{
  "id": "2109.15311",
  "version": "v2",
  "published": "2021-09-30T17:53:14.000Z",
  "updated": "2024-10-22T21:57:04.000Z",
  "title": "Simple zeros of $\\mathrm{GL}(2)$ $L$-functions",
  "authors": [
    "Alexandre de Faveri"
  ],
  "comment": "32 pages. To appear in JEMS",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f \\in S_k(\\Gamma_1(N))$ be a primitive holomorphic form of arbitrary weight $k$ and level $N$. We show that the completed $L$-function of $f$ has $\\Omega\\left(T^\\delta\\right)$ simple zeros with imaginary part in $\\left[-T, T\\right]$, for any $\\delta < \\frac{2}{27}$. This is the first power bound in this problem for $f$ of non-trivial level, where previously the best results were $\\Omega(\\log\\log\\log{T})$ for $N$ odd, due to Booker, Milinovich, and Ng, and infinitely many simple zeros for $N$ even, due to Booker. In addition, for $f$ of trivial level ($N=1$), we also improve an old result of Conrey and Ghosh on the number of simple zeros.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-10-22T21:57:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple zeros",
      "first power bound",
      "arbitrary weight",
      "best results",
      "non-trivial level"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}