Title: Computing isogeny classes of principally polarized abelian surfaces.

Abstract: Mazur's isogeny theorem gives an explicit finite set of primes that can occur as the degree of any irreducible rational isogeny of any elliptic curve over Q. Building on this powerful result, there is a complete classification of all possible isogeny classes/graphs of elliptic curves over Q. In this talk, we will investigate isogeny classes in dimension 2. Given a principally polarized abelian surface (PPAS) over Q with no extra endomorphisms, we will describe a practical algorithm to compute all other rational PPASs in its isogeny class. While there is no genus 2 analog of Mazur's theorem, one can still compute exceptional primes for torsion representations of any given PPAS, which is the starting point of our algorithm. Finally, I will discuss the results of running our algorithm on the database of genus 2 curves in the LMFDB, and show the isogeny graphs we obtain. This is joint work with Raymond van Bommel, Edgar Costa and Jean Kieffer.