{
  "id": "1703.09837",
  "version": "v1",
  "published": "2017-03-28T23:22:42.000Z",
  "updated": "2017-03-28T23:22:42.000Z",
  "title": "Kuznetsov, Petersson and Weyl on GL(3), I: The principal series forms",
  "authors": [
    "Jack Buttcane"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The Kuznetsov and Petersson trace formulae for $GL(2)$ forms may collectively be derived from Poincar\\'e series in the space of Maass forms with weight. Having already developed the spherical spectral Kuznetsov formula for $GL(3)$, the goal of this series of papers is to derive the spectral Kuznetsov formulae for non-spherical Maass forms and use them to produce the corresponding Weyl laws; this appears to be the first proof of the existence of such forms not coming from the symmetric-square construction. Aside from general interest in new types of automorphic forms, this is a necessary step in the development of a theory of exponential sums on $GL(3)$. We take the opportunity to demonstrate a sort of minimal method for developing Kuznetsov-type formulae, and produce auxillary results in the form of generalizations of Stade's formula and Kontorovich-Lebedev inversion. This first paper is limited to the non-spherical prinicpal series forms as there are some significant technical details associated with the generalized principal series forms, which will be handled in a separate paper. The best analog of this type of form on $GL(2)$ is the forms of weight one which sometimes occur on congruence subgroups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-28T23:22:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F72",
      "11F55"
    ],
    "keywords": [
      "maass forms",
      "generalized principal series forms",
      "spherical spectral kuznetsov formula",
      "non-spherical prinicpal series forms",
      "petersson trace formulae"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}