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Amicable Pairs and Aliquot Cycles for Elliptic Curves Over Number FieldsBrown, Jim, Heras, David, James, Kevin, Keaton, Rodney, Qian, Andrew 01 January 2016 (has links)
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.
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