{
  "id": "1701.00254",
  "version": "v1",
  "published": "2017-01-01T15:55:29.000Z",
  "updated": "2017-01-01T15:55:29.000Z",
  "title": "Generic Newton polygon for exponential sums in two variables with triangular base",
  "authors": [
    "Rufei Ren"
  ],
  "categories": [
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-01T15:55:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generic newton polygon",
      "exponential sums",
      "triangular base",
      "isosceles right triangle",
      "finite character"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}