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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).

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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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 · doi:10.2969/jmsj/1191418649
[3] D. E. Rohrlich, Galois conjugacy of unramified twists of Hecke characters, Duke Math. J. 47 (1980), no. 3, 695-703. · Zbl 0446.12011 · doi:10.1215/S0012-7094-80-04742-0
[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 · doi:10.2307/1970768
[5] T. Yang, On CM abelian varieties over imaginary quadratic fields, Math. Ann. 329 (2004), no. 1, 87-117. · Zbl 1088.11048 · doi:10.1007/s00208-004-0511-8
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