Abstract

Let E be an elliptic curve over Q. Silverman and Stange defined the set (p_1,...,p_L) of distinct primes to be an aliquot cycle of length L of E if each p_i is a prime of good reduction for E such that #E_{p_1)(F_{p_1})=p_2,...,#E_{p_{L-1}}(F_{p_{L-1}})=p_L, #E_{p_L}(F_{p_L})=p_1. Let \pi_{E,L}(X) denote the aliquot cycle counting function with p<=X. They conjectured for elliptic curves without complex multiplication that
\pi_{E,L}(X) ~ \sqrt{X}/(\log X)^{L}. Jones refined this conjecture to give an explicit constant,
namely \pi_{E,L}~C_{E,L}\sqrt{X}/(\log X)^L. In this thesis we will show that the conjectured upper bound holds for \pi_{E,L}(X) on average over the family of all elliptic curves with a short length for the average.