Reduction modulo p of crystalline Galois representations via μ_p-equivariance

Abstract

For a crystalline representation of the absolute Galois group of Q_p, with given Hodge-Tate weights, we obtain new constraints on the inertial weights of its mod p reduction. This allows us to formulate an explicit Serre weight conjecture, in the generality of L-parameters for unramified connected reductive groups over Q_p, and to prove the elimination direction of this conjecture. The proof uses prismatic techniques to show that the reductions modulo p of the Breuil-Kisin modules attached to crystalline Galois representations acquire a natural μ_p-equivariant structure. Combining this with results on the geometry of the μ_p-fixed points of affine Grassmannians leads to our new constraint.