Humbert surfaces and the Kummer plane

A Humbert surface is a hypersurface of the moduli space of principally polarized abelian surfaces defined by an equation of the form with integers . We give geometric characterizations of such Humbert surfaces in terms of the presence of certain curves on the associated Kummer plane. Intriguingly this shows that a certain plane configuration of lines and curves already carries all information about principally polarized abelian surfaces admitting a symmetric endomorphism with given discriminant.