Hida theory over some unitary Shimura varieties without ordinary locus

TL;DR

This paper extends Hida theory to certain unitary Shimura varieties lacking an ordinary locus, establishing boundedness of ordinary forms' dimension and constructing a finite type module of $\Lambda$-adic forms.

Contribution

It develops Hida theory for specific Shimura varieties without ordinary locus, introducing new bounds and module constructions.

Findings

Dimension of ordinary forms is uniformly bounded

Existence of a finite type $\Lambda$-adic module of ordinary forms

Hida theory applies to new classes of Shimura varieties

Abstract

We develop Hida theory for Shimura varieties of type A without ordinary locus. In particular we show that the dimension of the space of ordinary forms is bounded independently of the weight and that there is a module of $\Lambda$-adic cuspidal ordinary forms which is of finite type over $\Lambda$, where $\Lambda$ is a twisted Iwasawa algebra.