Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 2

Special Issue: In Honor of David Mumford

Guest Editors: Ching-Li Chai, Amnon Neeman

On Shimura varieties for unitary groups

Pages: 773 – 837

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n2.a8


M. Rapoport (Mathematisches Institut der Universität Bonn, Germany; and Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)

B. Smithling (Department of Mathematics, University of Maryland, College Park, Md., U.S.A.)

W. Zhang (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)


This is a largely expository article based on our paper [31] on arithmetic diagonal cycles on unitary Shimura varieties. We define a class of Shimura varieties closely related to unitary groups which represent a moduli problem of abelian varieties with additional structure, and which admit interesting algebraic cycles. We generalize to arbitrary signature type the results of loc. cit. valid under special signature conditions. We compare our Shimura varieties with other unitary Shimura varieties.


Shimura varieties, local models, Gan–Gross–Prasad cycles, Kudla–Rapoport divisors

2010 Mathematics Subject Classification

Primary 11G18. Secondary 14G35.

M.R. is supported by a grant from the Deutsche Forschungsgemeinschaft through the grant SFB/TR 45 and by funds connected with the Brin E-Nnovate Chair at the University of Maryland.

B.S. is supported by NSA Grant H98230-16-1-0024 and Simons Foundation Grant #585707.

W.Z. is supported by NSF DMS #1901642.

Received 29 June 2019

Accepted 16 April 2020

Published 12 May 2021