{
  "id": "2409.02589",
  "version": "v1",
  "published": "2024-09-04T10:14:37.000Z",
  "updated": "2024-09-04T10:14:37.000Z",
  "title": "Geometric realizations of representations for $\\text{PSL}(2, \\mathbb{F}_p)$ and Galois representations arising from defining ideals of modular curves",
  "authors": [
    "Lei Yang"
  ],
  "comment": "304 pages",
  "categories": [
    "math.NT",
    "math.AG",
    "math.RT"
  ],
  "abstract": "We construct a geometric realization of representations for $\\text{PSL}(2, \\mathbb{F}_p)$ by the defining ideals of rational models $\\mathcal{L}(X(p))$ of modular curves $X(p)$ over $\\mathbb{Q}$. Hence, for the irreducible representations of $\\text{PSL}(2, \\mathbb{F}_p)$, whose geometric realizations can be formulated in three different scenarios in the framework of Weil's Rosetta stone: number fields, curves over $\\mathbb{F}_q$ and Riemann surfaces. In particular, we show that there exists a correspondence among the defining ideals of modular curves over $\\mathbb{Q}$, reducible $\\mathbb{Q}(\\zeta_p)$-rational representations $\\pi_p: \\text{PSL}(2, \\mathbb{F}_p) \\rightarrow \\text{Aut}(\\mathcal{L}(X(p)))$ of $\\text{PSL}(2, \\mathbb{F}_p)$, and $\\mathbb{Q}(\\zeta_p)$-rational Galois representations $\\rho_p: \\text{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q}) \\rightarrow \\text{Aut}(\\mathcal{L}(X(p)))$ as well as their modular and surjective realization. This leads to a new viewpoint on the last mathematical testament of Galois by Galois representations arising from the defining ideals of modular curves, which leads to a connection with Klein's elliptic modular functions. It is a nonlinear and anabelian counterpart of the global Langlands correspondence among the $\\ell$-adic \\'{e}tale cohomology of modular curves over $\\mathbb{Q}$, i.e., Grothendieck motives ($\\ell$-adic system), automorphic representations of $\\text{GL}(2, \\mathbb{Q})$ and $\\ell$-adic representations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-04T10:14:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R39",
      "11F80",
      "20C33",
      "14C25",
      "14H42",
      "11G18",
      "14L30",
      "13A50",
      "14H37",
      "11G32"
    ],
    "keywords": [
      "modular curves",
      "defining ideals",
      "galois representations arising",
      "geometric realization",
      "kleins elliptic modular functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 304,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}