Amicable Pairs and Aliquot Cycles for Elliptic Curves

Abstract

An amicable pair for an elliptic curve $E/\mathbb{Q}$ is a pair of primes $(p, q)$ of good reduction for $E \sharp\tilde{E}_p(\mathbb{F}_p) = q$ and $\sharp\tilde{E}_q(\mathbb{F}q) = p$. In this paper we study elliptic amicable pairs and analogously defined longer elliptic aliquot cycles. We show that there exist elliptic curves with arbitrarily long aliquot cycles, but that CM elliptic curves (with $j \neq 0$) have no aliquot cycles of length greater than two.We give conjectural formulas for the frequency of amicable pairs. For CM curves, the derivation of precise conjectural formulas involves a detailed analysis of the values of the Grössencharacter evaluated at primes $\mathfrak{p}$ in $\operatorname{End}(E )$ having the property that $\sharp\tilde{E}_{\mathfrak{p}}(\mathbb{F}_{\mathfrak{p}})$ is prime. This is especially intricate for the family of curves with $j = 0$.