AGM and Jellyfish Swarms of Elliptic Curves. Issue 4 (21st April 2023)

• Record Type: Journal Article
• Authors: Griffin, Michael J.; Ono, Ken; Saikia, Neelam; Tsai, Wei-Lun
• Is Part Of: American Mathematical Monthly. Volume 130:Issue 4(2023)
• Page Start: 355
• Page End: 369
• Publication Date: 2023-04-21
• Subjects: 14H52 -- 33C05 -- 33C75; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2022.2160157
• Languages: English
• ISSNs: 0002-9890

Abstract

Abstract: The classical AGM produces wonderful infinite sequences of arithmetic and geometric means with common limit. For finite fields F q, with q ≡ 3 ( mod 4 ), we introduce a finite field analogue AGM F 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 F q -points, which can be used to prove that each swarm has at least ( 1 / 2 − ε ) q 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.