Abstract.
Let
E
$E$
be an elliptic curve over
${\mathbb {Q}}$
without complex
multiplication. For each prime
p
$p$
of good reduction,
let
|
E
(
p
)
|
$|E({\mathbb {F}}_p)|$
be the order of the group of points of the reduced
curve over
p
${\mathbb {F}}_p$
. According to a conjecture of Koblitz, there
should be infinitely many such primes
p
$p$
such that
|
E
(
p
)
|
$|E({\mathbb {F}}_p)|$
is prime,
unless there are some local obstructions predicted by the conjecture.
Suppose that
E
$E$
is a curve without local obstructions (which is
the case for most elliptic curves over
${\mathbb {Q}}$
).
We prove in this paper that, under the GRH, there
are at least
2
.
778
C
E
twin
x
/
(
log
x
)
2
$2.778 C_E^{\rm twin} x / (\log x)^2$
primes
p
$p$
such
that
|
E
(
p
)
|
$|E({\mathbb {F}}_p)|$
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
$C_E^{\rm twin}$
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
$p$
such that
|
E
(
p
)
|
$|E({\mathbb {F}}_p)|$
is prime.
Finally, we remark that our results still hold
under an hypothesis weaker than the GRH.