Amicable Pairs and Aliquot Cycles for Elliptic Curves Over Number Fields
Let E/ℚ be an elliptic curve. Silverman and Stange define primes p and q to be an elliptic, amicable pair if #E(Fp) = q and #E(Fq) = 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 (p1,⋯, pn) where the pi are primes of K satisfying mild conditions.
Brown, Jim; Heras, David; James, Kevin; Keaton, Rodney; and Qian, Andrew. 2016. Amicable Pairs and Aliquot Cycles for Elliptic Curves Over Number Fields. Rocky Mountain Journal of Mathematics. Vol.46(6). 1853-1866. https://doi.org/10.1216/RMJ-2016-46-6-1853 ISSN: 0035-7596