Isogeny graphs of abelian varieties and singular ideals in orders

TL;DR

This paper generalizes the structural understanding of isogeny graphs from elliptic curves to higher-dimensional abelian varieties with specific endomorphism rings, revealing new properties and broadening applicability.

Contribution

It introduces a new ideal-theoretic framework for analyzing isogeny graphs of abelian varieties, extending previous results to non-simple and non-ordinary cases.

Findings

Structural theorems for isogeny graphs of abelian varieties

Analysis of overorders in étale algebras

Examples of volcanoes with unexpected properties

Abstract

Famously, Kohel proved that isogeny graphs of ordinary elliptic curves are beautifully structured objects, now called volcanos. We prove graph structural theorems for abelian varieties of any dimension with commutative endomorphism ring and containing a fixed locally Bass order, leveraging an ideal-theoretic perspective on isogeny graphs. This generalizes previous results, which relied on restrictive additional assumptions, such as maximal real multiplication, ordinary, and absolutely simple (Brooks, Jetchev, Wesolowski 2017). In particular, our work also applies to non-simple and non-ordinary isogeny classes. To obtain our results, we first prove a structure theorem for the lattice of inclusion of the overorders of a locally Bass order in an \'etale algebra which is of independent interest. This analysis builds on a careful study of local singularities of the orders. We include several examples of volcanoes and isogeny graphs exhibiting unexpected properties ultimately due to our more general setting.