{
  "id": "1801.05380",
  "version": "v1",
  "published": "2018-01-16T17:32:44.000Z",
  "updated": "2018-01-16T17:32:44.000Z",
  "title": "On Hecke eigenvalues of Siegel modular forms in the Maass space",
  "authors": [
    "Sanoli Gun",
    "Biplab Paul",
    "Jyoti Sengupta"
  ],
  "comment": "10 pages, To appear Forum Math",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this article, we prove an omega-result for the Hecke eigenvalues $\\lambda_F(n)$ of Maass forms $F$ which are Hecke eigenforms in the space of Siegel modular forms of weight $k$, genus two for the Siegel modular group $Sp_2(\\Z)$. In particular, we prove $$ \\lambda_F(n)= \\Omega(n^{k-1}\\text{exp} (c \\frac{\\sqrt{\\log n}}{\\log\\log n})), $$ when $c>0$ is an absolute constant. This improves the earlier result $$ \\lambda_F(n)= \\Omega(n^{k-1} (\\frac{\\sqrt{\\log n}}{\\log\\log n})) $$ of Das and the third author. We also show that for any $n \\ge 3$, one has $$ \\lambda_F(n) \\leq n^{k-1}\\text{exp} \\left(c_1\\sqrt{\\frac{\\log n}{\\log\\log n}}\\right), $$ where $c_1>0$ is an absolute constant. This improves an earlier result of Pitale and Schmidt. Further, we investigate the limit points of the sequence $\\{\\frac{\\lambda_F(n)}{n^{k-1}}\\}_{n \\in \\N}$ and show that it has infinitely many limit points. Finally, we show that $\\lambda_F(n) >0$ for all $n$, a result earlier proved by Breulmann by a different technique.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-16T17:32:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F46",
      "11F30"
    ],
    "keywords": [
      "siegel modular forms",
      "hecke eigenvalues",
      "maass space",
      "limit points",
      "absolute constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}