A cohomological subspace Sample Clauses

A cohomological subspace. In this section we define a cohomological sub- space consisting of elements such that the Galois representation associated to the extension class has a crystalline lift of a certain shape. We will then study this space as we will use its properties later. Our definition of the space differs slightly from Definition 9.2 in [GLS14], as we allow for an extended range of Xxxxx–Xxxx weights. Let ρ˜ : GK → GL2(Zp) be a continuous representation such that ρ : GK → GL2(Fp) is reducible. Suppose that ρ˜ is crystalline with κ-Xxxxx–Xxxx weights {bκ,1, bκ,2} for each κ ∈ HomQp (K, Qp), and suppose further that 0 ≤ bκ,1 —bκ,2 ≤ p !~ for each κ. Write ρ = ψ1 ∗ , by Corollary 2.7 we find that there is a decomposition 0 ψ2 p K κ κ 2 K κ κ HomQ (K, Qp) = J Q Jc such that ψ |I = Y ωbκ,1 Y ωbκ,2 , ψ |I = Y ωbκ,1 Y ωbκ,2 , p κ∈J κ∈Jc κ∈Jc κ∈J there may be several such J but we temporarily fix one choice. Let ψ1, ψ2 : GK → Z× be crystalline lifts of ψ1, ψ2, with the properties that HTκ