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Endomorphism rings of abelian surfaces and projective models of their moduli spaces. (English) Zbl 0972.14017

From the text: Let \(A\) be a simple principally polarized complex abelian surface, \(\text{End}(A)\) its ring of endomorphisms and \(L=\text{End}(A) \otimes\mathbb{Q}\) the algebra of endomorphisms (= the algebra of complex multiplications). Then, as is well-known, the ring \(\text{End}(A)\) is of one of the following types: An order in a CM-field of degree four, an order in an indefinite rational quaternion algebra, an order in a real quadratic field, or \(\mathbb{Z}\). The dimensions of the corresponding moduli spaces – named Shimura varieties of PEL-type – is 0, 1, 2, 3, respectively. In the first three cases we refer to these Shimura varieties, together with their embeddings into the Satake compactification \({\mathcal A}_2= \text{Proj} (A(\Gamma_2))\) of \(\Gamma_2 \setminus \mathbb{H}_2\), as CM-points, QCM-curves (quaternionic complex multiplication), and Humbert surfaces. We construct projective models for Humbert surfaces and QCM-curves, i.e., Shimura curves together with their natural embedding into the coarse moduli space for principally polarized abelian surfaces. The points of a QCM-curve correspond to an abelian surface, such that its algebra of complex multiplications is an order in an indefinite rational quaternian algebra. Moreover, we determine the structure of such orders.

MSC:

14G35 Modular and Shimura varieties
14K22 Complex multiplication and abelian varieties
14K10 Algebraic moduli of abelian varieties, classification
14H50 Plane and space curves
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11G15 Complex multiplication and moduli of abelian varieties
11G18 Arithmetic aspects of modular and Shimura varieties
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