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The field of moduli of abelian surfaces with complex multiplication. (English) Zbl 1001.14016
grom the introduction: Let \(E\) be an elliptic curve defined over the complex number field \(\mathbb{C}\) whose endomorphism ring \(\text{End} (E)\) is isomorphic to \({\mathfrak O}_K\), the ring of integers of an imaginary quadratic field \(K\). By the theory of complex multiplication, we have:
(1.1) The following statements are equivalent:
(i) \(E\) has a model defined over \(\mathbb{Q}\),
(ii) The \(j\)-invariant \(j_E\) of \(E\) is contained in \(\mathbb{Q}\),
(iii) The class number \(h_K\) of \(K\) is equal to one.
We present the following questions:
\((Q:i,g)\) Generalize the statement \((1,i)\) to the case of \(g\)-dimensional abelian varieties of CM-type \((1\leq i\leq 4)\).
For a \(g\)-dimensional abelan variety \((g\geq 2)\), its defining equations are very complex. So we can not expect a good answer for \((Q:4,g)\) \((g\geq 2)\). But for the case of \(g=2\) or 3, since any simple principally polarized abelian variety of dimension \(g\) is isomorphic to the Jacobian variety of a curve of genus \(g\), we can try to determine a defining equation of the curve corresponding to the considered abelian variety. In particular in the case of \(g=2\) we have Igusa’s \(j\)-invariants and their representations by theta series. Therefore, if the answer for \((Q:1,2)\) and \((Q:2,2)\) are known, we can find those for \((Q:3,2)\) and \((Q:4,2)\).
In this paper we will give the answer for \((Q:1,2)\). More precisely our result (theorem 4.12) is the generalization of “(ii) \(\Leftrightarrow\) (iii) in (1.1)” to the case of \(g=2\) under the restriction to CM by the maximal order.

14K22 Complex multiplication and abelian varieties
11G15 Complex multiplication and moduli of abelian varieties
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