Abstract
Let $E/\mathbb{Q} $ be an elliptic curve. Silverman and Stange define primes $p$ and $q$ to be an elliptic, amicable pair if $\#E(\mathbb{F} _p) = q$ and $\#E(\mathbb{F} _q) = p$. More generally, they define the notion of aliquot cycles for elliptic curves. Here, we study the same notion in the case that the elliptic curve is defined over a number field~$K$. We focus on proving the existence of an elliptic curve~$E/K$ with aliquot cycle $(\mathfrak{p} _1, \ldots , \mathfrak{p} _{n})$ where the $\mathfrak{p} _{i}$ are primes of~$K$ satisfying mild conditions.
Citation
Jim Brown. David Heras. Kevin James. Rodney Keaton. Andrew Qian. "Amicable pairs and aliquot cycles for elliptic curves over number fields." Rocky Mountain J. Math. 46 (6) 1853 - 1866, 2016. https://doi.org/10.1216/RMJ-2016-46-6-1853
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