  The Satake compactification of the moduli space of principally polarized
abelian surfaces with a level two structure has a degree 8 endomorphism. The
aim of this paper is to show that this result can be extended to other modular
threefolds. The main tools are Siegel modular forms and Satake
compactifications of arithmetic quotients of the Siegel upper-half space.
Indeed, the construction of the degree 8 endomorphism on suitable modular
threefolds is done via an isomorphism of graded rings of modular forms. By
studying the action of the Fricke involution one gets a further extension of
the previous result to other modular threefolds. The possibility of a similar
situation in higher dimensions is also discussed.
