On a conjecture of Pappas on unitary Shimura varieties
Let $K$ be an imaginary quadratic field and let $V$ be a finite dimensional $K$-Hermitian space. Then there is a natural Shimura datum for the unitary similitude group of $V$, and the corresponding Shimura varieties admit moduli interpretations in terms of abelian varieties with additional structure. Given a prime number $p$, it is desirable (as for any Shimura variety) to define and study "good" p-integral models of these varieties. When $p$ is odd and the level subgroup at $p$ is of parahoric type, such models were proposed by Rapoport and Zink and, when $p$ ramifies in $K$, subsequently refined by Pappas, all in terms of an explicit moduli problem. In the case that the polarization in the moduli problem is principal at $p$, Pappas conjectured that his refined moduli problem is "good" in the sense that the resulting scheme is flat over the base, and he proved his conjecture when the signature of $V$ is $(n-1,1)$ or $(1,n-1)$. I will discuss a proof of his conjecture in general.