{
  "id": "2010.06898",
  "version": "v1",
  "published": "2020-10-14T09:18:40.000Z",
  "updated": "2020-10-14T09:18:40.000Z",
  "title": "A quaternionic construction of $p$-adic singular moduli",
  "authors": [
    "Xavier Guitart",
    "Marc Masdeu",
    "Xavier Xarles"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural $p$-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of $\\mathrm{SL}_2(\\mathbb{Z}[1/p])$ which can be evaluated at real quadratic irrationalities and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article we present a similar construction of cohomology casses in which $\\mathrm{SL}_2(\\mathbb{Z}[1/p])$ is replaced by an order in an indefinite quaternion algebra over a totally real number field $F$. These quaternionic cohomology classes can be evaluated at elements in almost totally complex extensions $K$ of $F$, and we conjecture that the corresponding values lie in algebraic extensions of $K$. We also report on extensive numerical evidence for this algebraicity conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-14T09:18:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "adic singular moduli",
      "quaternionic construction",
      "algebraic extensions",
      "cohomology classes",
      "real quadratic base fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}