We study the possible weights of an irreducible 2-dimensional modular mod p representation of Gal (F/F), where F is a totally real field in which p is allowed to ramify, and the representation is tamely ramified at primes above p. We describe a set of possible weights and completely determine the weights in some cases when e = 2, f = 2.

Let k be a perfect field of characteristic p > 2 and K a totally ramified extension of K₀ = Frac W(k) with uniformizer π. Let F ⊆ K be a subfield with ϖ, ring of integers O, and residue field k(F) ⊆ k with |k(F)| = q. Let W(F) = O⊗(W(k(F))) W(k) and consider the ring 𝔖 = W(F)⟦u⟧ with an endomorphism φ that lifts the q-power Frobenius of k on W(F) and satisfies φ(u) ≡ u^q mod ϖ and φ(u) ≡ 0 mod u. In this dissertation, we use O-divided powers to define the analogue of Breuil-Kisin modules over the rings 𝔖 and S, where S is an O-divided power envelope of the surjection 𝔖 ↠ O(K) sending u to π. We prove that these two module categories are equivalent, generalizing the case when F = Q(p) and ϖ - p. As an application of our theory, we generalize the results of Kisin [17] and Cais-Lau [8] to relate the Faltings Dieudonné crystal of a ϖ-divisible O-module, which gives a Breuil module over S in our sense, to the modules of Kisin-Ren, providing a geometric interpretation to the latter.

In this dissertation, we study semi-stable representations of G(Q(p)) and their mod p-reductions, which is a part of the problem in which we construct deformation spaces whose characteristic 0 closed points are the semi-stable lifts with Hodge-Tate weights (0, 1, 2) of a fixed absolutely irreducible residual representation ρ : G(Q(p)) → GL₃(F(p)). We first classify the isomorphism classes of semi-stable representations of G(Q(p)) with regular Hodge-Tate weights, by classifying admissible filtered (phi,N)-modules with Hodge-Tate weights (0, r, s) for 0 < r < s. We also construct a Galois stable lattice in some irreducible semi-stable representations with Hodge-Tate weights (0, 1, 2), by constructing strongly divisible modules, which is an analogue of Galois stable lattices on the filtered (ɸ, N)-module side. We compute the reductions mod p of the corresponding Galois representations to the strongly divisible modules we have constructed, by computing Breuil modules, which is, roughly speaking, mod p-reduction of strongly divisible modules. We also determine which Breuil modules corresponds to irreducible mod p representations of G(Q(p)).