Determination of simple CM abelian surfaces defined over $${\mathbb{Q}}$$

Abstract

It is known that in the moduli space \({\mathcal{A}_1}\) of elliptic curves, there exist precisely nine \({\mathbb{Q}}\)-rational points represented by an elliptic curve with complex multiplication by the maximal order of an imaginary quadratic field. In Murabayashi and Umegaki (J Algebra 235:267–274, 2001) and Umegaki [Determination of all \({\mathbb{Q}}\)-rational CM-points in the moduli spaces of polarized abelian surfaces, Analytic number theory (Beijng/Kyoto, 1999). Dev. Math., vol 6. Kluwer, Dordrecht, pp 349–357, 2002] we determined all \({\mathbb{Q}}\)-rational points in \({\mathcal{A}_2(d)}\) (the moduli space of d-polarized abelian surfaces) represented by a d-polarized abelian surface whose endomorphism ring is isomorphic to the maximal order of a quartic CM-field by using the result in Murabayashi (J Reine Angew Math 470:1–26, 1996). In this paper, we prove that polarized abelian surfaces corresponding to these \({\mathbb{Q}}\)-rational CM points have a \({\mathbb{Q}}\)-rational model by constructing certain Hecke characters.