HECKE FIELDS OF ANALYTIC FAMILIES OF MODULAR FORMS

摘要

Fix a prime p, and put p = 4 if p = 2 and p = p otherwise. For a Hecke eigenform f ∈ S_k(Γ_0(Np~rp),ψ) (p|N,r ≥ 0) and a subfield K of C, the Hecke field K(f) inside C is generated over K by the eigenvalues a_n = a(n, f) of f for the Hecke operators T(n) for all n. Then Q(f) is a finite extension of Q sitting inside the algebraic closure Q in C. Let F = 1 + pZ_p, which is a maximal torsion-free subgroup of Z~x_p . We choose and fix a generator γ := (1 + p) ∈ Γ so that Γ = γ~(Z_p) 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 Z_p). A p-adic slope 0 analytic family of eigenforms F = {f_P|P ∈ Spec(II)(C_p)} is indexed by points of Spec(II)(C_p), where Spec(II) is a finite fiat irreducible covering of Spec(Λ). For each P ∈ Spec(II), 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) is a p-adic analytic function on Spec(II). We call P arithmetic of weight k = k(P) ∈ Z with character εp : Γ →μ_p∞ (C_p) if P contains (1 + x — ε_P(γ)γ~k) ∈ Λ and k(P) ≥ 2. If P is arithmetic, fp is known to be a p-stabilized classical Hecke eigenform and has Neben character ψ_P whose restriction to Γ is given by ε_P. We write p~(r(P)) for the order of ε_P (then, the classical form f_P has level Np~(r(P))p). In order to make the introduction succinct, we put off, to Section 1, recalling the theory of analytic families of eigenforms including the definition and necessary properties of CM families. We define the following Hecke fields out of (V) For a fixed level Np~rp (0 ≤ r ≤ ∞), Q_(V,r)(F) is the composite of Q(f_P) for all arithmetic P ∈ Spec(II) (O_p) with k(P)≥ 2 and e p factoring through (H) For a fixed weight k≥ 2, Q_(H,k) (F) is the composite of Q(f_P) for all arithmetic P ∈ Spec(II)(Q_p) with k(P) = k. Here the composite is taken in the algebraic closure Q inside C. If r = 0, periodically in k(P), fp is old at p associated to a unique new form f°_P of level N prime to p; so, putting F° = {f°_P} for such arithmetic Ps, we can also define Q_V(F°) as the composite of Q(f°_P). Abusing the notation, we put Q_(V,-1_(F) := Q_V(F°).