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 $\epsilon >0$.

26 pages