AGM and jellyfish swarms of elliptic curves

Abstract

The classical $\mathrm{AGM}$ produces wonderful interdependent infinite sequences of arithmetic and geometric means with common limit. For finite fields ${\mathbb{F}}_q,$ with $q\equiv 3(\mathrm{mod}4),$ we introduce a finite field analogue ${\mathrm{AGM}}_{{\mathbb{F}}_q}$ that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a $jellyfishswarm,$ 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 ${\mathbb{F}}_q$-points, which can be used to prove that each swarm has at least $(1/2-\epsilon )\sqrt{q}$ jellyfish. Additionally, this interpretation gives a description of the $classnumbers$ of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish.

arXiv commentCorrects minor typos