## Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem (2001)

Citations: | 14 - 3 self |

### BibTeX

@TECHREPORT{Cojocaru01cyclicityof,

author = {Alina Carmen Cojocaru and M. Ram Murty},

title = {Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem},

institution = {},

year = {2001}

}

### OpenURL

### Abstract

1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.-P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1

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Citation Context ...ly, certain public-key cryptosystems based on the intractability of the discrete logarithm problem can be implemented using the group of points of an elliptic curve E defined over a finite field (see =-=[Ko87]-=-). Then one wants the cyclic group generated by a certain point a on E to have order divisible by a large prime. One way to accomplish this is to choose the elliptic curve E and the finite field so th... |

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Citation Context ...lations we need effective versions of this theorem (that is, versions with explicit error terms). They were first derived by J. Lagarias and A. Odlyzko in 1976 (see [LO]), refined by J.-P. Serre (see =-=[Se81]-=-), and subsequently improved by Kumar Murty, Ram Murty and N. Saradha (see [RM-KM-S] and [RM-KM]). Theorem 3.1 Assuming GRH for the Dedekind zeta function of L we have that π1(x, L/Q) = 1 � li x + O x... |

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Citation Context ...We also recall that, unconditionally, A(E) ≪ε N 1+ε for any ε > 0 (see [acC4]) and, moreover, if E is semistable (that is, N is square-free) or is a Frey curve, then A(E) is an absolute constant (see =-=[Maz]-=- and [DaMe]). Good estimates for the size of n(k) in the case of a CM elliptic curve can be obtained as a consequence of deep results in the theory of complex multiplication. Proposition 3.8 Let E be ... |

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Citation Context ...1(x, L/Q) ∼ 1 li x. #G In our calculations we need effective versions of this theorem (that is, versions with explicit error terms). They were first derived by J. Lagarias and A. Odlyzko in 1976 (see =-=[LO]-=-), refined by J.-P. Serre (see [Se81]), and subsequently improved by Kumar Murty, Ram Murty and N. Saradha (see [RM-KM-S] and [RM-KM]). Theorem 3.1 Assuming GRH for the Dedekind zeta function of L we ... |

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Citation Context ...call that, unconditionally, A(E) ≪ε N 1+ε for any ε > 0 (see [acC4]) and, moreover, if E is semistable (that is, N is square-free) or is a Frey curve, then A(E) is an absolute constant (see [Maz] and =-=[DaMe]-=-). Good estimates for the size of n(k) in the case of a CM elliptic curve can be obtained as a consequence of deep results in the theory of complex multiplication. Proposition 3.8 Let E be a CM ellipt... |

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Citation Context ...y(x) is some real parameter which is optimally chosen in each case. This approach emulates the one used by Hooley in his conditional treatment of 5sArtin’s primitive root conjecture (see chapter 3 of =-=[Ho]-=-). It was also used by the authors for obtaining the earlier results mentioned in Section 1. A more natural splitting, however, is f(x, Q) = � µ(k)π1(x, Q(E[k])/Q) + � µ(k)π1(x, Q(E[k])/Q) k≤y =: � ma... |

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Citation Context .... We are naturally led to the above 1976 question of Lang and Trotter on primitive points, or to the 1988 question of Koblitz of determining the density of the primes p for which #E(Fp) is prime (see =-=[Ko88]-=-). The cyclicity of E(Fp) is a common subproblem of both these questions. The precise goal in this paper is to determine an explicit asymptotic formula for f(x, Q) := # � p ≤ x : p ∤ N, E(Fp) cyclic �... |

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Citation Context ...rjective for any prime l /∈ SE. Moreover, if we set A(E) := 2 · 3 · 5 · � l, where the product is over l∈SE primes l, then φk is surjective for any positive integer k coprime to A(E) (see Appendix of =-=[acC4]-=-). We will refer to A(E) as Serre’s constant associated to E. The main results of the paper are as follows. Theorem 1 Let E be a non-CM elliptic curve defined over Q and of conductor N. Let A(E) be Se... |

14 |
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Citation Context ...is cyclic (see [BMP, pp. 963-964]). As we will see, this prediction is indeed true. In 1976, S. Lang and H. Trotter formulated an elliptic curve analogue of Artin’s conjecture on primitive roots (see =-=[LT]-=-): let E be an elliptic curve defined over Q, of conductor N and having arithmetic rank ≥ 1; let a ∈ E(Q) be a fixed rational point on E of infinite order; then the density of the primes p ∤ N for whi... |

