{
  "id": "1206.3186",
  "version": "v3",
  "published": "2012-06-14T17:17:10.000Z",
  "updated": "2012-08-30T15:48:01.000Z",
  "title": "The LS method for the classical groups in positive characteristic and the Riemann Hypothesis",
  "authors": [
    "Luis Alberto Lomelí"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We provide a definition for an extended system of $\\gamma$-factors for products of generic representations $\\tau$ and $\\pi$ of split classical groups or general linear groups over a non-archimedean local field of characteristic $p$. We prove that our $\\gamma$-factors satisfy a list of axioms (under the assumption $p \\neq 2$ when both groups are classical groups) and show their uniqueness (in general). This allows us to define extended local $L$-functions and root numbers. We then obtain automorphic $L$-functions $L(s,\\tau \\times \\pi)$, where $\\tau$ and $\\pi$ are globally generic cuspidal automorphic representations of split classical groups or general linear groups over a global function field. In addition to rationality and the functional equation, we prove that our automorphic $L$-functions satisfy the Riemann Hypothesis.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-08-30T15:48:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F70",
      "22E50",
      "22E55"
    ],
    "keywords": [
      "riemann hypothesis",
      "ls method",
      "positive characteristic",
      "general linear groups",
      "split classical groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.3186L"
    }
  }
}