The supersingular locus of the Shimura variety for $$\hbox {GU}(1, n-1)$$ GU ( 1 , n - 1 ) over a ramified prime

Abstract

Introduction 1 2. The moduli space 4 3. Vertex lattices 7 4. The pointwise Bruhat-Tits stratification of N 0 red 8 5. The Deligne-Lusztig variety for the symplectic group 10 6. Bruhat-Tits-strata 15 7. Applications to Shimura varieties 18 References 20

Regular formal moduli spaces and arithmetic transfer conjectures

“…Remark 7.9. For n even or odd, the space N n is an (open and closed formal subscheme of an) RZ space with "special maximal parahoric level structure" in the sense that, in the notation of The corresponding RZ space, in which the polarization λ in the moduli problem is principal (still with Kottwitz condition (4.1) of signature (1, n − 1)), is studied for n both even and odd in [26]. This space is not regular for n ≥ 3, and therefore it does not appear to fit into the framework of this paper.…”

“…In this section, continuing from Remark 7.9, we consider the formal moduli space of [26] in the special case n = 2, still with F/F 0 ramified; see also [12]. This is a moduli space for quadruples (X, ι, λ, ρ) as before, with (X, ι) satisfying the Kottwitz condition char ι(π) | Lie X = T 2 − ̟, (8.1) and where this time the polarization λ is principal.…”

“…In this section we give two closely related generalizations (different than in [26]) of the space N (1) 2 of the previous section to any even n. However the spaces we define are not regular, and they will play only an auxiliary role in the ATC we formulate in §12. Let n ≥ 2 be even, still with F/F 0 ramified.…”

“…Note that this terminology differs slightly from e.g. [11,26]. Of course, a vertex lattice is a vertex lattice of type r for some r. Let us single out the following special cases.…”

Some Results on Affine Deligne–lusztig Varieties

“…The work of [75], [64], [34], [35], [68] focused on specific Shimura varieties with certain maximal parahoric level structure. The work [15] studied the analogous group-theoretic question for arbitrary reductive groups.…”