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**Abelian varieties over \(\mathbb Q\) associated with an imaginary quadratic field.**
*(English)*
Zbl 1206.11076

Let \(K\) be an imaginary quadratic field with class number \(h\). In this paper the author characterizes \(h\)-dimensional CM abelian varieties over \(K\) which descend to abelian varieties over the rational number field \(\mathbb Q\) by their algebraic Hecke characters. If an abelian variety \(A\) over \(K\) has complex multiplication, then the dimension of \(A\) is \(h[H_g(\text{Im}\,\varepsilon: H_g]\) or \(2h[H_g(\text{Im}\,\varepsilon: H_g]\). Here \(H_g\) is the genus class field of \(K\) (Proposition 2). Hence these CM abelian varieties have minimal dimension \(h\) both over \(K\) and over \(\mathbb Q\).
Under the conditions that \(\text{End}_{\mathbb Q}(A)\otimes\mathbb Q\) are maximal real subfields of \(\text{End}_K(A)\otimes\mathbb Q\) and some
restrictions on the conductors of \(A\), such abelian varieties have been studied by T. Yang [Math. Ann. 329, No. 1, 87–117 (2004; Zbl 1088.11048)]. In this note removing the above conditions, the author gives a general treatment of these abelian varieties. He gives a characterization of the associated characters of them (Theorem 1) and finally he explicitly
determines such characters. (From the introduction).

Reviewer: Olaf Ninnemann (Berlin)

### MSC:

11G10 | Abelian varieties of dimension \(> 1\) |

11G05 | Elliptic curves over global fields |

11G15 | Complex multiplication and moduli of abelian varieties |

### Citations:

Zbl 1088.11048
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\textit{T. Nakamura}, Proc. Japan Acad., Ser. A 83, No. 8, 152--156 (2007; Zbl 1206.11076)

### References:

[1] | B. H. Gross, Arithmetic on elliptic curves with complex multiplication , Lecture Notes in Math. 776, Springer, Berlin, 1980. · Zbl 0433.14032 |

[2] | T. Nakamura, A classification of \(\mathbf{Q}\)-curves with complex multiplication, J. Math. Soc. Japan 56 (2004), no. 2, 635-648. · Zbl 1143.11327 |

[3] | D. E. Rohrlich, Galois conjugacy of unramified twists of Hecke characters, Duke Math. J. 47 (1980), no. 3, 695-703. · Zbl 0446.12011 |

[4] | G. Shimura, On the zeta-function of an abelian variety with complex multiplication, Ann. of Math. (2) 94 (1971), 504-533. · Zbl 0242.14009 |

[5] | T. Yang, On CM abelian varieties over imaginary quadratic fields, Math. Ann. 329 (2004), no. 1, 87-117. · Zbl 1088.11048 |

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