{
  "id": "1510.01219",
  "version": "v1",
  "published": "2015-10-05T16:39:26.000Z",
  "updated": "2015-10-05T16:39:26.000Z",
  "title": "Improved subconvexity bounds for GL(2)xGL(3) and GL(3) L-functions by weighted stationary phase",
  "authors": [
    "Mark McKee",
    "Haiwei Sun",
    "Yangbo Ye"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f$ be a fixed self-contragradient Hecke-Maass form for $SL(3,\\mathbb Z)$, and $u$ an even Hecke-Maass form for $SL(2,\\mathbb Z)$ with Laplace eigenvalue $1/4+k^2$, $k>0$. A subconvexity bound $O\\big(k^{4/3+\\varepsilon}\\big)$ in the eigenvalue aspect is proved for the central value at $s=1/2$ of the Rankin-Selberg $L$-function $L(s,f\\times u)$. Meanwhile, a subconvexity bound $O\\big((1+|t|)^{2/3+\\varepsilon}\\big)$ in the $t$ aspect is proved for $L(1/2+it,f)$. These bounds improved corresponding subconvexity bounds proved by Xiaoqing Li (Annals of Mathematics, 2011). The main technique in the proof, other than those used by Li, is an $n$th-order asymptotic expansion of a weighted stationary phase integral, for arbitrary $n\\geq1$. This asymptotic expansion sharpened the classical result for $n=1$ by Huxley.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-05T16:39:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "l-functions",
      "fixed self-contragradient hecke-maass form",
      "th-order asymptotic expansion",
      "weighted stationary phase integral",
      "eigenvalue aspect"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}