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

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and let $N$ be a positive integer. Now, $M_E(N)$ counts the number of primes $p$ such that the group $E_p(\mathbb{F}_p)$ is of order $N$. In an earlier joint work with Balasubramanian, we showed that $M_E(N)$ follows Poisson distribution when an average is taken over a family of elliptic curve with parameters $A$ and $B$ where $A,\, B\ge N^{\frac{\ell}{2}}(\log N)^{1+γ}$ and $AB>N^{\frac{3\ell}{2}}(\log N)^{2+γ}$ for a fixed integer $\ell$ 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 $\exp(N^{\frac{ε^2}{20\ell}})\ge A, B>N^ε$ and $AB>N^{\frac{3\ell}{2}}(\log N)^{6+γ}$ for any $ε>0$.

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