algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety fields of moduli; action of automorphism of reflex field on the torsion points; Abelian varieties with many complex multiplications; Abelian variety; cyclotomic fields; primes of good reduction; prime ideal decomposition of the endomorphism; Frobenius map; Riemann forms; field of definition; rank of a CM type; Langlands' conjecture; size of the Galois group of torsion points 