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

Local models are schemes, defined in terms of linear-algebraic moduli problems, which are used to model the étale-local structure 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${ \mathbb{Q} }_{p} $is ramified, quasi-split$G{U}_{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.