Aliquot Cycles for Elliptic Curves with Complex Multiplication

We review the history of elliptic curves and show that it is possible to form a group law using the points on an elliptic curve over some field L. We review various methods for computing the order of this group when L is finite, including the complex multiplication method. We then define and examine the properties of elliptic pairs, lists, and cycles, which are related to the notions of amicable pairs and aliquot cycles for elliptic curves, defined by Silverman and Stange. We then use the properties of elliptic pairs to prove that aliquot cycles of length greater than two exist for elliptic curves with complex multiplication, contrary to an assertion of Silverman and Stange, proving that such cycles only occur for elliptic curves of j-invariant equal to zero, and they always have length six. We explore the connection between elliptic pairs and several other conjectures, and propose limitations on the lengths of elliptic lists.