Generic Newton polygon for exponential sums in $n$ variables with parallelotope base

Abstract

Let $p$ be a prime number. Every $n$-variable polynomial $f(\underline{x})$ over a finite field of characteristic $p$ defines an Artin--Schreier--Witt tower of varieties whose Galois group is isomorphic to ${\mathbb{Z}}_p$. Our goal of this paper is to study the Newton polygon of the $L$-function associated to a finite character of ${\mathbb{Z}}_p$ and a generic polynomial whose convex hull is an $n$-dimensional paralleltope $\Delta$. We denote this polygon by $\mathrm{GNP}(\Delta )$. We prove a lower bound of $\mathrm{GNP}(\Delta )$, which is called the improved Hodge polygon $\mathrm{IHP}(\Delta )$. We show that $\mathrm{IHP}(\Delta )$ lies above the usual Hodge polygon $\mathrm{HP}(\Delta )$ at certain infinitely many points, and when $p$ is larger than a fixed number determined by $\Delta$, it coincides with $\mathrm{GNP}(\Delta )$ at these points. As a corollary, we roughly determine the distribution of the slopes of $\mathrm{GNP}(\Delta )$.

arXiv comment36 pages