Fabien Pazuki: Isogeny volcanoes: an ordinary inverse problem

Abstract:

Isogenies between elliptic curves have attracted a lot of attention, and over finite fields the structures that they generate are fascinating. For supersingular primes, isogeny graphs are very connected. For ordinary primes, isogeny graphs have a lot of connected components and each of these components has the shape… of a volcano! An \(\ell\)-volcano graph, to be precise, with \(\ell\) a prime. We study the following inverse problem: if we now start by considering a graph that has an \(\ell\)-volcano shape (we give a precise definition, of course), how likely is it that this abstract volcano can be realized as a connected component of an isogeny graph? We prove that any abstract \(\ell\)-volcano graph can be realized as a connected component of the \(\ell\)-isogeny graph of an ordinary elliptic curve defined over \(\mathbb{F}_p\), where \(\ell\) and \(p\) are two different primes, thereby solving the problem. On top of elliptic curves properties, the proof involves nice steps of algebraic number theory, diophantine equations, a Chebotarev argument… This is joint work with Henry Bambury and Francesco Campagna.