{
  "id": "2406.03617",
  "version": "v1",
  "published": "2024-06-05T20:26:30.000Z",
  "updated": "2024-06-05T20:26:30.000Z",
  "title": "Five-dimensional compatible systems and the Tate conjecture for elliptic surfaces",
  "authors": [
    "Lian Duan",
    "Xiyuan Wang",
    "Ariel Weiss"
  ],
  "comment": "52 pages. Comments welcome!",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $(\\rho_\\lambda\\colon G_{\\mathbb Q}\\to \\operatorname{GL}_5(\\overline{E}_\\lambda))_\\lambda$ be a strictly compatible system of Galois representations such that no Hodge--Tate weight has multiplicity $5$. We show that if $\\rho_{\\lambda_0}$ is irreducible for some $\\lambda_0$, then $\\rho_\\lambda$ is irreducible for all but finitely many $\\lambda$. More generally, if $(\\rho_\\lambda)_\\lambda$ is essentially self-dual, we show that either $\\rho_\\lambda$ is irreducible for all but finitely many $\\lambda$, or the compatible system $(\\rho_\\lambda)_\\lambda$ decomposes as a direct sum of lower-dimensional compatible systems. We apply our results to study the Tate conjecture for elliptic surfaces. For example, if $X_0\\colon y^2 + (t+3)xy + y= x^3$, we prove the codimension one $\\ell$-adic Tate conjecture for all but finitely many $\\ell$, for all but finitely many general, degree $3$, genus $2$ branched multiplicative covers of $X_0$. To prove this result, we classify the elliptic surfaces into six families, and prove, using perverse sheaf theory and a result of Cadoret--Tamagawa, that if one surface in a family satisfies the Tate conjecture, then all but finitely many do. We then verify the Tate conjecture for one representative of each family by making our irreducibility result explicit: for the compatible system arising from the transcendental part of $H^2_{\\mathrm{et}}(X_{\\overline{\\mathbb Q}}, \\mathbb{Q}_\\ell(1))$ for a representative $X$, we formulate an algorithm that takes as input the characteristic polynomials of Frobenius, and terminates if and only if the compatible system is irreducible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-05T20:26:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F80",
      "11F70",
      "14C25",
      "14D05",
      "14J27"
    ],
    "keywords": [
      "elliptic surfaces",
      "five-dimensional compatible systems",
      "perverse sheaf theory",
      "adic tate conjecture",
      "irreducibility result explicit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}