2008/12/15 by Chantal David, Jie Wu, David, Chantal +1 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #math.NT #msc:11N36 #msc:14H52
paper · pdf · doi:10.48550/arxiv.0812.2860
arxiv created 2008/12/15 · arxiv updated 2009/12/01
Let E be an elliptic curve over \Q without complex multiplication, and which is not isogenous to a curve with non-trivial 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 2.778 CE\rm twin x / (logx)2 primes p such that |E(\Fp)| has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng and Murty & Miri. This is also the first result where the dependence on the conjectural constant CE\rm twin 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 used by previous authors. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant 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.