It is known that in the moduli space A A of elliptic curves, there exist precisely 9 1 -ޑrational 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 A A of principally 2 polarized abelian surfaces, there exist precisely 19 -ޑrational 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. ᮊ
An elliptic curve E. defined over l is called a Q-curve if E and E are isogenous over Q for any ff in Gal(Q/Q). Many examples of Q-curves defined over quadratic fields have already been known. In this paper, we will give families of Q-curves defined over quartic and octic number fields.
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