AGM and jellyfish swarms of elliptic curves

TL;DR

This paper introduces a finite field analogue of the AGM, producing graph structures called jellyfish swarms that serve as tools in number theory, revealing insights into elliptic curves and class numbers.

Contribution

It develops a finite field AGM analogue that creates directed graphs resembling jellyfish swarms, linking graph structures to elliptic curve isogenies and class number interpretations.

Findings

Jellyfish swarms contain at least (1/2 - ε)√q jellyfish.

Each jellyfish corresponds to an isogeny class of elliptic curves.

The structure provides a new perspective on class numbers of quadratic fields.

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 $q\equiv 3\pmod 4,$ 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 $\mathit{jellyfish~swarm},$ as the 3D renderings of the connected components resemble $\mathit{jellyfish}$ (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 $(1/2-\varepsilon)\sqrt{q}$ jellyfish. Additionally, this interpretation gives a description of the $\mathit{class~numbers}$ of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish.