{
  "id": "2001.04956",
  "version": "v1",
  "published": "2020-01-14T18:28:31.000Z",
  "updated": "2020-01-14T18:28:31.000Z",
  "title": "Galois deformation spaces with a sparsity of automorphic points",
  "authors": [
    "Kevin Childers"
  ],
  "comment": "39 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $k/\\mathbb F_p$ denote a finite field. For any split connected reductive group $G/W(k)$ and certain CM number fields $F$, we deform certain Galois representations $\\overline\\rho:Gal(\\overline F/F) \\to G(k)$ to continuous families $X_{\\overline\\rho}$ of Galois representations $Gal(\\overline F/F) \\to G(\\overline{\\mathbb Q_p})$ lifting $\\overline\\rho$ such that the space of points of $X_{\\overline\\rho}$ which are geometric (in the sense of the Fontaine-Mazur conjecture) with parallel Hodge-Tate weights has positive codimension in $X_{\\overline\\rho}$. Thus the set of points in $X_{\\overline\\rho}$ which could (conjecturally) be associated to automorphic forms is sparse. This generalizes a result of Calegari and Mazur for $F/\\mathbb Q$ quadratic imaginary and $G = GL_2$. The sparsity of automorphic points for $F$ a CM field contrasts with the situation when $F$ is a totally real field, where automorphic points are often provably dense.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-14T18:28:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "galois deformation spaces",
      "automorphic points",
      "galois representations",
      "cm field contrasts",
      "cm number fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}