### 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 Np(E)=#E(Fp)=p+1-ap(E) is also a prime. We consider a variant of this question. For a newform f, without CM, of weight k≥4, on Γ _{0}(M) with trivial Nebentypus χ _{0} and with integer Fourier coefficients, let N _{p}(f)=χ _{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+log(k)] distinct prime factors. We give examples of about hundred forms to which our theorem applies. We also show, on GRH, that the number of distinct prime factors of N _{p}(f) is of normal order log(log(p)) and that the distribution of these values is asymptotically a Gaussian distribution ("Erdo{double acute}s-Kac type theorem").

Original language | English (US) |
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Pages (from-to) | 1314-1336 |

Number of pages | 23 |

Journal | Journal of Number Theory |

Volume | 132 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2012 |

### Keywords

- Hecke eigenvalues
- Koblitz conjecture
- Modular forms
- Normal orders

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Journal of Number Theory*,

*132*(6), 1314-1336. https://doi.org/10.1016/j.jnt.2011.10.004