{
  "id": "1912.09772",
  "version": "v1",
  "published": "2019-12-20T11:34:26.000Z",
  "updated": "2019-12-20T11:34:26.000Z",
  "title": "Analytic Twists of $\\rm GL_3\\times \\rm GL_2$ Automorphic Forms",
  "authors": [
    "Yongxiao Lin",
    "Qingfeng Sun"
  ],
  "comment": "35 pages; comments welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\pi$ be a Hecke--Maass cusp form for $\\rm SL_3(\\mathbb{Z})$ with normalized Hecke eigenvalues $\\lambda_{\\pi}(n,r)$ and let $f$ be a holomorphic or Maass cusp form for $\\rm SL_2(\\mathbb{Z})$ with normalized Hecke eigenvalues $\\lambda_f(n)$. In this paper, we are concerned with obtaining nontrivial estimates for the sum $$ \\sum_{r,n\\geq 1}\\lambda_{\\pi}(n,r)\\lambda_f(n)e\\left(t\\varphi(r^2n/N)\\right)V\\left(r^2n/N\\right), $$ where $e(x)=e^{2\\pi ix}$, $V(x)\\in \\mathcal{C}_c^{\\infty}(0,\\infty)$, $t\\geq 1$ is a large parameter and $\\varphi(x)$ is some nonlinear real-valued smooth function. As applications, we give an improved subconvexity bound for $\\rm GL_3\\times \\rm GL_2$ $L$-functions in the $t$-aspect, and under the Ramanujan conjecture we derive the following bound for sums of $\\rm GL_3\\times \\rm GL_2$ Fourier coefficients $$ \\sum_{r^2n\\leq x}\\lambda_{\\pi}(r,n)\\lambda_f(n)\\ll x^{5/7-1/364+\\varepsilon} $$ for any $\\varepsilon>0$, which breaks for the first time the barrier $O(x^{5/7+\\varepsilon})$ in a work of Friedlander--Iwaniec.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-20T11:34:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F30",
      "11L07",
      "11F66",
      "11M41"
    ],
    "keywords": [
      "automorphic forms",
      "analytic twists",
      "normalized hecke eigenvalues",
      "hecke-maass cusp form",
      "nonlinear real-valued smooth function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}