{
  "id": "1703.10411",
  "version": "v1",
  "published": "2017-03-30T11:16:08.000Z",
  "updated": "2017-03-30T11:16:08.000Z",
  "title": "On the ratios of Barnes' multiple gamma functions to the $p$-adic analogues",
  "authors": [
    "Tomokazu Kashio"
  ],
  "comment": "25pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $F$ be a totally real field. For each ideal class $c$ of $F$ and each real embedding $\\iota$ of $F$, Hiroyuki Yoshida defined an invariant $X(c,\\iota)$ as a finite sum of log of Barnes' multiple gamma functions with some correction terms. Then the derivative value of the partial zeta function $\\zeta(s,c)$ has a canonical decomposition $\\zeta'(0,c)=\\sum_{\\iota}X(c,\\iota)$, where $\\iota$ runs over all real embeddings of $F$. Yoshida studied the relation between $\\exp(X(c,\\iota))$'s, Stark units, and Shimura's period symbol. Yoshida and the author also defined and studied the $p$-adic analogue $X_p(c,\\iota)$: In particular, we discussed the relation between the ratios $[\\exp(X(c,\\iota)):\\exp_p(X_p(c,\\iota))]$ and Gross-Stark units. In a previous paper, the author proved the algebraicity of some products of $\\exp(X(c,\\iota))$'s. In this paper, we prove its $p$-adic analogue. Then, by using these algebraicity properties, we discuss the relation between the ratios $[\\exp(X(c,\\iota)):\\exp_p(X_p(c,\\iota))]$ and Stark units.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-30T11:16:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R27",
      "11R42",
      "11R80",
      "11S40",
      "11S80",
      "33B15"
    ],
    "keywords": [
      "multiple gamma functions",
      "adic analogue",
      "stark units",
      "partial zeta function",
      "shimuras period symbol"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}