{
  "id": "1111.2002",
  "version": "v1",
  "published": "2011-11-08T18:15:42.000Z",
  "updated": "2011-11-08T18:15:42.000Z",
  "title": "Maass spaces on U(2,2) and the Bloch-Kato conjecture for the symmetric square motive of a modular form",
  "authors": [
    "Krzysztof Klosin"
  ],
  "comment": "60 pages, incorporates an improved version of K. Klosin, Adelic Maass spaces on U(2,2), arxiv:0706.2828 - see section 5",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let K be an imaginary quadratic field of discriminant -D_K<0. We introduce a notion of an adelic Maass space S_{k, -k/2}^M for automorphic forms on the quasi-split unitary group U(2,2) associated with K and prove that it is stable under the action of all Hecke operators. When D_K is prime we obtain a Hecke-equivariant descent from S_{k,-k/2}^M to the space of elliptic cusp forms S_{k-1}(D_K, \\chi_K), where \\chi_K is the quadratic character of K. For a given \\phi \\in S_{k-1}(D_K, \\chi_K), a prime l >k, we then construct (mod l) congruences between the Maass form corresponding to \\phi and hermitian modular forms orthogonal to S_{k,-k/2}^M whenever the l-adic valuation of L^{alg}(\\Sym^2 \\phi, k) is positive. This gives a proof of the holomorphic analogue of the unitary version of Harder's conjecture. Finally, we use these congruences to provide evidence for the Bloch-Kato conjecture for the motives \\Sym^2 \\rho_{\\phi}(k-3) and \\Sym^2 \\rho_{\\phi}(k), where \\rho_{\\phi} denotes the Galois representation attached to \\phi.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-08T18:15:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F33",
      "11F55",
      "11F67",
      "11F80",
      "11F30"
    ],
    "keywords": [
      "bloch-kato conjecture",
      "symmetric square motive",
      "maass spaces",
      "hermitian modular forms orthogonal",
      "imaginary quadratic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.2002K"
    }
  }
}