{
  "id": "1910.09992",
  "version": "v1",
  "published": "2019-10-22T14:09:15.000Z",
  "updated": "2019-10-22T14:09:15.000Z",
  "title": "Power series expansions of modular forms and $p$-adic interpolation of the square roots of Rankin-Selberg special values",
  "authors": [
    "Andrea Mori"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f$ be a newform of even weight $2\\kappa$ for $D^\\times$, where $D$ is a possibly split indefinite quaternion algebra over $\\mathbb{Q}$. Let $K$ be a quadratic imaginary field splitting $D$ and $p$ an odd prime split in $K$. We extend our theory of $p$-adic measures attached to the power series expansions of $f$ around the Galois orbit of the CM point corresponding to an embedding $K\\hookrightarrow D$ to forms with any nebentypus and to $p$ dividing the level of $f$. For the latter we restrict our considerations to CM points corresponding to test objects endowed with an arithmetic $p$-level structure. Also, we restrict these $p$-adic measures to $\\mathbb{Z}_p^\\times$ and compute the corresponding Euler factor in the formula for the $p$-adic interpolation of the \"square roots\" of the Rankin-Selberg special values $L(\\pi_K\\otimes\\xi_r,\\frac12)$ where $\\pi_K$ is the base change to $K$ of the automorphic representation of $\\mathrm{GL}_2$ associated, up to Jacquet-Langland correspondence, to $f$ and $\\xi_r$ is a compatible family of gr\\\"ossencharacters of $K$ with infinite type $\\xi_{r,\\infty}(z)=(z/\\bar z)^{\\kappa+r}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-22T14:09:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F67"
    ],
    "keywords": [
      "rankin-selberg special values",
      "power series expansions",
      "square roots",
      "adic interpolation",
      "modular forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}