Let be an elliptic curve of rank and a rational point of infinite order. For a prime of good reduction, let be the order of the cyclic group generated by the reduction of modulo . We denote by the number of distinct prime divisors of . Assuming the GRH, we show that the normal order of is . We also prove conditionally that there exists a normal distribution for the quantity
The latter result can be viewed as an elliptic analogue of a conjecture of Erdös and Pomerance about the distribution of , where is a natural number and the order of modulo .
Cite this article
Yu-Ru Liu, A prime analogue of the Erdös--Pomerance conjecture for elliptic curves. Comment. Math. Helv. 80 (2005), no. 4, pp. 755–769DOI 10.4171/CMH/33