## AN ANALOGUE OF THE SIEGEL-WALFISZ THEOREM FOR THE CYCLICITY OF CM ELLIPTIC CURVES MOD p

Citations: | 2 - 1 self |

### BibTeX

@MISC{Akbary_ananalogue,

author = {Amir Akbary and V. Kumar Murty},

title = {AN ANALOGUE OF THE SIEGEL-WALFISZ THEOREM FOR THE CYCLICITY OF CM ELLIPTIC CURVES MOD p},

year = {}

}

### OpenURL

### Abstract

Abstract. Let E be a CM elliptic curve defined over Q and of conductor N. We establish an asymptotic formula, uniform in N and with improved error term, for the counting function of primes p for which the reduction mod p of E is cyclic. Our result resembles the classical Siegel-Walfisz theorem regarding the distribution of primes in arithmetic progressions. 1.

### Citations

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(Show Context)
Citation Context ...(x)/g(x) = 0; similarly f(x) = OA(g(x)) (or equivalently f(x) ≪A g(x)) means that the function |f(x)/g(x)| is bounded by a constant depending on A as x → ∞. In fact the proof of Theorem 2 of R. Murty =-=[10]-=- shows that, under GRH, ( x log log x C(x, E) = cE li(x) + OE (log x) 2 ) , where cE = ∞∑ k=1 µ(k) [Q(E[k]) : Q] . Here µ(.) is the Möbius function. One can show that the above series representing cE ... |

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17 |
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(Show Context)
Citation Context ...constant C = C(B) such that ∑ max (a,q)=1 N(q)≤Q 1 T (q) ∣πK(x; q, a) − li(x) h(q) ∣ ≪ x , (log x) B where Q = x 1 2 (log x) −C . The implied constant depends only on B and on the field K. Proof. See =-=[7]-=-, Theorem 1. □ We are ready to prove the main result of this paper. 3. Proof of Theorem 1.1 Proof. First of all note that in light of Lemma 2.2 we have C(x, N, E) = #{p ≤ x; p ∤ N, p does not split co... |

16 | Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem. Mathematische Annalen 330 - Cojocaru, Murty - 2004 |

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6 |
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(Show Context)
Citation Context ... positive if and only if Q(E[2]) ̸= Q (see [3, Section 6] for a proof). More precisely, if Q(E[2]) ̸= Q, in the CM case cE ≥ 1/4, and in the non-CM case cE ≫ 1/ log log N. Secondly Gupta and R. Murty =-=[5]-=- proved that the number of primes p ≤ x for which Ē(Fp) is cyclic is ≫ x/(log x) 2 . Stronger unconditional results are known for elliptic curves with complex multiplication. In [10], R. Murty removed... |

2 | On the greatest prime divisor of Np - Akbary - 2008 |

2 |
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(Show Context)
Citation Context ...re free. Since E[2] is isomorphic to Z/2Z ⊕ Z/2Z and prime to p-torsions have trivial intersection with the kernel of reduction mod p map, it is clear that if Q(E[2]) = Q then C(x, E) = 0 or 1. Serre =-=[12]-=- proved that, under the assumption of the GRH, if Q(E[2]) ̸= Q then C(x, E) = cE li(x) + o ( x log x where the density cE > 0. Here li(x) = ∫ x 2 dt/ log t, and we recall that f(x) = o(g(x)) means lim... |

1 |
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- C
(Show Context)
Citation Context ...he assumption of the Generalized Riemann Hypothesis for the Dedekind zeta functions of the extensions Q(E[m])/Q, where E[m] is the group of m-torsion points of E and m = 1 or m is square free. Since E=-=[2]-=- is isomorphic to Z/2Z ⊕ Z/2Z and prime to p-torsions have trivial intersection with the kernel of reduction mod p map, it is clear that if Q(E[2]) = Q then C(x, E) = 0 or 1. Serre [12] proved that, u... |