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

Abstract: 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 $A,\, B\ge N^{\frac{\ell}{2}}(\log N)^{1+\gamma}$ and $AB>N^{\frac{3\ell}{2}}(\log N)^{2+\gamma}$ for a fixed integer $\ell$ and any $\gamma>0$. In this paper, we show that for sufficiently large $N$, the same result holds even if we take $A$ and $B$ in the range $\exp(N^{\frac{\epsilon^2}{20\ell}})\ge A, B>N^\epsilon$ and $AB>N^{\frac{3\ell}{2}}(\log N)^{6+\gamma}$ for any $\epsilon>0$.