RISC Colloquium Talk and Algorithmic Combinatorics Seminar

Title: AGM and jellyfish swarms of elliptic curves

Speaker: Prof. Dr. Ken Ono
         (University of Virginia, U.S.A.)

Time and Location: Wednesday, March 16, 2022, 3 p.m.
                   Zoom presentation


Abstract: The classical AGM produces wonderful infinite sequences of
arithmetic and geometric means with common limit. For finite fields GF(q),
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 GF(q) -points. We
will describe this theory in this lecture, whose players include Gaussian
hypergeometric functions and Gauss’ class numbers of binary quadratic
forms.