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Invariant elements for $ p$-modular representations of $ {\mathbf{GL}}_{2}({\mathbf{Q}}_p)$

Author: Stefano Morra
Journal: Trans. Amer. Math. Soc. 365 (2013), 6625-6667
MSC (2010): Primary 22E50, 11F85
Published electronically: July 10, 2013
MathSciNet review: 3105765
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Abstract: Let $ p$ be an odd rational prime and $ F$ a $ p$-adic field. We give a realization of the universal $ p$-modular representations of $ {\mathbf {GL}}_{2}(F)$ in terms of an explicit Iwasawa module. We specialize our constructions to the case $ F={\mathbf {Q}}_p$, giving a detailed description of the invariants under principal congruence subgroups of irreducible admissible $ p$-modular representations of $ {\mathbf {GL}}_{2}({\mathbf {Q}}_p)$, generalizing previous work of Breuil and Paskunas. We apply these results to the local-global compatibility of Emerton, giving a generalization of the classical multiplicity one results for the Jacobians of modular curves with arbitrary level at $ p$.

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Additional Information

Stefano Morra
Affiliation: Laboratoire de Mathématiques de Montpellier, place Eugène Bataillon, Case courrier 051, 34095 Montpellier cedex 5, France
Address at time of publication: The Fields Institute, 222 College Street, Toronto, Ontario, Canada M5T 3J1

Keywords: $p$-modular Langlands programme, supersingular representations, socle filtration, congruence subgroup, multiplicity one
Received by editor(s): February 14, 2010
Received by editor(s) in revised form: April 11, 2012, May 6, 2012, July 2, 2012, and August 6, 2012
Published electronically: July 10, 2013
Additional Notes: The author was partially supported by a Fields-Ontario Postdoctoral Fellowship
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.