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

TL;DR

This paper proves topological flatness of certain local models for ramified unitary groups in even dimensions, confirming conjectures and characterizations related to their geometric properties.

Contribution

It establishes topological flatness for even-dimensional cases of local models, extending previous results for odd dimensions and confirming conjectures on admissibility.

Findings

Proved topological flatness for even n in local models.

Characterized mu-admissible sets in types B and D.

Confirmed vertexwise admissibility conjecture.

Abstract

Local models are schemes, defined in terms of linear-algebraic moduli problems, which give \'etale-local neighborhoods of integral models of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. In the case of a unitary similitude group whose localization at Q_p is ramified, quasi-split GU_n, Pappas has observed that the original local models are typically not flat, and he and Rapoport have introduced new conditions to the original moduli problem which they conjecture to yield a flat scheme. In a previous paper we proved that their new local models are topologically flat when n is odd. In the present paper we prove topological flatness when n is even. Along the way, we characterize the mu-admissible set for certain cocharacters mu in types B and D, and we show that for these cocharacters admissibility can be characterized in a vertexwise way, confirming a conjecture of Pappas and Rapoport.