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