Generic Newton polygon for exponential sums in two variables with triangular base

Abstract

Let $p$ be a prime number. Every two-variable polynomial $f(x_1,x_2)$ over a finite field of characteristic $p$ defines an Artin--Schreier--Witt tower of surfaces whose Galois group is isomorphic to ${\mathbb{Z}}_p$. Our goal of this paper is to study the Newton polygon of the $L$-functions associated to a finite character of ${\mathbb{Z}}_p$ and a generic polynomial whose convex hull is a fixed triangle $\Delta$. We denote this polygon by $\text{GNP}(\Delta )$. We prove a lower bound of $\text{GNP}(\Delta )$, which we call the improved Hodge polygon $\text{IHP}(\Delta )$, and we conjecture that $\text{GNP}(\Delta )$ and $\text{IHP}(\Delta )$ are the same. We show that if $\text{GNP}(\Delta )$ and $\text{IHP}(\Delta )$ coincide at a certain point, then they coincide at infinitely many points. When $\Delta$ is an isosceles right triangle with vertices $(0,0)$, $(0,d)$ and $(d,0)$ such that $d$ is not divisible by $p$ and that the residue of $p$ modulo $d$ is small relative to $d$, we prove that $\text{GNP}(\Delta )$ and $\text{IHP}(\Delta )$ coincide at infinitely many points. As a corollary, we deduce that the slopes of $\text{GNP}(\Delta )$ roughly form an arithmetic progression with increasing multiplicities.

References in corpus (1)

• Generic Newton Slopes for Artin-Schreier-Witt Tower in two variables