Let $$\mathcal {Q}_D$$ Q D be the set of moduli points on Siegel’s modular threefold whose corresponding principally polarized abelian surfaces have quaternionic multiplication by a maximal order $$\mathcal {O}$$ O in an indefinite quaternion algebra of discriminant D over $$\mathbb {Q}$$ Q such that the Rosati involution coincides with a positive involution of the form $$\alpha \mapsto \mu ^{-1}\overline{\alpha }\mu $$ α ↦ μ - 1 α ¯ μ on $$\mathcal {O}$$ O for some $$\mu \in \mathcal {O}$$ μ ∈ O with $$\mu ^2+D=0$$ μ 2 + D = 0 . In this paper, we first give a formula for the number of irreducible components in $$\mathcal {Q}_D$$ Q D , strengthening an earlier result of Rotger. Then for each irreducible component of genus 0, we determine its rational parameterization in terms of a Hauptmodul of the associated Shimura curve.

中文翻译：

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Siegel 模三重中的四元数位点

令 $$\mathcal {Q}_D$$ QD 是 Siegel 模三重模上的一组模点，其对应的主要极化阿贝尔表面在不定四元数代数中具有最大阶数的四元数乘法 $$\mathcal {O}$$ O $$\mathbb {Q}$$ Q 上的判别式 D 使得 Rosati 对合与 $$\alpha \mapsto \mu ^{-1}\overline{\alpha }\mu $$ 形式的正对合一致α ↦ μ - 1 α ¯ μ on $$\mathcal {O}$$ O 对于某些 $$\mu \in \mathcal {O}$$ μ ∈ O with $$\mu ^2+D=0$$ μ 2 + D = 0。在本文中，我们首先给出了 $$\mathcal {Q}_D$$ QD 中不可约分量数的公式，加强了 Rotger 的早期结果。然后对于属 0 的每个不可约分量，我们根据相关 Shimura 曲线的 Hauptmodul 确定其合理参数化。