Hodge-Stickelberger polygons for L-functions of exponential sums of P(x^s)

2007/06/15 by Régis Blache, Blache, Regis, Éric Férard +3
Mathematics · #11 #14 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0706.2340

openalex publication_date 2007/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P(x) be a one-variable Laurent polynomial of degree (d₁,d₂) over a finite field of characteristic p. For any fixed positive integer s not divisible by p, we prove that the (normalized) p-adic Newton polygon of the L-functions of exponential sums of P(x^s) has a tight lower bound which we call `Hodge-Stickelberger polygon', depending only on d<sub>1</sub>,d<sub>2</sub>,s, and (p mod s). This Hodge-Stickelberger polygon is a weighted convolution of a `Hodge polygon' for L-function of exponential sum of P(x) and the `Newton polygon' for L-function of exponential sum of x^s (given by the classical Stickelberger theory). We prove an analogous Hodge-Stickelberger lower bound for multivariable Laurent polynomials as well. We prove this Hodge-Stickelberger polygon is the limit of generic Newton polygons of P(x^s) in a sense that was made explicit in the paper.

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