We prove a comparison isomorphism between certain moduli spaces of $p$-divisible groups and strict ${\mathcal{O}}_{K}$-modules (RZ-spaces). Both moduli problems are of PEL-type (polarization, endomorphism, level structure) and the difficulty lies in relating polarized $p$-divisible groups and polarized strict ${\mathcal{O}}_{K}$-modules. We use the theory of relative displays and frames, as developed by Ahsendorf, Lau and Zink, to translate this into a problem in linear algebra. As an application of these results, we verify new cases of the arithmetic fundamental lemma (AFL) of Wei Zhang: The comparison isomorphism yields an explicit description of certain cycles that play a role in the AFL. This allows, under certain conditions, to reduce the AFL identity in question to an AFL identity in lower dimension.

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相对单位 RZ-空间和算术基本引理

我们证明了某些模空间之间的比较同构$p$- 可分组和严格${\mathcal{O}}_{K}$-模块（RZ 空间）。两个模量问题都是 PEL 类型的（极化、自同态、能级结构），难点在于关联极化$p$- 可分的群体和两极分化的严格${\mathcal{O}}_{K}$-模块。我们使用由 Ahsendorf、Lau 和 Zink 开发的相对显示和框架理论，将其转化为线性代数中的问题。作为这些结果的应用，我们验证了 Wei Zhang 的算术基本引理 (AFL) 的新案例：比较同构产生了对在 AFL 中起作用的某些循环的明确描述。这允许在某些条件下将所讨论的 AFL 身份减少为较低维度的 AFL 身份。