Benjamin Wesolowski

Fix a prime number l. Graphs of isogenies of degree a power of l are  
well-understood for elliptic curves, but not for higher-dimensional  
abelian varieties. We study the case of absolutely simple ordinary  
abelian varieties over a finite field. We analyse graphs of so-called  
L-isogenies (where L is an ideal in the real endomorphism ring),  
resolving that, in arbitrary dimension, their structure is similar,  
but not identical, to the "volcanoes" occurring as graphs of isogenies  
of elliptic curves. Specializing to the case of principally  
polarizable abelian surfaces, we derive a few algorithmic consequences.