{
  "id": "2403.15795",
  "version": "v1",
  "published": "2024-03-23T10:51:18.000Z",
  "updated": "2024-03-23T10:51:18.000Z",
  "title": "Special values of $L$-functions for $\\mathrm{GL}_5 \\times \\mathrm{GL}_4$",
  "authors": [
    "Ankit Rai",
    "Gunja Sachdeva"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\Pi$ be a cuspidal automorphic representation of $\\mathrm{GL}_5(\\mathbb{A}_\\mathbb{Q})$, and let $\\Sigma = \\mathrm{Ind}(\\pi, \\chi_2|\\cdot|^{1/2}, \\chi_3|\\cdot|^{-1/2})$ be an automorphic representation of $\\mathrm{GL}_4(\\mathbb{A}_\\mathbb{Q})$ induced from the standard parabolic subgroup of the form $(2, 1, 1)$ where $\\pi$ is a cuspidal automorphic representation of $\\mathrm{GL}_2(\\mathbb{A}_\\mathbb{Q})$. Assume that $\\Pi_{\\infty}$ and $\\Sigma_{\\infty}$ are cohomological with respect to the trivial representation in which case $s=1/2$ is a critical point for the Rankin-Selberg $L$-function $L(s, \\Pi \\times \\Sigma)$. Following Mahnkopf, we prove a result about $L(\\tfrac{1}{2}, \\Pi \\times \\Sigma)$, and as a corollary, obtain an algebraicity result for the ratio $L(1, \\Pi \\times \\chi)/L(1, \\Pi \\times \\chi')$, where $\\chi,\\chi'$ are finite order Hecke characters such that $\\chi_{\\infty} = \\chi'_{\\infty} = \\mathrm{sgn}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-23T10:51:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "special values",
      "cuspidal automorphic representation",
      "finite order hecke characters",
      "standard parabolic subgroup",
      "algebraicity result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}