## Reductions of an elliptic curve with almost prime orders

Citations: | 3 - 0 self |

### BibTeX

@MISC{Cojocaru_reductionsof,

author = {Alina Carmen Cojocaru},

title = {Reductions of an elliptic curve with almost prime orders},

year = {}

}

### OpenURL

### Abstract

1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exist infinitely many primes p for which #Ep(Fp) has at most 5 prime factors. We also obtain upper bounds for the number of primes p ≤ x for which #Ep(Fp) is a prime. 1

### Citations

181 | Sieve Methods - Halberstam, Richert - 1974 |

118 |
Die Typen der Multiplikatorenringe elliptischer Funktionenkörper
- Deuring
- 1941
(Show Context)
Citation Context ...ersingular primes ≤ x (see [El, pp. 25–26]). Hence in this case it is unnecessary to distinguish between supersingular and ordinary primes in our sievings. If E is with CM, then by results of Deuring =-=[De]-=-, half of the primes are supersingular, half are ordinary. To be more precise, the supersingular primes of E are the primes inert in the CM field K of E, and the ordinary primes of E are the primes sp... |

101 |
Effective versions of the Chebotarev density theorem, Algebraic
- Lagarias, Odlyzko
- 1977
(Show Context)
Citation Context ...ly with non-CM elliptic curves, but could be easily modified to handle CM elliptic curves, as well. It makes use of an effective version of the Chebotarev Density Theorem, due to Lagarias and Odlyzko =-=[LaOd]-=-, which requires the assumption of GRH. Theorem 2 (S. Ali Miri -V. Kumar Murty, 2001 [MiMu]) Let E be a non-CM elliptic curve defined over Q, of conductor N, and such that the finitely many elliptic c... |

51 | Le grand crible dans la théorie analytique des nombres - Bombieri |

29 | Modular forms and Chebotarev density theorem
- Murty, Murty, et al.
- 1988
(Show Context)
Citation Context ...d sieve of Richert [Ri], can, however, be generalized. Elaborating on the ideas introduced in [MiMu], and using Richert’s sieve, an improved Chebotarev Density Theorem due to Murty, Murty and Saradha =-=[MuMuSa]-=-, and a reduction method due to Serre [Se2], J. Steuding and A. Weng [StWe] showed: Theorem 3 (J. Steuding -A. Weng, 2003 [StWe]) Let E be an elliptic curve defined over Q, of conductor N, and such th... |

25 |
Primality of the number of points on an elliptic curve over a finite field
- Koblitz
- 1988
(Show Context)
Citation Context ... of infinite order, for how many primes p ≤ x do we have that Ep(Fp) = 〈α(mod p)〉? Clearly, if #Ep(Fp) is prime, then Ep(Fp) = 〈α(mod p)〉 is satisfied for any α. Based on this observation, N. Koblitz =-=[Ko]-=- formulated the stronger question: for how many primes p do we have that #Ep(Fp) is prime? More precisely, with the convention (kept throughout the paper) that p denotes a rational prime, we have: Con... |

25 | Elliptic curves with complex multiplication and the Conjecture of Birch and Swinnerton-Dyer. Arithmetic theory of elliptic curves - Rubin - 1997 |

21 |
On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Sci. Sinica 16
- Chen
- 1973
(Show Context)
Citation Context ... + 2 is almost a prime is that there are infinitely many primes p for which p + 2 has at most 2 prime divisors (counted with multiplicities) and was obtained by J. Chen using sieves with weights (see =-=[Ch]-=- or [HaRi, pp. 320-338]). Chen’s method does not seem to be amenable to generalizations to treat the situation of elliptic curves. The method of proof of the slightly weaker result that there are infi... |

20 | An Introduction to Sieve Methods and their Applications - Cojocaru, Murty - 2006 |

20 |
Primitive points on elliptic curves
- Gupta, Murty
- 1986
(Show Context)
Citation Context ...about the cyclicity of Ep(Fp) may be 2sviewed as a subproblem of an elliptic curve version of Artin’s problem about primitive roots, formulated by Lang and Trotter [LaTr] in 1976, and investigated in =-=[GuMu]-=-: given an elliptic curve E over Q with rank ≥ 1, and given α ∈ E(Q) a point of infinite order, for how many primes p ≤ x do we have that Ep(Fp) = 〈α(mod p)〉? Clearly, if #Ep(Fp) is prime, then Ep(Fp)... |

17 | The large sieve inequality for algebraic number fields - Huxley - 1971 |

16 |
Primitive points on elliptic curves
- Lang, Trotter
(Show Context)
Citation Context ... [Co1] for general d). The question about the cyclicity of Ep(Fp) may be 2sviewed as a subproblem of an elliptic curve version of Artin’s problem about primitive roots, formulated by Lang and Trotter =-=[LaTr]-=- in 1976, and investigated in [GuMu]: given an elliptic curve E over Q with rank ≥ 1, and given α ∈ E(Q) a point of infinite order, for how many primes p ≤ x do we have that Ep(Fp) = 〈α(mod p)〉? Clear... |

