{
  "id": "1806.02244",
  "version": "v1",
  "published": "2018-06-06T15:29:57.000Z",
  "updated": "2018-06-06T15:29:57.000Z",
  "title": "Construction of elliptic $\\mathfrak{p}$-units",
  "authors": [
    "Werner Bley",
    "Martin Hofer"
  ],
  "comment": "26 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $L/k$ be a finite abelian extension of an imaginary quadratic number field $k$. Let $\\mathfrak{p}$ denote a prime ideal of $\\mathcal{O}_k$ lying over the rational prime $p$. We assume that $\\mathfrak{p}$ splits completely in $L/k$ and that $p$ does not divide the class number of $k$. If $p$ is split in $k/\\mathbb{Q}$ the first named author has adapted a construction of Solomon to obtain elliptic $\\mathfrak{p}$-units in $L$. In this paper we generalize this construction to the non-split case and obtain in this way a pair of elliptic $\\mathfrak{p}$-units depending on a choice of generators of a certain Iwasawa algebra (which here is of rank 2). In our main result we express the $\\mathfrak{p}$-adic valuations of these $\\mathfrak{p}$-units in terms of the $p$-adic logarithm of an explicit elliptic unit. The crucial input for the proof of our main result is the computation of the constant term of a suitable Coleman power series, where we rely on recent work of T. Seiriki.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-06T15:29:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R27",
      "11G16",
      "11G15"
    ],
    "keywords": [
      "construction",
      "imaginary quadratic number field",
      "main result",
      "finite abelian extension",
      "explicit elliptic unit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}