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

TL;DR

This paper investigates the Newton polygons of L-functions associated with two-variable polynomials over finite fields, establishing bounds and conjecturing their equality with a constructed Hodge polygon, with specific results for certain triangular cases.

Contribution

It introduces an improved lower bound for the Newton polygon, conjectures their equality with the Hodge polygon, and proves this in special cases involving isosceles right triangles.

Findings

Established a lower bound called the improved Hodge polygon.

Conjectured the Newton polygon and Hodge polygon are equal.

Proved the coincidence at infinitely many points for specific triangles.

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.