Wildly ramified unitary local models for special parahorics. The odd dimensional case

TL;DR

This paper constructs and analyzes local models for wildly ramified unitary groups of odd dimension with special parahoric level structure, revealing their geometric properties and connections to v-sheaf models.

Contribution

It provides a lattice-theoretic description of parahoric subgroups in residue characteristic two and establishes the smoothness, normality, and Cohen-Macaulay properties of the associated local models.

Findings

Local models are smooth for one conjugacy class of parahoric subgroups.

Local models are normal and Cohen-Macaulay for the other class.

They represent the v-sheaf local models of Scholze-Weinstein.

Abstract

We construct local models for wildly ramified unitary similitude groups of odd dimension $n\geq 3$ with special parahoric level structure and signature $(n-1,1)$. We first give a lattice-theoretic description for parahoric subgroups using Bruhat-Tits theory in residue characteristic two, and apply them to define local models following the lead of Rapoport-Zink and Pappas-Rapoport. In our case, there are two conjugacy classes of special parahoric subgroups. We show that the local models are smooth for the one class and normal, Cohen-Macaulay for the other class. We also prove that they represent the v-sheaf local models of Scholze-Weinstein. Under some additional assumptions, we obtain an explicit moduli interpretation of the local models.