{
  "id": "1909.06905",
  "version": "v1",
  "published": "2019-09-15T23:00:55.000Z",
  "updated": "2019-09-15T23:00:55.000Z",
  "title": "$p$-adic estimates of exponential sums on curves",
  "authors": [
    "Joe Kramer-Miller"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "The purpose of this article is to prove a ``Newton over Hodge'' result for exponential sums on curves. Let $X$ be a smooth proper curve over a finite field $\\mathbb{F}_q$of characteristic $p\\geq 5$ and let $V \\subset X$ be an affine curve. For a regular function $\\overline{f}$ on $V$, we may form the $L$-function $L(\\overline{f},V,s)$ associated to the exponential sums of $\\overline{f}$. In this article, we prove a lower estimate on the Newton polygon of $L(\\overline{f},V,s)$. This estimate depends on the local monodromy of $f$ around each point $x \\in X-V$. As a corollary, we obtain a lower estimate on the Newton polygon of a curve with an action of $\\mathbb{Z}/p\\mathbb{Z}$ in terms of local monodromy invariants.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-15T23:00:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F30",
      "11T23",
      "11G20"
    ],
    "keywords": [
      "exponential sums",
      "adic estimates",
      "newton polygon",
      "lower estimate",
      "smooth proper curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}