Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows

We prove the weight part of Serre’s conjecture in generic situations for forms of U(3) which are compact at infinity and split at places dividing p as conjectured by Herzig (Duke Math J 149(1):37–116, 2009). We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge–Tate weights (2, 1, 0) for K/ Q_p unramified combined with patching techniques. Our results show that the (geometric) Breuil–Mézard conjectures hold for these deformation rings.