{
  "id": "1903.08996",
  "version": "v1",
  "published": "2019-03-20T12:56:22.000Z",
  "updated": "2019-03-20T12:56:22.000Z",
  "title": "A zig-zag conjecture and local constancy for Galois representations",
  "authors": [
    "Eknath Ghate"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1901.01728",
  "categories": [
    "math.NT"
  ],
  "abstract": "We make a zig-zag conjecture describing the reductions of irreducible crystalline two-dimensional representations of $G_{{\\mathbb{Q}}_p}$ of half-integral slopes and exceptional weights. Such weights are two more than twice the slope mod $(p-1)$. We show that zig-zag holds for half-integral slopes at most $\\frac{3}{2}$. We then explore the connection between zig-zag and local constancy results in the weight. First we show that known cases of zig-zag force local constancy to fail for small weights. Conversely, we explain how local constancy forces zig-zag to fail for some small weights and half-integral slopes at least $2$. However, we expect zig-zag to be qualitatively true in general. We end with some compatibility results between zig-zag and other results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-20T12:56:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "zig-zag conjecture",
      "galois representations",
      "half-integral slopes",
      "zig-zag force local constancy",
      "local constancy forces zig-zag"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}