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

### Yu-Ru Liu

University of Waterloo, Waterloo, Canada

## 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 $g_{b}(p)$ be the order of the cyclic group generated by the reduction $bˉ$ of $b$ modulo $p$. We denote by $ω(g_{b}(p))$ the number of distinct prime divisors of $g_{b}(p)$. Assuming the GRH, we show that the normal order of $ω(g_{b}(p))$ is $ggp$. 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 $ω(f_{a}(n))$, where $a$ is a natural number $>1$ and $f_{a}(n)$ the order of $a$ modulo $n$.

## 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. 755–769

DOI 10.4171/CMH/33