15 March 2018 The Breuil–Mézard conjecture when lp
Jack Shotton
Duke Math. J. 167(4): 603-678 (15 March 2018). DOI: 10.1215/00127094-2017-0039


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

ρ¯:GFGLn(F) be a continuous representation. Let R(ρ¯) be the universal framed deformation ring for ρ¯. If l=p, then the Breuil–Mézard conjecture (as recently formulated by Emerton and Gee) relates the mod l reduction of certain cycles in R(ρ¯) to the mod l reduction of certain representations of GLn(OF). We state an analogue of the Breuil–Mézard conjecture when lp, and we prove it whenever l>2 using automorphy lifting theorems. We give a local proof when l is “quasibanal” for F and ρ¯ is tamely ramified. We also analyze the reduction modulo l of the types σ(τ) defined by Schneider and Zink.


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Jack Shotton. "The Breuil–Mézard conjecture when lp." Duke Math. J. 167 (4) 603 - 678, 15 March 2018. https://doi.org/10.1215/00127094-2017-0039


Received: 18 April 2016; Revised: 20 July 2017; Published: 15 March 2018
First available in Project Euclid: 23 December 2017

zbMATH: 06857027
MathSciNet: MR3769675
Digital Object Identifier: 10.1215/00127094-2017-0039

Primary: 11S37
Secondary: 11S20

Keywords: Breuil–Mézard conjecture , Galois representations , local Langlands , number theory , Taylor–Wiles method

Rights: Copyright © 2018 Duke University Press


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Vol.167 • No. 4 • 15 March 2018
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