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The Breuil–Mézard conjecture when l≠p

Shotton, Jack

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Abstract

Let l and p be primes, let F=Qp be a finite extension with absolute Galois group GF , let F be a finite field of characteristic l, and let W GF ! GLn.F/ be a continuous representation. Let R./ be the universal framed deformation ring for . If l D p, then the Breuil–Mézard conjecture (as recently formulated by Emerton and Gee) relates the mod l reduction of certain cycles in R./ to the mod l reduction of certain representations of GLn.OF /. We state an analogue of the Breuil–Mézard conjecture when l ¤ p, and we prove it whenever l>2 using automorphy lifting theorems. We give a local proof when l is “quasibanal” for F and is tamely ramified. We also analyze the reduction modulo l of the types . / defined by Schneider and Zink.

Journal Article Type Article
Acceptance Date Jul 26, 2017
Online Publication Date Dec 23, 2017
Publication Date Mar 15, 2018
Deposit Date Sep 20, 2018
Publicly Available Date Oct 10, 2018
Journal Duke Mathematical Journal
Print ISSN 0012-7094
Electronic ISSN 1547-7398
Publisher Duke University Press
Peer Reviewed Peer Reviewed
Volume 167
Issue 4
Pages 603-678
DOI https://doi.org/10.1215/00127094-2017-0039
Public URL https://durham-repository.worktribe.com/output/1314057
Related Public URLs https://arxiv.org/abs/1608.01784

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