The Breuil–Mézard conjecture when l≠p

Shotton, Jack

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
• Public URL: https://durham-repository.worktribe.com/output/1314057
• Related Public URLs: https://arxiv.org/abs/1608.01784