{
  "id": "1802.08527",
  "version": "v1",
  "published": "2018-02-23T13:50:13.000Z",
  "updated": "2018-02-23T13:50:13.000Z",
  "title": "Reductions of points on algebraic groups, II",
  "authors": [
    "Peter Bruin",
    "Antonella Perucca"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $A$ be the product of an abelian variety and a torus over a number field $K$, and let $m$ be a positive integer. If $\\alpha \\in A(K)$ is a point of infinite order, we consider the set of primes $\\mathfrak p$ of $K$ such that the reduction $(\\alpha \\bmod \\mathfrak p)$ is well defined and has order coprime to $m$. This set admits a natural density, which we are able to express as a finite sum of products of $\\ell$-adic integrals, where $\\ell$ varies in the set of prime divisors of $m$. We deduce that the density is a rational number, whose denominator is bounded (up to powers of $m$) in a very strong sense. This extends the results of the paper \"Reductions of points on algebraic groups\" by Davide Lombardo and the second author, where the case $m$ prime is established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-23T13:50:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G10",
      "11F80",
      "11R44",
      "11Y40"
    ],
    "keywords": [
      "algebraic groups",
      "infinite order",
      "abelian variety",
      "order coprime",
      "natural density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}