i for every i. We prove that there exists a Hodge polygon, depending only on d_i's, which is a lower bound to the Newton polygon of L functions of exponential sums of f(x). Moreover, we show that these two polygons coincide if p=1 mod d_i for every i=1,...,n. As a corollary, we obtain a tight lower bound of Newton polygon of Artin-Schreier curve.

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