Analytic continuation on Shimura varieties with $\mu$-ordinary locus

TL;DR

This paper investigates the geometry of unitary Shimura varieties without an ordinary locus, constructs canonical subgroups near the $mu$-ordinary locus, and proves a classicality theorem for overconvergent modular forms using analytic continuation.

Contribution

It introduces a new approach to canonical subgroups and classicality results on Shimura varieties lacking an ordinary locus.

Findings

Existence of canonical subgroups near the $mu$-ordinary locus with explicit bounds.

Definition of overconvergent modular forms and Hecke operators in this setting.

Proof that eigenforms are classical under certain conditions.

Abstract

We study the geometry of unitary Shimura varieties without assuming the existence of an ordinary locus. We prove, by a simple argument, the existence of canonical subgroups on a strict neighborhood of the $\mu$-ordinary locus (with an explicit bound). We then define the overconvergent modular forms (of classical weight), as well as the relevant Hecke operators. Finally, we show how an analytic continuation argument can be adapted to this case to prove a classicality theorem, namely that an overconvergent modular form which is an eigenform for the Hecke operators is classical under certain assumptions.