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

Abstract

Let E be an elliptic curve defined over Q and let N be a positive integer. Then, ME(N) counts the number of primes p such that the group Ep(Fp) is of order N. For a given positive integer ℓ, we study the probability of the event {ME(N)=ℓ}. In an earlier joint work with Balasubramanian, we showed that ME(N) follows the Poisson distribution when an average is taken over a family of elliptic curves with parameters A and B where A,B≥Nℓ+12(log⁡N)1+γ and AB>N3(ℓ+1)2(log⁡N)2+γ 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 exp⁡(Nϵ220(ℓ+1))≥A,B>Nϵ and AB>N3(ℓ+1)2(log⁡N)3+2γ(log⁡log⁡N)ℓ+8 for any ϵ>0.