AI 中文总结
本文移除Ribet相关研究中的可允许假设，完成$(\boldsymbol{\boldsymbol{O}}_p, \boldsymbol{\boldsymbol{O}}_p)$双格的完整分类，推导完美四元数厄米特型存在的充要条件，并将结果应用于超特殊阿贝尔簇的主极化存在性研究。
AI 中文摘要
在关于Shimura曲线坏约化的一篇有影响力的论文[K. Ribet, Bimodules and abelian surfaces, 载于《代数数论》, 359-407, Adv. Stud. Pure Math., 第17卷, 1989]中，Ribet研究了特征p的代数闭域$\boldsymbol{\bar{\boldsymbol{F}}}_p$上的某些超特殊阿贝尔曲面，这些曲面带有在p处分歧的不定四元数$\boldsymbol{\boldsymbol{Q}}$代数中的极大阶$\boldsymbol{\boldsymbol{O}}$的四元数乘法。特别地，他在额外的可允许假设下，通过分类在$\boldsymbol{\boldsymbol{Z}}_p$上自由的$(\boldsymbol{\boldsymbol{O}}_p, \boldsymbol{\boldsymbol{O}}_p)$双模$\boldsymbol{\boldsymbol{L}}_p$（见该文献的网址），对这类$\boldsymbol{\boldsymbol{O}}$阿贝尔曲面的p可除群进行了分类。本文中，我们通过移除该可允许假设并给出$(\boldsymbol{\boldsymbol{O}}_p, \boldsymbol{\boldsymbol{O}}_p)$双格$\boldsymbol{\boldsymbol{L}}_p$的完整分类，推广了Ribet的结果。为右阶$\boldsymbol{\boldsymbol{O}}_p$配备典范对合，且假设左阶$\boldsymbol{\boldsymbol{O}}_p$还配备正交对合$*$。我们推导了在右$\boldsymbol{\boldsymbol{O}}_p$格$\boldsymbol{\boldsymbol{L}}_p$上存在完美四元数厄米特型$\boldsymbol{\boldsymbol{\text{〈 , 〉}}}_p:\boldsymbol{\boldsymbol{L}}_p\times\boldsymbol{\boldsymbol{L}}_p\to \boldsymbol{\boldsymbol{O}}_p$的充要条件，该型在左阶$\boldsymbol{\boldsymbol{O}}_p$上诱导给定的对合$*$，并对这类自对偶四元数厄米特$(\boldsymbol{\boldsymbol{O}}_p, *, \boldsymbol{\boldsymbol{O}}_p)$双格$(\boldsymbol{\boldsymbol{L}}_p, \boldsymbol{\boldsymbol{\text{〈 , 〉}}}_p)$给出完整分类。在全局层面，我们将这些分类结果应用于研究带有$\boldsymbol{\boldsymbol{O}}$作用的$\boldsymbol{\boldsymbol{\bar{\boldsymbol{F}}}}_p$上超特殊阿贝尔簇的主极化存在性问题。
英文摘要
In an influential paper [K. Ribet, Bimodules and abelian surfaces, in Algebraic number theory, 359-407, Adv. Stud. Pure Math., Vol. 17, 1989] on the bad reduction of Shimura curves, Ribet studies certain superspecial abelian surfaces over $\overline{\mathbb{F}}_p$ with quaternion multiplication by a maximal order $\mathcal{O}$ in an indefinite quaternion $\mathbb{Q}$-algebra ramified at $p$. In particular, he classifies the $p$-divisible groups of such $\mathcal{O}$-abelian surfaces by classifying $(\mathcal{O}_p, \mathcal{O}_p)$-bimodules $L_p$ that are free over $\mathbb{Z}_p$ (i.e.bilattices) under an additional admissible assumption. In this paper, we generalize Ribet's result by removing the admissible assumption and producing a complete classification of $(\mathcal{O}_p, \mathcal{O}_p)$-bilattices $L_p$. Equip the right order $\mathcal{O}_p$ with the canonical involution, and suppose additionally that the left order $\mathcal{O}_p$ is equipped with an orthogonal involution $*$. We derive the necessary and sufficient condition for the existence of a perfect quaternion hermitian form $\langle~,~\rangle_p:L_p\times L_p\to \mathcal{O}_p$ on the right $\mathcal{O}_p$-lattice $L_p$ inducing the given involution $*$ on the left order $\mathcal{O}_p$, and give a complete classification of such self-dual quaternion hermitian $(\mathcal{O}_p, *, \mathcal{O}_p)$-bilattices $(L_p, \langle~,~\rangle_p)$. Globally, we apply these classification results to the study of the existence of principal polarizations on superspecial abelian varieties over $\overline{\mathbb{F}}_p$ equipped with $\mathcal{O}$-action.
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