Serre weights Sample Clauses

Serre weights. It turns out that, instead of considering the rep- resentations Symk´2 Fp, it is possible to pass to Jordan–Ho¨lder factors, which will lead us naturally to the definition of a Serre weight. Let us describe how to do this now. Suppose that ρ is cohomologically modular of weight k ě 2 and level N > 4, where N is the Artin conductor away from p. Since Γ := Γ1(N ) is sufficiently small, the quotient map H → H/Γ gives a universal cover allowing us to relate group cohomology of Γ to the cohomol- ogy of the modular curve Y := H/Γ. For example, it is explained in [DI95, p. 108] that for any subquotient A of Symk´2F2 we can identify H2(Γ, A) with H2(Y, A) for a locally constant sheaf of sections A constructed from c A. Then its follows from Poincar´e duality that H2(Y, A) – H0(Y, A), which vanishes by the non-compactness of the modular curve Y . So H2(Γ, A) = 0 for any subquotient A of Symk´2