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Arithmetic diagonal cycles on unitary Shimura varieties

Part of: Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  27 October 2020

M. Rapoport ,
B. Smithling  and
W. Zhang
M. Rapoport
Affiliation:
Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115Bonn, Germanyrapoport@math.uni-bonn.de Department of Mathematics, University of Maryland, College Park, MD20742, USA
B. Smithling
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD20742, USAcorgcorgcorg@gmail.com
W. Zhang
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA02139, USAweizhang@mit.edu
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Abstract

We define variants of PEL type of the Shimura varieties that appear in the context of the arithmetic Gan–Gross–Prasad (AGGP) conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension.

Keywords

arithmetic Gan–Gross–Prasad conjectureunitary Shimura varietiesarithmetic diagonal cyclesrelative trace formulaGross–Zagier formulaspecial values of L-functions

MSC classification

Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 14G35: Modular and Shimura varieties 14G40: Arithmetic varieties and schemes; Arakelov theory; heights
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 9 , September 2020 , pp. 1745 - 1824
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Copyright
Copyright © The Author(s) 2020

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