{
  "id": "1711.06283",
  "version": "v1",
  "published": "2017-11-16T19:02:18.000Z",
  "updated": "2017-11-16T19:02:18.000Z",
  "title": "On the equation $N_{p_1}(E)\\cdot N_{p_2}(E)\\cdots N_{p_k}(E)=n$",
  "authors": [
    "Kirti Joshi"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "For a given elliptic curve $E/\\mathbb{Q}$, set $N_p(E)$ to be the number of points on $E$ modulo $p$ for a prime of good reduction for $E$. Given integer $n$, let $G_k(E,n)$ be the number of $k$-tuples of pairwise distinct primes $p_1,\\ldots,p_k$ of good reduction for $E$, for which equation in the title holds, then on assuming the Generalized Riemann Hypothesis for elliptic curves without CM (and unconditionally if the curves have complex multiplication), I show that $\\varlimsup_{n\\to\\infty} G_k(E,n)=\\infty$ for any integer $k\\geq 3$. I also conjecture that this result also holds for $k=1$ and $k=2$. In particular for $k=1$ this conjecture says that there are \"elliptic progressions of primes\" i.e. sequences of primes $p_1<p_2\\cdots <p_m$ of arbitrary lengths $m$ such that $N_{p_1}(E)=N_{p_2}(E)=\\cdots =N_{p_m}(E)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-16T19:02:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G20"
    ],
    "keywords": [
      "elliptic curve",
      "pairwise distinct primes",
      "arbitrary lengths",
      "generalized riemann hypothesis",
      "complex multiplication"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}