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Murabayashi, Naoki

Determination of simple CM abelian surfaces defined over \(\mathbb{Q}\). (English) Zbl 1210.14047

Math. Ann. 342, No. 3, 657-671 (2008).
Summary: It is known that in the moduli space \({\mathcal{A}}_1\) of elliptic curves, there exist precisely nine \({\mathbb{Q}}\)- represented by an elliptic curve with complex multiplication by the maximal order of an imaginary . N. Murabayashi and A. Umegaki [J. Algebra 235, No.1, 267–274 (2001; Zbl 0974.14015)] and A. Umegaki [in: Analytic number theory (Beijing/Kyoto 1999). Dev. Math. 6, 349–357 (2002; Zbl 1027.11043)] determined all \({\mathbb{Q}}\)-rational points in \({\mathcal{A}}_2(d)\) (the moduli space of \(d\)-polarized ) represented by a \(d\)-polarized abelian surface whose is isomorphic to the maximal order of a quartic CM-field by using the result of N. Murabayashi [J. Reine Angew. Math. 470, 1–26 (1996; Zbl 1001.14016)]. 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.

MSC:

14K22 Complex multiplication and abelian varieties
11G15 Complex multiplication and moduli of abelian varieties
14K15 Arithmetic ground fields for abelian varieties

Citations:

Zbl 0974.14015; Zbl 1027.11043; Zbl 1001.14016
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References:

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