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

Daniel Le , Bao V Le Hung , Brandon Levin , Stefano Morra (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

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 [Her09]. 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/Qp unramified combined with patching techniques. Our results show that the (geometric) Breuil-Mézard conjectures hold for these deformation rings.

Domaines: Théorie des nombres [math.NT]

Daniel Le, Bao V Le Hung, Brandon Levin, Stefano Morra. Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows. Inventiones Mathematicae, 2018, 212 (1), pp.1-107. ⟨10.1007/s00222-017-0762-0⟩. ⟨hal-02097176⟩