{
  "id": "1106.4249",
  "version": "v4",
  "published": "2011-06-21T16:11:14.000Z",
  "updated": "2019-03-07T11:56:57.000Z",
  "title": "On the arithmetic of Shalika models and the critical values of $L$-functions for ${\\rm GL}(2n)$",
  "authors": [
    "Harald Grobner",
    "A. Raghuram"
  ],
  "comment": "This is an extended and slightly modified version of an article, which appeared in Amer. J. Math. 136 (2014)",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let $\\Pi$ be a cohomological cuspidal automorphic representation of ${\\rm GL}_{2n}(\\mathbb A)$ over a totally real number field $F$. Suppose that $\\Pi$ has a Shalika model. We define a rational structure on the Shalika model of $\\Pi_f$. Comparing it with a rational structure on a realization of $\\Pi_f$ in cuspidal cohomology in top-degree, we define certain periods $\\omega^{\\epsilon}(\\Pi_f)$. We describe the behaviour of such top-degree periods upon twisting $\\Pi$ by algebraic Hecke characters $\\chi$ of $F$. Then we prove an algebraicity result for all the critical values of the standard $L$-functions $L(s, \\Pi \\otimes \\chi)$; here we use the work of B. Sun on the non-vanishing of a certain quantity attached to $\\Pi_\\infty$. As an application, we obtain new algebraicity results in the following cases: Firstly, for the symmetric cube $L$-functions attached to holomorphic Hilbert modular cusp forms; we also discuss the situation for higher symmetric powers. Secondly, for Rankin-Selberg $L$-functions for ${\\rm GL}_3 \\times {\\rm GL}_2$; assuming Langlands Functoriality, this generalizes to Rankin-Selberg $L$-functions of ${\\rm GL}_n \\times {\\rm GL}_{n-1}$. Thirdly, for the degree four $L$-functions for ${\\rm GSp}_4$. Moreover, we compare our top-degree periods with periods defined by other authors. We also show that our main theorem is compatible with conjectures of Deligne and Gross.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-04-23T13:41:56.000Z",
      "title": "On the arithmetic of Shalika models and the critical values of L-functions for GL(2n)",
      "abstract": "Let \\Pi be a cohomological cuspidal automorphic representation of GL_2n(A) over a totally real number field F. Suppose that \\Pi has a Shalika model. We define a rational structure on the Shalika model of \\Pi_f. Comparing it with a rational structure on a realization of \\Pi_f in cuspidal cohomology in top-degree, we define certain periods \\omega^{\\epsilon}(\\Pi_f). We describe the behaviour of such top-degree periods upon twisting \\Pi by algebraic Hecke characters \\chi of F. Then we prove an algebraicity result for all the critical values of the standard L-functions L(s, \\Pi \\otimes \\chi); here we use the recent work of B. Sun on the non-vanishing of a certain quantity attached to \\Pi_\\infty. As an application, we obtain new algebraicity results in the following cases: Firstly, for the symmetric cube L-functions attached to holomorphic Hilbert modular cusp forms; we also discuss the situation for higher symmetric powers. Secondly, for Rankin-Selberg L-functions for GL_3 \\times GL_2; assuming Langlands Functoriality, this generalizes to Rankin-Selberg L-functions of GL_n \\times GL_{n-1}. Thirdly, for the degree four L-functions for GSp_4. Moreover, we compare our top-degree periods with periods defined by other authors. We also show that our main theorem is compatible with conjectures of Deligne and Gross.",
      "comment": "This is the final version. Accepted for publication in the American Journal of Mathematics. 37 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2019-03-07T11:56:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F67",
      "11F41",
      "11F70",
      "11F75",
      "22E55"
    ],
    "keywords": [
      "shalika model",
      "critical values",
      "holomorphic hilbert modular cusp forms",
      "rational structure",
      "arithmetic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.4249G"
    }
  }
}