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

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve of rank $\ge 1$ and $b\in E(\mathbb{Q})$ 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 $\bar{b}$ of $b$ modulo $p$. We denote by $\omega (g_b(p))$ the number of distinct prime divisors of $g_b(p)$. Assuming the GRH, we show that the normal order of $\omega (g_b(p))$ is $\log \log p$. 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 $\omega (f_a(n))$, where $a$ is a natural number $>1$ and $f_a(n)$ the order of $a$ modulo $n$.