JournalscmhVol. 80, No. 4pp. 755–769

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

  • Yu-Ru Liu

    University of Waterloo, Waterloo, Canada
A prime analogue of the Erdös--Pomerance conjecture for elliptic curves cover

Abstract

Let E/QE/{\mathbb Q} be an elliptic curve of rank 1\ge 1 and bE(Q)b\in E({\mathbb Q}) a rational point of infinite order. For a prime pp of good reduction, let gb(p)g_b(p) be the order of the cyclic group generated by the reduction bˉ\bar b of bb modulo pp. We denote by ω(gb(p))\omega(g_b(p)) the number of distinct prime divisors of gb(p)g_b(p). Assuming the GRH, we show that the normal order of ω(gb(p))\omega(g_b(p)) is loglogp\log \log p. We also prove conditionally that there exists a normal distribution for the quantity

ω(gb(p))loglogploglogp.\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 ω(fa(n))\omega(f_a(n)), where aa is a natural number >1> 1 and fa(n)f_a(n) the order of aa modulo nn.

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