Abstract.
Let 
E be an elliptic curve over 
without complex
multiplication. For each prime 
p of good reduction,
let 
|E( p )| be the order of the group of points of the reduced
curve over 
p . According to a conjecture of Koblitz, there
should be infinitely many such primes 
p such that 
|E( p )| is prime,
unless there are some local obstructions predicted by the conjecture.
Suppose that 
E 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 
2.778C E twin x/(logx) 2 primes 
p such
that 
|E( p )| 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 
C E twin 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 
p such that 
|E( p )| is prime.
Finally, we remark that our results still hold
under an hypothesis weaker than the GRH.
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