We give a formula for the class number of an arbitrary complex mutliplication (CM) algebraic torus over $\mathbb {Q}$ . This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.

中文翻译：

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CM ALGEBRAIC TORI、CM ABELIAN 品种的分类号和单一 SHIMURA 品种的成分

我们给出了$\mathbb {Q}$ 上任意复数乘法 (CM) 代数环的类数的公式。这是根据 Ono 和 Shyr 的结果证明的。作为应用，我们给出了有限域上极化 CM 阿贝尔簇数、单一 Shimura 簇的连通分量和某些极化阿贝尔簇数的公式。我们还给出了我们主要结果的第二个证明。