{
  "id": "1809.02584",
  "version": "v1",
  "published": "2018-09-07T17:21:36.000Z",
  "updated": "2018-09-07T17:21:36.000Z",
  "title": "Galois representations attached to elliptic curves with complex multiplication",
  "authors": [
    "Álvaro Lozano-Robledo"
  ],
  "comment": "54 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary quadratic field, and let $\\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\\geq 1$. Let $E$ be an elliptic curve with CM by $\\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\\mathbb{Q}(j(E))$. Let $p\\geq 2$ be a prime, let $G_{\\mathbb{Q}(j(E))}$ be the absolute Galois group of $\\mathbb{Q}(j(E))$, and let $\\rho_{E,p^\\infty}\\colon G_{\\mathbb{Q}(j(E))}\\to \\operatorname{GL}(2,\\mathbb{Z}_p)$ be the Galois representation associated to the Galois action on the Tate module $T_p(E)$. The goal is then to describe, explicitly, the groups of $\\operatorname{GL}(2,\\mathbb{Z}_p)$ that can occur as images of $\\rho_{E,p^\\infty}$, up to conjugation, for an arbitrary order $\\mathcal{O}_{K,f}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-07T17:21:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "14H52",
      "11G15"
    ],
    "keywords": [
      "elliptic curve",
      "complex multiplication",
      "absolute galois group",
      "imaginary quadratic field",
      "adic galois representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}