The reducible case Clause Samples

The reducible case. We are ready to provide a generalised version of one of the main results in [GLS14]. Theorem 2.17 (Generalised Theorem 9.1). Suppose p > 2. Let ρ˜ : GK → GL2(Zp) be a continuous representation such that its reduction ρ : GK → GL2(Fp) is reducible. Suppose that ρ˜ is crystalline with κ-▇▇▇▇▇–▇▇▇▇ weights {bκ,1, bκ,2} for each κ ∈ HomQp (K, Qp), and suppose further that 0 ≤ bκ,1 — bκ,2 ≤ p for each κ. Then there is a reducible crystalline representation ρ˜′ : GK → GL2(Zp) with the same κ-▇▇▇▇▇–▇▇▇▇ weights as ρ˜ for each κ such that ρ ~= ρ′, where ρ′ is the reduction of ρ˜′. The proof is almost identical to the proof of [GLS14, Theorem 9.1] using our generalised results and our adjusted notions of types of Kisin modules in Definition 2.9 and the set Jmax from Definition 2.12. We give it in full as we will refer back to it later. 0 ψ2 2 IK