Almost prime values of the order of elliptic curves over finite fields.

Let be an elliptic curve over without complex multiplication. For each prime of good reduction, let be the order of the group of points of the reduced curve over . According to a conjecture of Koblitz, there should be infinitely many such primes such that is prime, unless there are some local obstructions predicted by the conjecture. Suppose that is a curve without local obstructions (which is the case for most elliptic curves over ). We prove in this paper that, under the GRH, there are at least primes such that has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng [20, 21] and Miri & Murty [15]. This is also the first result where the dependence on the conjectural constant appearing in Koblitz's conjecture (also called the twin prime conjecture for elliptic curves) is made explicit. This is achieved by sieving a slightly different sequence than the one of [20] and [15]. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant of Zywina [24] appearing in the upper bound for the number of primes such that is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH. [ABSTRACT FROM AUTHOR]