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

algebraic geometrynumber theory

Audience: researchers in the discipline

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SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca

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