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

TL;DR

This paper refines the understanding of how often amicable pairs occur for elliptic curves by establishing an asymptotic formula for their average count across a family of such curves.

Contribution

The paper improves previous bounds to an asymptotic formula for the average number of amicable pairs for elliptic curves over a family, advancing theoretical understanding.

Findings

Established an asymptotic for the average number of amicable pairs

Refined previous bounds to precise asymptotic results

Contributed to the theoretical framework of elliptic curve pairings

Abstract

Amicable pairs for a fixed elliptic curve defined over $\mathbb{Q}$ 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.