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Journal für die reine und angewandte Mathematik

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Conjecture de type de Serre et formes compagnons pour GSp4

1Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA

2Département de Mathématiques, UMR 7539, Institut Galilée, Université de Paris 13, 93430 Villetaneuse, France

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2013, Issue 676, Pages 1–32, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2011.190, January 2012

Publication History

Received:
2008-12-09
Revised:
2011-05-28
Published Online:
2012-01-21

Abstract

We present a Serre-type conjecture on the modularity of four-dimensional symplectic mod p Galois representations. We assume that the Galois representation is irreducible and odd (in the symplectic sense). The modularity condition is formulated using the étale and the algebraic de Rham cohomology of Siegel modular varieties of level prime to p. We concentrate on the case when the Galois representation is ordinary at p and we give a corresponding list of Serre weights. When the representation is moreover tamely ramified at p, we conjecture that all weights of this list are modular, otherwise we describe a subset of weights on the list that should be modular. We propose a construction of de Rham cohomology classes using the dual BGG complex, which should realise some of these weights.

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[1]
Jim Brown and Rodney Keaton
Journal of Number Theory, 2013, Volume 133, Number 5, Page 1492

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