SFU Number Theory and Algebraic Geometry Seminar: Fabien Pazuki

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. If time permits, we will touch upon some new applications and new challenges. This is joint work with Henry Bambury and Francesco Campagna.