On the Moduli Description of Local Models for Ramified Unitary Groups

Abstract

Abstract. Local models are schemes which are intended to model theétale-local structure of p-adic integral models of Shimura varieties. Pappas and Zhu have recently given a general group-theoretic construction of flat local models with parahoric level structure for any tamely ramified group, but it remains an interesting problem to characterize the local models, when possible, in terms of an explicit moduli problem. In the setting of local models for ramified, quasi-split GUn, work towards an explicit moduli description was initiated in the general framework of Rapoport and Zink's book and was subsequently advanced by Pappas and Pappas-Rapoport. In this paper we propose a further refinement to their moduli problem, which we show is both necessary and sufficient to characterize the (flat) local model in a certain special maximal parahoric case with signature (n − 1, 1).

Regular formal moduli spaces and arithmetic transfer conjectures

“…We continue to use the notation of §7. We closely follow the analysis in [32]. Similarly to §4 in loc.…”

“…In the particular situation of Remark 7.3 (which is for n odd, (r, s) = (n−1, 1), and I = {m}), condition (7) also implies the wedge condition, but we do not know if this implication holds in general. In all cases, the following is conjectured in [32]. 12 here we drop this requirement.…”

“…, 2n onto the complement of S in increasing order. By [32,Lem. 2.8], sgn(σ S ) = (−1) ΣS+n/2 , where ΣS denotes the sum of the elements in S. If S is of type (n − 1, 1), then S = {1, .…”

“…We recall the notion of "worst term" from [32,Def. 4.8], which we take with respect to the e T -basis for V n .…”

“…The remaining step, which is more difficult, is to show the the corresponding morphism on local models is an isomorphism, and hence the two sides have the same deformation theories. Here the "refined" spin condition introduced in [32] plays an important role, insofar as it is necessary for flatness of the local models, cf. Remarks 9.10 and 9.14.…”