{
  "id": "1608.07527",
  "version": "v1",
  "published": "2016-08-26T17:20:13.000Z",
  "updated": "2016-08-26T17:20:13.000Z",
  "title": "Period relations and special values of Rankin-Selberg $L$-functions",
  "authors": [
    "Michael Harris",
    "Jie Lin"
  ],
  "comment": "To appear in the Proceedings of the conference \"Representation theory, number theory, and invariant theory\" in honor of Roger Howe.|",
  "categories": [
    "math.NT"
  ],
  "abstract": "This is a survey of recent work on values of Rankin-Selberg $L$-functions of pairs of cohomological automorphic representations that are {\\it critical} in Deligne's sense. The base field is assumed to be a CM field. Deligne's conjecture is stated in the language of motives over $\\QQ$, and express the critical values, up to rational factors, as determinants of certain periods of algebraic differentials on a projective algebraic variety over homology classes. The results that can be proved by automorphic methods express certain critical values as (twisted) period integrals of automorphic forms. Using Langlands functoriality between cohomological automorphic representations of unitary groups, which can be identified with the de Rham cohomology of Shimura varieties, and cohomological automorphic representations of $GL(n)$, the automorphic periods can be interpreted as motivic periods. We report on recent results of the two authors, of the first-named author with Grobner, and of Guerberoff.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-26T17:20:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F67",
      "11G18",
      "11F75"
    ],
    "keywords": [
      "period relations",
      "cohomological automorphic representations",
      "special values",
      "rankin-selberg",
      "automorphic methods express"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}