AGM and Jellyfish Swarms of Elliptic Curves

Abstract

The classical produces wonderful infinite sequences of arithmetic and geometric means with common limit. For finite fields with we introduce a finite field analogue that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a jellyfish swarm, as the 3D renderings of the connected components resemble jellyfish (i.e., tentacles connected to a bell head). These swarms turn out to be more than the stuff of child’s play; they are taxonomical devices in number theory. Each jellyfish is an isogeny graph of elliptic curves with isomorphic groups of -points, which can be used to prove that each swarm has at least jellyfish. This interpretation also gives a description of the class numbers of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish.