{
  "id": "2103.12533",
  "version": "v1",
  "published": "2021-03-23T13:38:55.000Z",
  "updated": "2021-03-23T13:38:55.000Z",
  "title": "Multiplicity one bound for cohomological automorphic representations with a fixed level",
  "authors": [
    "Dohoon Choi"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $F$ be a totally real field, and $\\mathbb{A}_F$ be the adele ring of $F$. Let us fix $N$ to be a positive integer. Let $\\pi_1=\\otimes\\pi_{1,v}$ and $\\pi_2=\\otimes\\pi_{2,v}$ be distinct cohomological cuspidal automorphic representations of $\\mathrm{GL}_n(\\mathbb{A}_{F})$ with levels less than or equal to $N$. Let $\\mathcal{N}(\\pi_1,\\pi_2)$ be the minimum of the absolute norm of $v \\nmid \\infty$ such that $\\pi_{1,v} \\not \\simeq \\pi_{2,v}$ and that $\\pi_{1,v}$ and $\\pi_{2,v}$ are unramified. We prove that there exists a constant $C_N$ such that for every pair $\\pi_1$ and $\\pi_2$, $$\\mathcal{N}(\\pi_1,\\pi_2) \\leq C_N.$$ This improves known bounds $$ \\mathcal{N}(\\pi_1,\\pi_2)=O(Q^A) \\;\\;\\; (\\text{some } A \\text{ depending only on } n), $$ where $Q$ is the maximum of the analytic conductors of $\\pi_1$ and $\\pi_2$. This result applies to newforms on $\\Gamma_1(N)$. In particular, assume that $f_1$ and $f_2$ are Hecke eigenforms of weight $k_1$ and $k_2$ on $\\mathrm{SL}_2(\\mathbb{Z})$, respectively. We prove that if for all $p \\in \\{2,7\\}$, $$\\lambda_{f_1}(p)/\\sqrt{p}^{(k_1-1)} = \\lambda_{f_2}(p)/\\sqrt{p}^{(k_2-1)},$$ then $f_1=cf_2$ for some constant $c$. Here, for each prime $p$, $\\lambda_{f_i}(p)$ denotes the $p$-th Hecke eigenvalue of $f_i$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-23T13:38:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cohomological automorphic representations",
      "fixed level",
      "multiplicity",
      "distinct cohomological cuspidal automorphic representations",
      "th hecke eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}