Duke Math. J. 167 (4), 603-678, (15 March 2018) DOI: 10.1215/00127094-2017-0039
KEYWORDS: number theory, Galois representations, Breuil–Mézard conjecture, Taylor–Wiles method, local Langlands, 11S37, 11S20
Let and be primes, let be a finite extension with absolute Galois group , let be a finite field of characteristic , and let
be a continuous representation. Let be the universal framed deformation ring for . If , then the Breuil–Mézard conjecture (as recently formulated by Emerton and Gee) relates the mod reduction of certain cycles in to the mod reduction of certain representations of . We state an analogue of the Breuil–Mézard conjecture when , and we prove it whenever using automorphy lifting theorems. We give a local proof when is “quasibanal” for and is tamely ramified. We also analyze the reduction modulo of the types defined by Schneider and Zink.