Seminar in Algebraic Geometry and Number Theory

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. The proof involves nice steps of algebraic number theory and diophantine equations. The talk should be rather accessible. This is joint work with Henry Bambury and Francesco Campagna.