{
  "id": "2305.07809",
  "version": "v1",
  "published": "2023-05-13T00:31:55.000Z",
  "updated": "2023-05-13T00:31:55.000Z",
  "title": "On $p$-adic $L$-functions for symplectic representations of GL(N) over number fields",
  "authors": [
    "Chris Williams"
  ],
  "comment": "24 pages, comments welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $F$ be a number field, and $\\pi$ a regular algebraic cuspidal automorphic representation of $\\mathrm{GL}_N(\\mathbb{A}_F)$ of symplectic type. We construct a $p$-adic $L$-function attached to any non-critical $p$-refinement $\\tilde\\pi$ of $\\pi$ to $(n,n)$-parahoric level. More precisely, we construct a distribution $L_p(\\tilde\\pi)$ on the Galois group $\\mathrm{Gal}_p$ of the maximal abelian extension of $F$ unramified outside $p\\infty$, and show that it interpolates all the standard critical $L$-values of $\\pi$ at $p$ (including, for example, cyclotomic and anticyclotomic variation when $F$ is imaginary quadratic). We show that $L_p(\\tilde\\pi)$ satisfies a natural growth condition; in particular, when $\\tilde\\pi$ is ordinary, $L_p(\\tilde\\pi)$ is a (bounded) measure on $\\mathrm{Gal}_p$. As a corollary, when $\\pi$ is unitary, has regular weight, and admits an ordinary refinement, we deduce non-vanishing $L(\\pi\\times(\\chi\\circ N_{F/\\mathbb{Q}}),1/2) \\neq 0$ of the twisted central value for all but finitely many Dirichlet characters $\\chi$ of $p$-power conductor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-13T00:31:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "number field",
      "symplectic representations",
      "regular algebraic cuspidal automorphic representation",
      "maximal abelian extension",
      "natural growth condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}