{
  "id": "1902.10911",
  "version": "v1",
  "published": "2019-02-28T06:01:45.000Z",
  "updated": "2019-02-28T06:01:45.000Z",
  "title": "Analytic continuation of differential operators and applications to Galois representations",
  "authors": [
    "Ellen E. Eischen",
    "Max Flander",
    "Alexandru Ghitza",
    "Elena Mantovan",
    "Angus McAndrew"
  ],
  "comment": "25 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The main goal of this paper is to describe the effect of certain differential operators on mod $p$ Galois representations associated to automorphic forms on unitary and symplectic groups. As intermediate steps, we analytically continue the mod $p$ reduction of certain $p$-adic differential operators, defined a priori only over the ordinary locus in some of the authors' earlier work, to the whole Shimura variety associated to a unitary or symplectic group; and we explicitly describe the commutation relations between Hecke operators and our differential operators. The motivation for this investigation comes from the special case of $\\mathrm{GL}_2$, where similar operators were used to study the weight part of Serre's conjecture. In that case, one has a convenient description in terms of $q$-expansions, but such a formula-driven approach is not readily accessible in our settings. So, building on ideas of Gross and Katz, we pursue an intrinsic approach that avoids a need for $q$-expansions or analogues.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-28T06:01:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F60",
      "11F80",
      "11G18",
      "11F46",
      "11F55"
    ],
    "keywords": [
      "galois representations",
      "analytic continuation",
      "applications",
      "symplectic group",
      "adic differential operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}