{
  "id": "1310.2988",
  "version": "v3",
  "published": "2013-10-10T23:54:16.000Z",
  "updated": "2014-10-01T23:46:42.000Z",
  "title": "From the function-sheaf dictionary to quasicharacters of $p$-adic tori",
  "authors": [
    "Clifton Cunningham",
    "David Roe"
  ],
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "We introduce the category of quasicharacter sheaves on smooth commutative group schemes $G$ over finite fields $k$ and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We show that the group of isomorphism classes of quasicharacter sheaves is an extension of the group of characters of $G(k)$ by a cohomology group determined by the component group scheme of $G$. As an application, we find a category of sheaves for which isomorphism classes correspond to quasicharacters in the case of algebraic tori, abelian varieties and unipotent wound groups over local fields. In this geometrization of quasicharacters, our sheaves are local systems on Greenberg transforms of locally finite type N\\'eron models.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-11T12:23:02.000Z",
      "title": "A function-sheaf dictionary for algebraic tori over local fields",
      "abstract": "We introduce and study the category of quasicharacter sheaves on smooth, commutative group schemes G locally of finite type over finite fields k. Assuming that the geometric component group of G is finitely-generated, we show that the group of isomorphism classes of quasicharacter sheaves on G is canonically isomorphic to the group of characters of G(k). Then, for any non-archimedean local field K and any algebraic torus T over K, we use this result to produce a functorial isomorphism between quasicharacters of T(K) and isomorphism classes of quasicharacter sheaves on the Greenberg transform of the N\\'eron model of T.",
      "comment": "Fix typos and error in proof of Lemma 14.1",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-10-01T23:46:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14L15",
      "11G25",
      "14F05",
      "14D24",
      "11S31",
      "22E50"
    ],
    "keywords": [
      "algebraic torus",
      "function-sheaf dictionary",
      "quasicharacter sheaves",
      "isomorphism classes",
      "geometric component group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.2988C"
    }
  }
}