{
  "id": "1805.00209",
  "version": "v1",
  "published": "2018-05-01T06:47:41.000Z",
  "updated": "2018-05-01T06:47:41.000Z",
  "title": "Non-vanishing of central values of certain $L$-functions on ${\\rm GL}(2)\\times {\\rm GL}(3)$",
  "authors": [
    "Shingo Sugiyama",
    "Masao Tsuzuki"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\phi$ be an even Hecke-Maass cusp form on ${\\rm SL}_2(\\mathbb{Z})$ whose $L$-function does not vanish at the center of the functional equation. In this article, we obtain an exact formula of the average of triple products of $\\phi$, $f$ and $\\bar f$, where $f$ runs over the normalized Hecke eigen elliptic cusp forms on ${\\rm SL}_2(\\mathbb{Z})$ of a fixed weight $l \\ge 4$. As an application, we prove an infinitude of pairs $(\\phi, F)$ of $\\phi$ as above and a cohomological cusp form $F$ on ${\\rm SL}_3(\\mathbb{Z})$ such that $L(1/2, \\phi \\times F)= 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-01T06:47:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F67",
      "11F72"
    ],
    "keywords": [
      "central values",
      "hecke eigen elliptic cusp forms",
      "normalized hecke eigen elliptic cusp",
      "hecke-maass cusp form",
      "non-vanishing"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}