General notation. In the following we will study continuous Galois rep- resentations ρ : GQ → GL2(Fp). We will fix an algebraic closure Qp and an em- bedding Q ‹→ Qp, so that we can view GQp as a subgroup of GQ. We will identify GQp with Gp the decomposition subgroup at p and write Ip for the inertia group at
General notation. For the reader’s convenience, we collect here the notation that will be used in the thesis. For a noetherian ring R, we write D(R) for the derived category of R-modules. For an abelian group N , we write Ntor for its torsion submodule and set Ntf := N/Ntor, which we regard as embedded in the associated space Q ⊗Z N . Let E be a field containing R. If A is a R-module, we sometimes denote E⊗R A by simply AE or E· A. For any complex C = (Ci)i∈Z in D(R) and an integer j, we define the ‘shifted complex’ C(j) to be the complex such that the module Ci+j is placed at degree i. For any R-module M and integer a, we write M [a] to be the complex such that M is placed at degree −a and the zero module is placed at any other degree. Fix an algebraic closure Qc of Q. For any non-negative integer m, we denote by µm the subgroup of all m-th roots of unity in (Qc)×. For a rational prime p, we denote the inverse limit lim µpn by Zp(1). For j > 0, we set Zp(j) := Zp(1)⊗j and for j < 0, we set Zp(j) := HomZp (Zp(j), Zp). For any Zp-module T and integer j, we denote T ⊗Zp Zp(j) by T (j). For any finite group Γ and any Zp[Γ] (resp. Z[Γ])-module N we write N V for the Pontryagin dual HomZp (N, Qp/Zp) (resp. HomZ(N, Q/Z)) which we endow with the usual contragredient action of Zp[Γ] (resp. Z[Γ]). To be more specific, the contragradient action is defined by setting σ · f (x) = f (σ—1 · x) for any σ ∈ Γ and f (x) ∈ HomZp (N, Qp/Zp) (resp. HomZ(N, Q/Z)). For a number field k, we denote the set of archimedean and p-adic places of k by S∞(k) and Sp(k) respectively. Sometimes we write these sets as S∞ and Sp if it is clear from the context what the underlying field is. For any place v of k, we denote its completion of k at ^ the place v by kv. We denote the maximal totally real subfield of k by k+. For an extension L/k, we write Sram(L/k) for the set of places of k that ramify in L. For any subset of places S of k, we denote by SL the set of places in L above those in S and OL,S for the ring of SL-integers in L. For any place w of L above v of k, we denote its residue field by κ(w), set Nw := |κ(w)| and identify the decomposition subgroup of w in Gal(L/k) with Gal(Lw/kv) in the usual way. ^ For an abelian group G, we write G for the set of irreducible complex (linear) characters of G of finite order. If G is finite, then for each χ in G we define an idempotent of C[G] by setting eχ := |G|—1 Σσ∈G χ(σ)σ—1. Chapter 2 Preliminaries
