Prime analogues of the Erdős–Kac theorem for elliptic curves

Let E/QE/Q be an elliptic curve. For a prime p   of good reduction, let E(Fp)E(Fp) be the set of rational points defined over the finite field FpFp. We denote by ω(#E(Fp))ω(#E(Fp)), the number of distinct prime divisors of #E(Fp)#E(Fp). We prove that the quantity (assuming the GRH if E is non-CM)ω(#E(Fp))−loglogploglogp distributes normally. This result can be viewed as a “prime analogue” of the Erdős–Kac theorem. We also study the normal distribution of the number of distinct prime factors of the exponent of E(Fp)E(Fp).