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RELATIVE UNITARY RZ-SPACES AND THE ARITHMETIC FUNDAMENTAL LEMMA

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  24 March 2020

Andreas Mihatsch Open the ORCID record for Andreas Mihatsch [Opens in a new window]
Andreas Mihatsch*
Affiliation:
Rheinische Friedrich-Wilhelms-Universitat Bonn, Mathematisches Institut, Endenicher Allee 60, Bonn, 53115, Germany (mihatsch@math.uni-bonn.de)
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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.

Keywords

moduli spacep-divisible groupdisplayarithmetic fundamental lemma

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties
Secondary: 14G35: Modular and Shimura varieties
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 1 , January 2022 , pp. 241 - 301
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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