A prime analogue of the Erdös-Pomerance conjecture for elliptic curves

Abstract

Let E/Q be an elliptic curve of rank ≥ 1 and b ∈ E(Q) a rational point of infinite order. For a prime p of good reduction, let gb(p) be the order of the cyclic group generated by the reduction b of b modulo p. We denote by ω(gb(p)) the number of distinct prime divisors of gb(p). Assuming the GRH, we show that the normal order of ω(gb(p)) is log log p. We also prove conditionally that there exists a normal distribution for the quantity ω(gb(p)) − log log p √log log p . The latter result can be viewed as an elliptic analogue of a conjecture of Erdös and Pomerance about the distribution of ω(fa(n)), where a is a natural number > 1 and fa(n) the order of a modulo n.