The Breuil–Mézard conjecture when l≠p

Shotton, Jack

doi:10.1215/00127094-2017-0039

The Breuil–Mézard conjecture when l≠p

Shotton, Jack

Authors

Dr Jack Shotton jack.g.shotton@durham.ac.uk
Assistant Professor

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.

Citation

Shotton, J. (2018). The Breuil–Mézard conjecture when l≠p. Duke Mathematical Journal, 167(4), 603-678. https://doi.org/10.1215/00127094-2017-0039

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
Related Public URLs	https://arxiv.org/abs/1608.01784