Amicable pairs and aliquot cycles for elliptic curves

TL;DR

This paper investigates amicable pairs and aliquot cycles for elliptic curves, establishing their existence, limitations for CM curves, and conjectural frequency formulas, with detailed analysis for curves with j=0.

Contribution

It demonstrates the existence of arbitrarily long aliquot cycles for elliptic curves and characterizes the absence of such cycles in CM curves with j not 0, providing conjectural frequency formulas.

Findings

Elliptic curves can have arbitrarily long aliquot cycles.

CM elliptic curves with j ≠ 0 have no aliquot cycles longer than two.

Conjectural formulas for the frequency of amicable pairs are proposed.

Abstract

An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good reduction for E satisfying #E(F_p) = q and #E(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 aliqout cycles, but that CM elliptic curves (with j not 0) have no aliqout 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 Grossencharacter evaluated at a prime ideal P in End(E) having the property that #E(F_P) is prime. This is especially intricate for the family of curves with j = 0.