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Fix a prime p, and put p =4ifp=2andp = p otherwise. For a Hecke eigenform f ∈ Sk(Γ0(Nprp),ψ)(p ∤ N,r ≥ 0) and a subfield K of C, theHecke field K(f) inside C is generated over K by the eigenvalues an = a(n, f) off for the Hecke operators T (n) for all n. ThenQ(f) is a finite extension of Q sitting inside the algebraic closure Q in C. Let Γ = 1 + pZp, which is a maximal torsion-free subgroup of Z × p. We choose and fix a generator γ: = (1 + p) ∈ ΓsothatΓ=γZp and identify the Iwasawa algebra Λ = W [[Γ]] with the power series ring W [[x]] by Γ ∋ γ ↦ → (1 + x) (for a discrete valuation ring W finite flat over Zp). A p-adic slope 0 analytic family of eigenforms F = {fP |P ∈ Spec(I)(Cp)} is indexed by points of Spec(I)(Cp), where Spec(I) is a finite flat irreducible covering of Spec(Λ). For each P ∈ Spec(I), fP is a p-adic modular form of slope 0 of level Np ∞ for a fixed prime to p-level N. The family is called analytic because P ↦ → a(n, fP)isap-adic analytic function on Spec(I). We call P arithmetic of weight k = k(P) ∈ Z with character εP:Γ → μp∞(Cp) ifP contains (1 + x − εP (γ)γk) ∈ Λandk(P) ≥ 2. If P is arithmetic, fP is known to be a p-stabilized classical Hecke eigenform and r(P

Year: 2011

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