{
  "id": "1308.2261",
  "version": "v1",
  "published": "2013-08-09T23:55:04.000Z",
  "updated": "2013-08-09T23:55:04.000Z",
  "title": "On the rank one abelian Gross-Stark conjecture",
  "authors": [
    "Kevin Ventullo"
  ],
  "comment": "21 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $F$ be a totally real number field, $p$ a rational prime, and $\\chi$ a finite order totally odd abelian character of Gal$(\\bar{F}/F)$ such that $\\chi(\\mathfrak{p})=1$ for some $\\mathfrak{p}|p$. Motivated by a conjecture of Stark, Gross conjectured a relation between the derivative of the $p$-adic $L$-function associated to $\\chi$ at its exceptional zero and the $\\mathfrak{p}$-adic logarithm of a $p$-unit in the $\\chi$ component of $F_\\chi^\\times$. In a recent work, Dasgupta, Darmon, and Pollack have proven this conjecture assuming two conditions: that Leopoldt's conjecture holds for $F$ and $p$, and that if there is only one prime of $F$ lying above $p$, a certain relation holds between the $\\mathscr{L}$-invariants of $\\chi$ and $\\chi^{-1}$. The main result of this paper removes both of these conditions, thus giving an unconditional proof of the conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-09T23:55:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R42",
      "11F41",
      "11F33",
      "11F80"
    ],
    "keywords": [
      "abelian gross-stark conjecture",
      "order totally odd abelian character",
      "finite order totally odd abelian",
      "totally real number field",
      "leopoldts conjecture holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.2261V"
    }
  }
}