{
  "id": "2010.07247",
  "version": "v1",
  "published": "2020-10-14T17:09:41.000Z",
  "updated": "2020-10-14T17:09:41.000Z",
  "title": "Computing L-Polynomials of Picard curves from Cartier-Manin matrices",
  "authors": [
    "Sualeh Asif",
    "Francesc Fité",
    "Dylan Pentland"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $\\mathbb{Q}$ at primes $p$ of good reduction for $C$. By determining the density of the set of primes of ordinary reduction, we prove that, for all but a density zero subset of primes, the Zeta function $Z(C_p,T)$ is uniquely determined by the Cartier--Manin matrix $A_p$ of $C$ modulo $p$, the irreducibility of $f$ modulo $p$ (or the failure thereof), and the exponent of the Jacobian of $C$ modulo $p$; we also show that for primes $\\equiv 1 \\pmod{3}$ the matrix $A_p$ suffices and that for primes $\\equiv 2 \\pmod{3}$ the genericity assumption on $C$ is unnecessary. By combining this with recent work of Sutherland, we obtain a practical probabilistic algorithm of Las Vegas type that computes $Z(C_p,T)$ for almost all primes $p \\le N$ using $N\\log(N)^{3+o(1)}$ expected bit operations. This is the first practical result of this type for curves of genus greater than 2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-14T17:09:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M38",
      "14G10",
      "11Y16",
      "11G40"
    ],
    "keywords": [
      "cartier-manin matrix",
      "computing l-polynomials",
      "zeta function",
      "density zero subset",
      "generic picard curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}