15 | On the surjectivity of the galois representations associated to non-cm elliptic curves
- Cojocaru, Kani
(Show Context)
Citation Context ...ntegers OK of an imaginary quadratic field K, then for any integer d coprime to 6N we have that Gal(Q(E[d])/K) � �∗ . dOK � OK For proofs of (or more details on) these results, we refer the reader to =-=[Co2]-=-, [Ru, p.187], and [Se1]. From here we deduce that for integers d composed of sufficiently large primes, the densities δ(d) and δ o (d) are multiplicative in d. Moreover, we can calculate the followin... |

15 |
Divisors of Fourier coefficients of modular forms
- Gun, Murty
(Show Context)
Citation Context ...Theorem 5 In this section we prove Theorem 5 by using Turán’s normal order method. These investigations actually go back to a general formalism of the normal order method due to K. Murty and R. Murty =-=[MuMu1]-=-. We keep the convention that p and l denote rational primes. We proceed along classical lines and show that � (ν(p + 1 − ap) − log log x) 2 � � x log log x = O , log x p≤x 22sunder a quasi-GRH if E i... |

10 |
Questions About the Reductions Modulo Primes of an Elliptic Curve
- Cojocaru
- 2004
(Show Context)
Citation Context ...ay ask: given a fixed integer d �= 0, for how many primes p ≤ x do we have dp = d? The particular case that dp = d = 1 (i.e. Ep(Fp) is cyclic) has been studied extensively over the past 30 years (see =-=[Co1]-=- and the references therein; also see [Co1] for general d). The question about the cyclicity of Ep(Fp) may be 2sviewed as a subproblem of an elliptic curve version of Artin’s problem about primitive r... |

6 |
On the normal number of prime factors of p&1 and some related problems concerning Euler's ,-function
- Erdos
- 1935
(Show Context)
Citation Context ...the above implies � (ν(p + 1 − ap) − log log p) 2 � � x log log x = O , log x p≤x p∤N ap�=0 which is exactly what is claimed in the statement of Theorem 5. We recall that a classical theorem of Erdös =-=[Er]-=- asserts that � (ν(p + 1) − log log p) 2 � � x log log x = O . (13) log x p≤x Therefore, in view of Deuring’s formula for the number of supersingular primes of a CM elliptic curve E, for such E it rem... |

6 |
An application of sieve methods to elliptic curves
- Miri, Murty
(Show Context)
Citation Context ...lar to the ones of Hardy and Littlewood on the twin prime conjecture led Koblitz to the above formula. No progress was made on Koblitz’s Conjecture until recently, when S. Ali Miri and V. Kumar Murty =-=[MiMu]-=- exploited the similarity between the primality of #Ep(Fp) = p + 1 − ap and that of p + 2. More precisely, they carried out an elliptic curve version of the classical result that there are infinitely ... |

5 |
The Chebotarev density theorem and pair correlation of zeros of Artin L-functions , preprint
- Murty, Murty
(Show Context)
Citation Context ...x #{p ≤ x : p ∤ N, #Ep(Fp) has at most 3 prime factors} ≥ C(E) . (14) (log x) 2 This is a consequence of the improved effective versions of the Chebotarev Density Theorem due to K. Murty and R. Murty =-=[MuMu2]-=-. Namely, under GRH and PCC one would obtain that, in the non-CM case, Rd = ON � d 1/2 x 1/2 log(dx) � . Then, using this estimate in the Richert’s sieve, one gets (14). In the CM case the assumptions... |

4 | Supersingular primes of a given elliptic curve over a number field - Elkies - 1987 |

2 |
Selberg’s sieve with weights
- Richert
- 1969
(Show Context)
Citation Context ...e slightly weaker result that there are infinitely many primes p for which p + 2 has at most 3 prime divisors (counted with multiplicities) [HaRi, pp. 247-252], based on the weighted sieve of Richert =-=[Ri]-=-, can, however, be generalized. Elaborating on the ideas introduced in [MiMu], and using Richert’s sieve, an improved Chebotarev Density Theorem due to Murty, Murty and Saradha [MuMuSa], and a reducti... |

1 |
Elliptic curve analogues of the twin prime conjecture, Senior Thesis 2005
- Lane
(Show Context)
Citation Context ...ume the θ-quasi-GRH for Dedekind zeta functions for some arbitrary 1/2 ≤ θ < 1. Then B(E) := � p∤N #Ep(Fp) prime 1 p < ∞. The constant B(E) has been computed for a few particular elliptic curves E in =-=[La]-=-. 2 Divisors of #Ep(Fp) 2.1. A general sieve problem. Let E be an elliptic curve defined over Q and of conductor N. We keep the notation introduced in Section 1, and we recall that our 6sprincipal goa... |