Filtered F-crystals on Shimura varieties of abelian type

Abstract

In this paper, we define and construct canonical filtered F -crystals with Gstructure over the integral models for Shimura varieties of abelian type at hyperspecial level defined by Kisin [12]. We check that these are related by p-adic comparison theorems to the usual lisse sheaves, and as an application we also use this to show that the Galois representations generated from the p-adic étale cohomology of Shimura varieties with nontrivial coefficient sheaves are crystalline, at least in the case of proper abelian type Shimura varieties. Contents1. Introduction 2.2. Filtered G-bundles 2.3. Filtered G-bundles with connection 2.4. Filtered F -crystals with G-structure 2.5. Lattices and S-modules 2.6. Some p-adic comparison results with G-structure 3. Filtered F -crystals on Shimura Varieties 3.1. Canonical G-bundles on integral models of Shimura Varieties 3.2. Special type case 3.3. Hodge type case 3.4. Abelian type case 3.5. Comparison with étale G-bundles 3.6. Application to Galois representations References

Eichler-Shimura Relations for Shimura Varieties of Hodge Type

“…For any geometric point x ∈ S κ ( Fp ), while the p-divisible group A x [p ∞ ] and the crystalline tensors (s α,0,x ) depends on some choices made during the construction of S K (G, X), it induces an F -crystal with G-structure D(G x ) over Fp which is independent of them. Moreover, [Lov17] (c.f. [KMPS, Appendix A]) constructed a universal F -isocrystal with G-structure over S K (G, X), which we denote by D 0 .…”

Stratifications in good reductions of Shimura varieties of abelian type

“…on S Kp (G, X) constructed by Lovering in [29], which may be viewed as a crystalline model of the universal de Rham bundle ω dR : Rep Qp (G c ) → Fil ∇ S Kp (G,X) rig , see [27].…”

“…We discuss the example of quaternionic Shimura varieties in each section. In section 5, we revisit our constructions of stratifications using the filtered F -crystal with G c -structure of [29]. In section 6, we study the relations between the Newton stratification, the Ekedahl-Oort stratification, and the central leaves both in the general and special setting.…”

“…By [1] Theorem 1.2, to give a G-bundle on X is the same as to give a G-torsor on X. As explained in [29] 2.2.8, by putting together Propositions 2.1.5 and 2.2.7 of loc. cit., we find that to give a filtered G-bundle on X is the same as to give a G-torsor I/X together with a G-equivariant morphism I → P. Here P is the scheme of parabolic subgroups of G.…”

An alternative construction of zip period maps for Shimura varieties

“…By construction the Dieudonné module L comes with Frobenius-invariant tensors which we denote by s cris . We know from the proof of [Kis10, 1.5.4] (see also [Lov17,Thm. 3.3.12] for more details) that there exists a W (κ)-algebra homomorphism (in fact an isomorphism) g : B → A such that the tuple N, s dR,A is obtained as the pull back along g of the tuple L, s cris , expect that one has to use the integrable connection ∇ L , as we did for ∇ N , to deal with the possible incompatibility of Frobenius lifts between σ A and σ B .…”