{
  "id": "1803.03698",
  "version": "v1",
  "published": "2018-03-09T21:33:57.000Z",
  "updated": "2018-03-09T21:33:57.000Z",
  "title": "Almost prime values of the order of abelian varieties over finite fields",
  "authors": [
    "Samuel Bloom"
  ],
  "comment": "27 pages; comments welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E/\\mathbb Q$ be an elliptic curve, and denote by $N(p)$ the number of $\\mathbb{F}_p$-points of the reduction modulo $p$ of $E$. A conjecture of Koblitz, refined by Zywina, states that the number of primes $p \\leq X$ at which $N(p)$ is also prime is asymptotic to $C_E \\cdot X / \\log(X)^2$, where $C_E$ is an arithmetically-defined non-negative constant. Following Miri-Murty (2001) and others, Y.R. Liu (2006) and David-Wu (2012) study the number of prime factors of $N(p)$. We generalize their arguments to abelian varieties $A / \\mathbb Q$ whose adelic Galois representation has open image in $\\textrm{GSp}_{2g} \\widehat{\\mathbb Z}$. Our main result, after David-Wu, finds a conditional lower bound on the number of primes at which $\\# A_p ( \\mathbb{F}_p)$ has few prime factors. We also present some experimental evidence in favor of a generalization of Koblitz's conjecture to this context.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-09T21:33:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G10",
      "11N36",
      "11N45"
    ],
    "keywords": [
      "abelian varieties",
      "prime values",
      "finite fields",
      "prime factors",
      "adelic galois representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}