Isogeny Graphs and Endomorphism Rings of Ordinary Abelian Varietes

We consider the graph whose nodes are abelian varieties over a finite field and whose edges are isogenies of a given kernel type. Its structure is deeply related to several key invariants and properties of the varieties. We will see that it can be made entirely explicit for elliptic curves. Over the past decade this explicitness has unlocked significant advances in computational number theory. Turning our attention to abelian surfaces, we will explain what obstructs a straightforward generalization of these results, and discuss what has nevertheless been achieved.