AMICABLE PAIRS AND ALIQUOT CYCLES FOR ELLIPTIC CURVES OVER NUMBER FIELDS

摘要

Let E/Q be an elliptic curve. Silverman and Stange define primes p and q to be an elliptic amicable pair if #E(F-p) = q and #E(F-q) = p. More generally, they define the notion of aliquot cycles for elliptic curves. Here we study the same notion in the case that the elliptic curve is defined over a number field K. We focus on proving the existence of an elliptic curve E/K with aliquot cycle (p(1),..., p(n)) where the pi are primes of K satisfying mild conditions.