$j$-invariants of CM curves $E$ over (say) the complex numbers are known as singular moduli. As the theory of complex multiplication explains, singular moduli are algebraic integers of great arithmetical significance, for they generate the maximal abelian extensions of imaginary quadratic fields.

I would be interested in reading about generalizations of this, where $E$ is replaced by an abelian variety of higher dimension with a big endomorphism ring. What are the singular moduli in this case, and what are their features? Can you recommend some references please? Or better address this question? Thanks!

$j$-invariants of CM curves $E$ over (say) the complex numbers are known as singular moduli. As the theory of complex multiplication explains, singular moduli are algebraic integers of great arithmetical significance, for they generate the maximal abelian extensions of imaginary quadratic fields.

I would be interested in reading about generalizations of this, where $E$ is replaced by an abelian variety of higher dimension with a big endomorphism ring. What are the singular moduli in this case, and what are their features? Can you recommend some references please? Or better address this question? Thanks!

[ERRATUM: singular moduli only generate the maximal abelian extensions of imaginary quadratic fields K over which ${\rm Gal}(K/\mathbf{Q})$ acts by inversion.]