{
  "id": "2009.07194",
  "version": "v1",
  "published": "2020-09-15T16:04:39.000Z",
  "updated": "2020-09-15T16:04:39.000Z",
  "title": "Theta functions, fourth moments of eigenforms, and the sup-norm problem I",
  "authors": [
    "Ilya Khayutin",
    "Raphael S. Steiner"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We give sharp point-wise bounds in the weight-aspect on fourth moments of modular forms on arithmetic hyperbolic surfaces associated to Eichler orders. Therefore we strengthen a result of Xia and extend it to co-compact lattices, where we improve upon work of Das--Sengupta. We realize this fourth moment by constructing a holomorphic theta kernel on $\\mathbf{G} \\times \\mathbf{G} \\times \\mathbf{SL}_{2}$, for $\\mathbf{G}$ an indefinite inner-form of $\\mathbf{SL}_2$ over $\\mathbb{Q}$, based on the Bergman kernel, and considering its $L^2$-norm in the Weil variable. The constructed theta kernel further gives rise to new elementary theta series for integral quadratic forms of signature $(2,2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-15T16:04:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F72"
    ],
    "keywords": [
      "fourth moment",
      "theta functions",
      "sup-norm problem",
      "eigenforms",
      "arithmetic hyperbolic surfaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}