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 $\textrm{GNP}(\Delta)$. We prove a lower bound of $\textrm{GNP}(\Delta)$, which we call the improved Hodge polygon $\textrm{IHP}(\Delta)$, and we conjecture that $\textrm{GNP}(\Delta)$ and $\textrm{IHP}(\Delta)$ are the same. We show that if $\textrm{GNP}(\Delta)$ and $\textrm{IHP}(\Delta)$ 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 $\textrm{GNP}(\Delta)$ and $\textrm{IHP}(\Delta)$ coincide at infinitely many points. As a corollary, we deduce that the slopes of $\textrm{GNP}(\Delta)$ roughly form an arithmetic progression with increasing multiplicities.