AGM and Jellyfish Swarms of Elliptic Curves

Date: 16/03/2022
Time: 15:00 - 16:00

Location: Zoom

The classical AGM produces wonderful infinite sequences of arithmetic and geometric means with common limit. For finite fields Fq, with q ≡ 3 (mod 4), we introduce a finite field analogue AGMFq 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 Fq -points. We will describe this theory in this lecture, whose players include Gaussian hypergeometric functions and Gauss’class numbers of binary quadratic forms. Zoom-Login: Meeting-ID: 945 0781 4483 Password: HaveFun