A cohomological reformulation Sample Clauses

A cohomological reformulation. We will now explain how to reformulate the weight k(ρ) in Serre’s conjecture above in terms of Galois cohomology. This point of view appears first in the work of Xxx and Xxxxxxx [AS86b]. Suppose ρ : Gal(Q/Q) → GL2(Fp) is odd and irreducible as above. Let us also fix N to be the Artin conductor of ρ away from p. In particular, N is coprime to p. For any k ě 2, we denote the (k — 2)th symmetric power of the standard representation of GL (F ) k´2F2. Hence, 1 Fp via its reduction to a subgroup Γ (N ) Ď SL (Z) also acts on Symk´2 p on Fp by Sym p of SL2(Fp). Recall that we have the Hecke algebra of Γ1(N ), defined in terms of double coset operators, acting on the group cohomology groups H1(Γ (N ), Symk´2F2). We will say that ρ occurs in H1(Γ (N ), Symk´2F2) p 1 p 1 p if we can find an eigenvector for the Hecke operators in this vector space with eigenvalues Tr ρ(Frobl) for all l - pN. We can then use the Xxxxxxx–Shimura isomorphism to prove that ρ is modular of weight k and level N (in the sense of Conjecture 2.2.2) if and only if ρ occurs in H1(Γ (N ), Symk´2F2). Since we do not know of a good reference for this classical result, let us give a proof here. p