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

• Autores: Yu-Ru Liu
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 80, Nº 4, 2005, págs. 755-769
• Idioma: inglés
• DOI: 10.4171/cmh/33
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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 $$ \frac{\omega(g_b(p)) - \log \log p}{\sqrt{\log \log p}}. $$ 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 $n$