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Almost prime values of the order of elliptic curves over finite fields
, 2008
"... Abstract. Let E be an elliptic curve over Q without complex multiplication, and which is not isogenous to a curve with nontrivial rational torsion. For each prime p of good reduction, let E(Fp)  be the order of the group of points of the reduced curve over Fp. We prove in this paper that, under t ..."
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Abstract. Let E be an elliptic curve over Q without complex multiplication, and which is not isogenous to a curve with nontrivial rational torsion. For each prime p of good reduction, let E(Fp)  be the order of the group of points of the reduced curve over Fp. We prove in this paper that, under the GRH, there are at least 2.778Ctwin E x/(log x)2 primes p such that E(Fp)  has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng [18] and Murty & Miri [13]. This is also the first result where the dependence on the conjectural constant Ctwin E appearing in the twin prime conjecture for elliptic curves (also known as Koblitz’s conjecture) is made explicit. This is achieved by sieving a slightly different sequence than the one of [18] and [13]. By sieving the same sequence and using Selberg’s linear sieve, we can also improve the constant of Zywina [22] appearing in the upper bound for the number of primes p such that E(Fp)  is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH. 1.
AVERAGES OF THE NUMBER OF POINTS ON ELLIPTIC CURVES
"... ABSTRACT. If E is an elliptic curve defined over Q and p is a prime of good reduction for E, let E(Fp) denote the set of points on the reduced curve modulo p. Define an arithmetic function ME(N) by setting ME(N): = #{p: #E(Fp) = N}. Recently, David and the third author studied the average of ME(N) ..."
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ABSTRACT. If E is an elliptic curve defined over Q and p is a prime of good reduction for E, let E(Fp) denote the set of points on the reduced curve modulo p. Define an arithmetic function ME(N) by setting ME(N): = #{p: #E(Fp) = N}. Recently, David and the third author studied the average of ME(N) over certain “boxes ” of elliptic curves E. Assuming a plausible conjecture about primes in short intervals, they showed the following: for odd N, the average of ME(N) over a box with sufficiently large sides is ∼ K ∗ (N) log N for an explicitlygiven function K ∗ (N). The function K ∗ (N) is somewhat peculiar: defined as a product over the primes dividing N, it resembles a multiplicative function at first glance. But further inspection reveals that it is not, and so one cannot directly investigate its properties by the usual tools of multiplicative number theory. In this paper, we overcome these difficulties and prove a number of statistical results about K ∗ (N). For example, we determine the mean value of K ∗ (N) over odd N and over prime N, and we show that K ∗ (N) has a distribution function. We also explain how our results relate to existing theorems and conjectures on the multiplicative properties of #E(Fp), such as Koblitz’s conjecture. 1.