#### Vol. 10, No. 4, 2016

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

### Stéphane Bijakowski

Vol. 10 (2016), No. 4, 843–885
DOI: 10.2140/ant.2016.10.843
##### 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.

##### Keywords
Shimura variety, overconvergent modular forms, $\mu$-ordinary locus, canonical subgroups
##### Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 11F55, 14G35
##### Milestones
Received: 29 May 2015
Revised: 11 April 2016
Accepted: 12 May 2016
Published: 20 June 2016
##### Authors
 Stéphane Bijakowski Department of Mathematics Imperial College 180 Queen’s Gate London SW7 2AZ United Kingdom