The Buzzard–Diamond–Xxxxxx recipe Sample Clauses

The Buzzard–Diamond–Xxxxxx recipe. We define a set W (ρ) con- sisting of all Serre weights such that ρ is conjectured to be modular with respect to those weights in the sense of [BDJ10, Definition 2.1]. We first define Wv(ρ) for each v dividing p depending on ρ|GFv where Fv is the completion of F at v. The set W (ρ) then consists of v|p Vv with Vv ∈ Wv(ρ). We distinguish between irreducible and reducible ρ|GFv . 1. ALGEBRAIC MODULARITY AND THE BUZZARD–DIAMOND–XXXXXX CONJECTURE 33 1.2.1. Weights in the irreducible case. Let Fv′ be the unramified quadratic ex- tension of Fv in Fv. We let F′v denote the residue field and Σv′ denote the set of embeddings F′v → Fp and define a map π : F′v → Fv by τ′ '→ τ′|Fv . For τ′ ∈ Σv′ , we let ωτ' denote the corresponding fundamental character of IFv' = IFv . Now since ρ|GFv is irreducible it is tamely ramified and in fact induced ! from a character of GF'v (see e.g. [Bar20, Lemma 2.1.2]). We define Wv(ρ) by the rule V → ∈ Wv(ρ) ⇐⇒ ρ|I ~ Y ωaτ τ∈Σv Qτ'∈J' bπ(τ') τ' π(τ') τ'/∈J' Fv for some J′ ⊂ Σ′v such that π : J′ —∼→ Σv. 1.2.2. Weights in the reducible case. Suppose we have ρ|GFv ~= ψ1 ∗ ! , 0 ψ for some characters ψ1, ψ2 : GFv → Fp×. Then we first define a set IFv IFv W′(ψ1, ψ2) = {(V →, J) | J ⊂ Σv, ψ1| = Y ωaτ Y ωbτ , ψ2| = Y ωaτ Y ωbτ }. τ∈Σv τ∈J τ∈Σv τ/∈J Following [DDR16] we give a bit of background before we define a certain distinguished subspace of H1(GF , Fp(ψ)). To ease notation, we write K = Fv and write ρK instead of ρGFv . The representation ρK gives a short exact sequence F [G ]p K 0 → Fp(ψ1) → Tρ → Fp(ψ2) → 0, which defines a class cρ in Ext Fp[GK ] (Fp(ψ2), Fp(ψ1)) =~ Ext1 (Fp, Fp(ψ)) ~= H1(GK, Fp(ψ)),