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Citation Context ...mposed of primes which are divisors of A(E) and k2 composed of primes which are coprime to A(E). Then n(k) ≥ φ(k1)n(k2) ≫ φ(k)k 3 2. Proof. 1. For the first statement of part 1 we refer the reader to =-=[Se68]-=- or [acC4, Appendix]. Now for k coprime to A(E) let us show that Q(ζk) is the maximal abelian extension contained in Q(E[k]). We have Gal(Lk/Q) � GL2(Z/kZ) and Gal(Q(ζk)/Q) � (Z/kZ) ∗ , GL2(Z/kZ) SL2(... |

7 |
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(Show Context)
Citation Context ...nd Ram Murty showed that, under no hypothesis and for any elliptic curve E which has an irrational 2-division point, we have f(x, Q) ≫E x . (5) (log x) 2 The implied ≫E-constant depends on E. In 2000 =-=[acC1]-=- A.C. Cojocaru showed that if E is a non-CM elliptic curve, then Serre’s result holds under the assumption of a quasi-GRH, namely a zero-free region of Re s > 3/4 for the Dedekind zeta functions of th... |

7 | Cojocaru, Cyclicity of Elliptic Curves modulo p - C - 2002 |

7 |
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Citation Context ...allest prime p = pE for which E(Fp) is cyclic. This latter problem can be viewed as an elliptic curve analogue of Linnik’s question about the size of the least prime in an arithmetic progression (see =-=[Li44a]-=- and [Li44b]). In 1976 J.-P. Serre adapted Hooley’s conditional proof of Artin’s conjecture [Ho, chapter 3] to obtain an asymptotic formula for the number of primes p ≤ x for which the group of points... |

5 |
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Citation Context ...r the assumption of a quasi-GRH, namely a zero-free region of Re s > 3/4 for the Dedekind zeta functions of the division fields of E, and with error(E, x) = ON � x log log x (log x) 2 � . (6) In 2001 =-=[acC2]-=- she also gave a new simpler unconditional proof for formula (2) in the case of a CM elliptic curve and obtained � � � � x x log log x error(E, x) = ON = O · . (7) (log x)(log log log x) (log x)(log l... |

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5 |
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(Show Context)
Citation Context ..., Q) = fE li x + O x 7/10 (log(Nx)) 4/5 � A(E) + O � x 7/10 (log x) 4/5� , The ON-constants above depend on N, and the O-constants are absolute. For formulations of AHC and PCC we refer the reader to =-=[RM01]-=-. � (log log x)(log(Nx)) 6/5 x1/5 A(E) log x 3 � . Theorem 2 Let E be an elliptic curve defined over Q, of conductor N, and with complex multiplication by the full ring of integers of an imaginary qua... |

5 |
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(Show Context)
Citation Context ...). They were first derived by J. Lagarias and A. Odlyzko in 1976 (see [LO]), refined by J.-P. Serre (see [Se81]), and subsequently improved by Kumar Murty, Ram Murty and N. Saradha (see [RM-KM-S] and =-=[RM-KM]-=-). Theorem 3.1 Assuming GRH for the Dedekind zeta function of L we have that π1(x, L/Q) = 1 � li x + O x #G 1/2 � �� log |dL| + log x . The implied O-constant is absolute. 6 nL p.sThis version of the ... |

4 | The linear homogeneous group - Brenner - 1960 |

4 |
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(Show Context)
Citation Context ...s cyclic for any supersingular prime p of Ea. By 1987 results of N. Elkies there are infinitely many supersingular primes, hence there are infinitely many primes p for which Ea(Fp) is cyclic. In 1990 =-=[RG-RM90]-=- Rajiv Gupta and Ram Murty showed that, under no hypothesis and for any elliptic curve E which has an irrational 2-division point, we have f(x, Q) ≫E x . (5) (log x) 2 The implied ≫E-constant depends ... |

4 |
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Citation Context ...ed CM). His proof uses class field theoretical properties of CM elliptic curves and the large sieve for number fields in the form of a number field version of the Bombieri-Vinogradov Theorem. In 1987 =-=[RM87]-=- he also demonstrated unconditionally the existence of infinitely many primes p for which E(Fp) is cyclic for certain elliptic curves E without complex multiplication (denoted non-CM). More precisely,... |

4 |
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(Show Context)
Citation Context ...ley’s conditional proof of Artin’s conjecture [Ho, chapter 3] to obtain an asymptotic formula for the number of primes p ≤ x for which the group of points modulo p of an elliptic curve is cyclic (see =-=[Se77]-=- or [RM83, pp. 159-161]). Before stating Serre’s result, let us introduce some more notation. For any positive integer k let E[k] denote the group of points of E that are annihilated by k (called the ... |

4 | Collected papers. volume III - Serre - 1985 |

3 | Cyclicity of CM elliptic curves modulo p - Cojocaru |

3 | Cyclicity of elliptic curves modulo p - Cojocaru - 2002 |

1 | Artin’s conjecture and elliptic analogues - Murty - 1996 |