LAUSR

Abstract 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.