AGM and jellyfish swarms of elliptic curves

2021/10/23 by Michael J. Griffin, Griffin, Michael J., Ken Ono +5
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Mathematical Dynamics and Fractals #Cellular Automata and Applications

paper · pdf · doi:10.48550/arxiv.2110.12226

Abstract

The classical AGM produces wonderful interdependent infinite sequences of arithmetic and geometric means with common limit. For finite fields \mathbbF_q, with q≡ 3\pmod 4, we introduce a finite field analogue AGM_\mathbbF_q 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 \mathbbF_q-points, which can be used to prove that each swarm has at least (1/2-ε)√(q) jellyfish. Additionally, this interpretation gives a description of the class~numbers of Gauss, Hurwitz, and Kronecker which is akin to counting types of spots on jellyfish.

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