Let $E$ be an elliptic curve over the rationals.
Koblitz conjectured that the number of primes $p \le x $ for which
$|E(\mathbb{F}_p)|$ is prime, is asymptotic
to
$$C_E \frac{x}{\log^2(x)},$$
for some constant $C_E$.
In this talk, we will try to discuss a result of S. Ali Miri and
V. Kumar Murty which addresses the above conjecture.
The result states that we
considers elliptic curves with complex multiplication
and assume the
Riemann hypothesis for all Dedekind zeta functions,
then it can be shown that
$$
|\{p \le x :
\nu (|E(\mathbb{F}_p)|) \le 16\}| \ll \frac{x}{\log^2 x}.
$$
Here $\nu(n)$ is used to denote the number of
distinct prime divisors of $n$.