Topological flatness of local models for ramified unitary groups. I. The odd dimensional case

Abstract

Local models are certain schemes, defined in terms of linear-algebraic moduli problems, which giveétale-local neighborhoods of integral models of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. When the group defining the Shimura variety ramifies at p, the local models (and hence the Shimura models) as originally defined can fail to be flat, and it becomes desirable to modify their definition so as to obtain a flat scheme. In the case of unitary similitude groups whose localizations at Qp are ramified, quasi-split GUn, Pappas and Rapoport have added new conditions, the so-called wedge and spin conditions, to the moduli problem defining the original local models and conjectured that their new local models are flat. We prove a preliminary form of their conjecture, namely that their new models are topologically flat, in the case n is odd.

Topological flatness of local models for ramified unitary groups. II. The even dimensional case

“…The analog of Theorem 1.4 for odd n is proved in [24]. However, the proof for even n that we give here is considerably more laborious.…”

“…Thus it is of interest to describe, when possible, the corrected models and local models via a refinement of the original moduli problem. This is the problem of concern in this paper and its prequel [24], in the case of a unitary similitude group attached to an imaginary quadratic number field ramified at the prime p = 2.…”

“…, e n , and consider the quasi-split reductive group G := GU n := GU(φ) over F 0 . In this paper, we consider exclusively the case that n is even; the case of odd n was treated in [24]. Let m := n/2.…”

“…Recall that a scheme over a regular, integral, one-dimensional base scheme is topologically flat if its generic fiber is topologically dense. For odd n, the analog of Theorem 1.3 is proved in [24], where it is also shown that the wedge local model M ∧ I is topologically flat (though typically not flat, as its scheme structure typically differs from that of M Following Görtz [3] (see also [4, 5, 13-16, 22, 24]), the key technique in the proof of Theorem 1.3 is to embed the geometric special fiber of M naive See ğ 5.3 for the definition of the {µ r,s }-admissible set, and ğ 7.5 for the proof of the theorem. The {µ r,s }-admissible set was defined by Kottwitz and Rapoport [11] and Rapoport [19].…”

Vertexwise criteria for admissibility of alcoves

“…In the cases where the equality Adm(µ) = Perm(µ) could be proved, it has been used to establish the topological flatness of M loc (cf. Görtz [Go1,Go2,Go3] and Smithling [Sm1,Sm2,Sm3,Sm4]). Recently Pappas and Zhu [PZ] defined group-theoretic "local models" attached to any pair (G, {µ}) where G is a tamely ramified group over a p-adic field, and showed that the strata in the special fiber are parametrized by the {µ}-admissible set.…”