Short average distribution of a prime counting function over families of elliptic curves

Let E be an elliptic curve defined over Q and let N be a positive integer. Then, M_E(N) counts the number of primes p such that the group E_p(F_p) is of order N. For a given positive integer ℓ, we study the probability of the event {M_E(N)=ℓ}. In an earlier joint work with Balasubramanian, we showed that M_E(N) follows the Poisson distribution when an average is taken over a family of elliptic curves with parameters A and B where [Formula presented] and [Formula presented] for a positive integer ℓ and any γ>0. In this paper, we show that for sufficiently large N, the same result holds even if we take A and B in the range [Formula presented] and [Formula presented] for any ϵ>0.