{
  "id": "1704.04606",
  "version": "v1",
  "published": "2017-04-15T08:01:13.000Z",
  "updated": "2017-04-15T08:01:13.000Z",
  "title": "$p$-adic measures associated with zeta values and $p$-adic $\\log$ multiple gamma functions",
  "authors": [
    "Tomokazu Kashio"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study a relation between two refinements of the rank one abelian Gross-Stark conjecture: For a suitable abelian extension $H/F$ of number fields, a Gross-Stark unit is defined as a $p$-unit of $H$ satisfying some proporties. Let $\\tau \\in \\mathrm{Gal}(H/F)$. Yoshida and the author constructed the symbol $Y_p(\\tau)$ by using $p$-adic $\\log$ multiple gamma functions, and conjectured that the $\\log_p$ of a Gross-Stark unit can be expressed by $Y_p(\\tau)$. Dasgupta constructed the symbol $u_T(\\tau)$ by using the $p$-adic multiplicative integration, and conjectured that a Gross-Stark unit can be expressed by $u_T(\\tau)$. In this paper, we give an explicit relation between $Y_p(\\tau)$ and $u_T(\\tau)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-15T08:01:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R27",
      "11R37",
      "11R42",
      "11R80",
      "11S40",
      "11S80",
      "33B15"
    ],
    "keywords": [
      "multiple gamma functions",
      "zeta values",
      "adic measures",
      "gross-stark unit",
      "abelian gross-stark conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}