{
  "id": "1802.09849",
  "version": "v1",
  "published": "2018-02-27T12:21:55.000Z",
  "updated": "2018-02-27T12:21:55.000Z",
  "title": "Bilinear forms with Kloosterman sums, II",
  "authors": [
    "E. Kowalski",
    "Ph. Michel",
    "W. Sawin"
  ],
  "comment": "44 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We prove non-trivial bounds for bilinear forms with hyper-Kloosterman sums modulo a prime $q$ which, for both variables of length $M$, are non-trivial as soon as $M\\geq q^{3/8+\\delta}$ for any $\\delta>0$. This range is identical with the best-known results known for simpler monomial exponentials. The proof combines refinements of the analytic tools from our previous paper, and new geometric results. The key geometric idea is a comparison statement that shows that even when the \"sum-product\" sheaves that appear in the analysis fail to be irreducible, their decomposition reflects that of the \"input\" sheaves, except for parameters in a high-codimension subset. This property is proved by a subtle interplay between \\'etale cohomology in its algebraic and diophantine incarnations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-27T12:21:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11L05",
      "11N37",
      "11N75",
      "11F66",
      "14F20",
      "14D05"
    ],
    "keywords": [
      "bilinear forms",
      "hyper-kloosterman sums modulo",
      "simpler monomial exponentials",
      "diophantine incarnations",
      "best-known results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}