algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety group schemes; arithmetic groundfields; degenerations of abelian varieties; arithmetic compactifications for the moduli spaces; principally polarized abelian varieties; geometric invariant theory; deformation theory; algebraization of formal moduli; Satake-Baily-Borel compactification; toroidal compactification; degenerations of polarized abelian varieties; polarizations; algebraic stack; Shimura varieties; equivariant sheaves; Hodge theory; De Rham cohomology; crystalline cohomology; action of Hecke operators; conformal quantum field theory 