Abstract:We study a form of refined class number formula (resp. type number formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups of right ideal classes (resp. unit groups of maximal orders). For each finite noncyclic group $G$, we give an explicit formula for the number of conjugacy classes of maximal orders whose unit groups modulo center are isomorphic to $G$, and write down a representative for each conjugacy class. This leads to a complete recipe (even explicit formulas in special cases) for the refined class number formula for all finite groups. As an application, we prove the existence of superspecial abelian surfaces whose endomorphism algebras coincide with $\mathbb {Q}( \sqrt {p} )$ in all positive characteristic $p\not \equiv 1\pmod {24}$.

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中文翻译：

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实二次域上全定四元数代数的最大阶单位群

摘要：通过考虑正确理想类的自同构群（最大阶数的单位群），我们研究了实二次域上全定四元数代数中最大阶数的一种形式的细化类数公式（resp. type number formula） ）。对于每个有限非循环群$G$，我们给出了单位群模中心与$G$同构的最大阶共轭类的数量的明确公式，并写出每个共轭类的代表。这导致了所有有限群的精炼类数公式的完整配方（在特殊情况下甚至是显式公式）。作为应用，我们证明了超特殊阿贝尔曲面的存在，其自同态代数与 $\mathbb {Q}( \sqrt {p} )$ 在所有正特征 $p\not \equiv 1\pmod {24}$ 中重合。

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