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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.

##### MSC:
 14K22 Complex multiplication and abelian varieties 11G15 Complex multiplication and moduli of abelian varieties
##### Keywords:
$$j$$-invariant; complex multiplication; abelian variety
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