1. AGM and Jellyfish Swarms of Elliptic Curves.
- Author
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Griffin, Michael J., Ono, Ken, Saikia, Neelam, and Tsai, Wei-Lun
- Subjects
- *
JELLYFISHES , *DIRECTED graphs , *FINITE fields , *NUMBER theory , *GEOMETRIC series - 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. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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