Abelian surfaces admitting an (l,l)-endomorphism

We give a classification of all principally polarized abelian surfaces that admit an (l,l)-isogeny to themselves, and show how to compute all the abelian surfaces that occur. We make the classification explicit in the simplest case l=2. As part of our classification, we also show how to find all principally polarized abelian surfaces with multiplication by a given imaginary quadratic order.