Prime analogues of the Erdős–Kac theorem for elliptic curves

Abstract

Let E / Q be an elliptic curve. For a prime p of good reduction, let E ( F p ) be the set of rational points defined over the finite field F p . We denote by ω ( # E ( F p ) ) , the number of distinct prime divisors of # E ( F p ) . We prove that the quantity (assuming the GRH if E is non-CM) ω ( # E ( F p ) ) − log log p log log p 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 ( F p ) .