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Original Article

Fully Hodge–Newton Decomposable Shimura Varieties

Fakultät für Mathematik/Institut für Experimentelle Mathematik, Universität Duisburg-Essen, 45117, Essen, Germany
Department of Mathematics, University of Maryland, College Park, MD, 20742, USA
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
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Abstract

The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of Rapoport–Zink spaces and of affine Deligne–Lusztig varieties. We prove a Hodge–Newton decomposition for affine Deligne–Lusztig varieties and for the special fibers of Rapoport–Zink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (Hodge–Newton decomposability) which can be phrased in combinatorial terms. Second, we study the Shimura varieties in which every non-basic σ-isogeny class is Hodge–Newton decomposable. We show that (assuming the axioms of He and Rapoport in Manuscr. Math. 152(3–4): 317–343, 2017) this condition is equivalent to nice conditions on either the basic locus or on all the non-basic Newton strata of the Shimura varieties. We also give a complete classification of Shimura varieties satisfying these conditions. While previous results along these lines often have restrictions to hyperspecial (or at least maximal parahoric) level structure, and/or quasi-split underlying group, we handle the cases of arbitrary parahoric level structure and of possibly non-quasi-split underlying groups. This results in a large number of new cases of Shimura varieties where a simple description of the basic locus can be expected. As a striking consequence of the results, we obtain that this property is independent of the parahoric subgroup chosen as level structure. We expect that our conditions are closely related to the question whether the weakly admissible and admissible loci coincide.

Keywords

Reduction of Shimura varietiesAffine Deligne–Lusztig varietiesNewton stratificationHodge–Newton decomposition

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Peking Mathematical Journal
Volume 2 Issue 2,
June 2019
Pages 99-154
DOI: 10.1007/s42543-019-00013-2
Cite this article:
Görtz, U., He, X. & Nie, S. Fully Hodge–Newton Decomposable Shimura Varieties. Peking Math J 2, 99-154 (2019). https://doi.org/10.1007/s42543-019-00013-2

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Received: 03 April 2018
Revised: 10 January 2019
Accepted: 05 March 2019
Published: 05 August 2019
© Peking University 2019
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