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 μ-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