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. 
Reinier Broker, Kristin Lauter, Marco Streng
math.AG math.NT