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The average number of amicable pairs and aliquot cycles for a family of elliptic curves

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The average number of amicable pairs and aliquot cycles for a family of elliptic curves

Parks, James (2013) The average number of amicable pairs and aliquot cycles for a family of elliptic curves. PhD thesis, Concordia University.

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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.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (PhD)
Authors:Parks, James
Institution:Concordia University
Degree Name:Ph. D.
Program:Mathematics
Date:27 August 2013
Thesis Supervisor(s):David, Chantal
Keywords:elliptic curves, aliquot cycles, amicable pairs
ID Code:977734
Deposited By: JAMES PARKS
Deposited On:13 Jan 2014 15:11
Last Modified:18 Jan 2018 17:45
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