{
  "id": "2006.07819",
  "version": "v1",
  "published": "2020-06-14T06:46:31.000Z",
  "updated": "2020-06-14T06:46:31.000Z",
  "title": "Sub-convexity bound for $GL(3) \\times GL(2)$ $L$-functions: $GL(3)$-spectral aspect",
  "authors": [
    "Sumit Kumar",
    "Kummari Mallesham",
    "Saurabh Kumar Singh"
  ],
  "comment": "First draft",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\phi$ be a Hecke-Maass cusp form for $SL(3, \\mathbb{Z})$ with Langlands parameters $({\\bf t}_{i})_{i=1}^{3}$ satisfying $$|{\\bf t}_{3} - {\\bf t}_{2}| \\leq T^{1-\\xi -\\epsilon}, \\quad \\, {\\bf t}_{i} \\approx T, \\quad \\, \\, i=1,2,3$$ with $1/2 < \\xi <1$ and any $\\epsilon>0$. Let $f$ be a holomorphic or Maass Hecke eigenform for $SL(2,\\mathbb{Z})$. In this article, we prove a sub-convexity bound $$L(\\phi \\times f, \\frac{1}{2}) \\ll \\max \\{ T^{\\frac{3}{2}-\\frac{\\xi}{4}+\\epsilon} , T^{\\frac{3}{2}-\\frac{1-2 \\xi}{4}+\\epsilon} \\} $$ for the central values $L(\\phi \\times f, \\frac{1}{2})$ of the Rankin-Selberg $L$-function of $\\phi$ and $f$, where the implied constants may depend on $f$ and $\\epsilon$. Conditionally, we also obtain a subconvexity bound for $L(\\phi \\times f, \\frac{1}{2})$ when the spectral parameters of $\\phi$ are in generic position, that is $${\\bf t}_{i} - {\\bf t}_{j} \\approx T, \\quad \\, \\text{for} \\, i \\neq j, \\quad \\, {\\bf t}_{i} \\approx T , \\, \\, i=1,2,3.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-14T06:46:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sub-convexity bound",
      "spectral aspect",
      "maass hecke eigenform",
      "hecke-maass cusp form",
      "generic position"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}