On the reduction modulo p of certain modular p-adic Galois representations

We determine the possible Serre weights associated to certain Hilbert modular forms when the rational prime p is totally ramified in the totally real field F. Our weight lowering method for arbitrarily large weight is applicable when the slope is sufficiently small, enabling us to compute the mod p reduction at inertia from the known results in the small weight range. In the case of elliptic modular forms (and for certain Hilbert modular forms in the p totally split case) we obtain a unique mod p reduction when the slope is in (0,1/2).