AGM and Jellyfish Swarms of Elliptic Curves

Description

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.

Period	16 Mar 2022
Event type	Guest talk
Location	AustriaShow on map