Topological flatness of local models for ramified unitary groups. I. The odd dimensional case

TL;DR

This paper proves that certain modified local models for ramified unitary groups in odd dimensions are topologically flat, advancing understanding of their geometric properties in the context of Shimura varieties.

Contribution

It establishes topological flatness of the new local models for ramified unitary groups in the odd-dimensional case, supporting conjectures on their flatness.

Findings

New local models are topologically flat in the odd-dimensional case.

Supports conjecture that wedge and spin conditions lead to flat models.

Advances understanding of integral models of ramified unitary Shimura varieties.

Abstract

Local models are certain 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. When the group defining the Shimura variety ramifies at p, the local models (and hence the Shimura models) as originally defined can fail to be flat, and it becomes desirable to modify their definition so as to obtain a flat scheme. In the case of unitary similitude groups whose localizations at Q_p are ramified, quasi-split GU_n, Pappas and Rapoport have added new conditions, the so-called wedge and spin conditions, to the moduli problem defining the original local models and conjectured that their new local models are flat. We prove a preliminary form of their conjecture, namely that their new models are topologically flat, in the case n is odd.