{
  "id": "1005.2998",
  "version": "v1",
  "published": "2010-05-17T19:11:10.000Z",
  "updated": "2010-05-17T19:11:10.000Z",
  "title": "Remarks on the Fourier coefficients of modular forms",
  "authors": [
    "Kirti Joshi"
  ],
  "comment": "22 pages",
  "journal": "Journal of Number Theory 132 Number 6 (2012) Pages 1314-1336",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We consider a variant of a question of N. Koblitz. For an elliptic curve $E/\\Q$ which is not $\\Q$-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes $p$ such that $N_p(E)=#E(\\F_p)=p+1-a_p(E)$ is also a prime. We consider a variant of this question. For a newform $f$, without CM, of weight $k\\geq 4$, on $\\Gamma_0(M)$ with trivial Nebentypus $\\chi_0$ and with integer Fourier coefficients, let $N_p(f)=\\chi_0(p)p^{k-1}+1-a_p(f)$ (here $a_p(f)$ is the $p^{th}$-Fourier coefficient of $f$). We show under GRH and Artin's Holomorphy Conjecture that there are infinitely many $p$ such that $N_p(f)$ has at most $[5k+1+\\sqrt{\\log(k)}]$ distinct prime factors. We give examples of about hundred forms to which our theorem applies.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-05-17T19:11:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "modular forms",
      "elliptic curve",
      "artins holomorphy conjecture",
      "integer fourier coefficients",
      "distinct prime factors"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.2998J"
    }
  }
}