  Given an elliptic curve $E$ and a positive integer $N$, we consider the
problem of counting the number of primes $p$ for which the reduction of $E$
modulo $p$ possesses exactly $N$ points over $\mathbb F_p$. On average (over a
family of elliptic curves), we show bounds that are significantly better than
what is trivially obtained by the Hasse bound. Under some additional
hypotheses, including a conjecture concerning the short interval distribution
of primes in arithmetic progressions, we obtain an asymptotic formula for the
average.
