Prime analogues of the Erdős–Kac theorem for elliptic curves

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Abstract

Let E/Q be an elliptic curve. For a prime p of good reduction, let E(Fp) be the set of rational points defined over the finite field Fp. We denote by ω(#E(Fp)), the number of distinct prime divisors of #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).

MSC

11N60
11G20

Keywords

Prime divisors
Rational points
Elliptic curves

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1

Research partially supported by an NSERC discovery grant.