On the moduli description of ramified unitary local models of signature $(n-1,1)$

Abstract

We provide a moduli description of the ramified unitary local model of signature $(n-1,1)$ with arbitrary parahoric level structure, assuming the residue field has characteristic not equal to $2$, thereby confirming a conjecture of Smithling. Our approach involves writing down explicit equations for the special fiber and proving that they define a normal, Cohen-Macaulay scheme, which is also of independent interest. As applications, we obtain moduli descriptions for: (1) ramified unitary Pappas-Zhu local models with arbitrary parahoric level; (2) the irreducible components of their special fiber in the maximal parahoric case; (3) integral models of ramified unitary Shimura varieties with arbitrary (quasi-)parahoric level.