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An asymptotic for the average number of amicable pairs for elliptic curves

Published online by Cambridge University Press:  26 October 2017

JAMES PARKS
JAMES PARKS*
Affiliation:
Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, AB, T1K 3M4, Canada.
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Abstract

Amicable pairs for a fixed elliptic curve defined over ℚ were first considered by Silverman and Stange where they conjectured an order of magnitude for the function that counts such amicable pairs. This was later refined by Jones to give a precise asymptotic constant. The author previously proved an upper bound for the average number of amicable pairs over the family of all elliptic curves. In this paper we improve this result to an asymptotic for the average number of amicable pairs for a family of elliptic curves.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 1 , January 2019 , pp. 33 - 59
Copyright
Copyright © Cambridge Philosophical Society 2017 

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Footnotes

with an appendix by Sumit Giri

Present address: Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany e-mail: james.william.a.parks@gmail.com

†

This work was supported by a Pacific Institute for the Mathematical Sciences Postdoctoral Fellowship.

References

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