{
  "id": "1701.08733",
  "version": "v1",
  "published": "2017-01-30T18:05:11.000Z",
  "updated": "2017-01-30T18:05:11.000Z",
  "title": "On slopes of $L$-functions of $\\mathbb{Z}_p$-covers over the projective line",
  "authors": [
    "Michiel Kosters",
    "Hui June Zhu"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $P: \\cdots \\rightarrow C_2\\rightarrow C_1\\rightarrow {\\mathbb P}^1$ be a $\\mathbb{Z}_p$-cover of the projective line over a finite field of cardinality $q$ and characteristic $p$ which ramifies at exactly one rational point, and is unramified at other points. In this paper, we study the $q$-adic valuations of the reciprocal roots in $\\mathbb{C}_p$ of $L$-functions associated to characters of the Galois group of $P$. We show that for all covers $P$ such that the genus of $C_n$ is a quadratic polynomial in $p^n$ for $n$ large, the valuations of these reciprocal roots are uniformly distributed in the interval $[0,1]$. Furthermore, we show that for a large class of such covers $P$, the valuations of the reciprocal roots in fact form a finite union of arithmetic progressions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-30T18:05:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T23",
      "11L07",
      "13F35",
      "11R58"
    ],
    "keywords": [
      "projective line",
      "reciprocal roots",
      "finite field",
      "adic valuations",
      "rational point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}