Fully Hodge–Newton Decomposable Shimura Varieties

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 fibres 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 [25]) 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.

On the Bruhat–Tits stratification of a quaternionic unitary Rapoport–Zink space

“…A general understanding of the Bruhat-Tits stratification is achieved in the powerful work of [GH15] and the subsequent work of [GHN16]. There the problem is studied in the setting of affine Deligne-Lusztig varieties and a general group theoretic method is employed.…”

Some Results on Affine Deligne–lusztig Varieties

“…Part (3) is the most mysterious one. In [17], we state that "We do not see any reason why this independence of the parahoric could be expected a priori, but it is an interesting parallel with the question when the weakly admissible and admissible loci in the rigid analytic period domain coincide. "…”

“…We refer to [17, Definition 2.1] for the precise definition. The following Hodge-Newton decomposition for X(µ, b) was established in a joint work with Görtz and Nie [17].…”

“…We refer to [17,Theorem 3.16] for the precise statement. Note that an essential new feature is that unlike the Hodge-Newton decomposition of a single affine Deligne-Lusztig variety (e.g.…”

“…For quasi-split groups, it means that for any σ-orbit O on the set of simple roots, we have i∈O µ, ω i 1. For non quasi-split groups, the condition is more involved and we refer to [17,Definition 2.2] for the precise definition. It is also worth mentioning that it is not very difficult to classify the pairs (G, µ) with the minute condition.…”