Determination of All -Rational CM-Points in the Moduli Space of Principally Polarized Abelian Surfaces,

It is known that in the moduli space 1 of elliptic curves, there exist precisely 9 -rational points corresponding to the isomorphism class of elliptic curves with complex multiplication by the ring of algebraic integers of a principal imaginary quadratic number field. Here, we prove that in the moduli space 2 of principally polarized abelian surfaces, there exist precisely 19 -rational points corresponding to the isomorphism class of abelian surfaces whose endomorphism rings are isomorphic to the rings of algebraic integers of some imaginary cyclic quartic number fields.