On the Bloch-Kato conjecture for adjoint motives of modular forms

Report on work of F. Diamond, M. Flach and L. Guo

November 17, 1998

Suppose f is a newform of weight k ≥ 2, level N and character ψ^{−1} with coefficients in the ring of integers O of a number field K. For ∗ = B, dr or λ (a prime of K), we let M_{∗} denote the corresponding realization of the motive attached to f. This is a two-dimensional K_{∗}-subspace (where K_{B} = K_{dr} = K) of the corresponding cohomology group of the modular curve X_{1}(N) with coefficients in a sheaf depending on k. We have certain additional structure on each M_{∗} (an involution, filtration or G_{Q}-action) and comparison isomorphisms relating certain pairs of realizations. Let S be a set of primes containing those dividing Nk!, and let O_{B} = O_{dr} = O_{S} denote the S-integers in K. For ∗ = B, dr or λ for λ ∉ S, we also have natural O_{∗}-lattices M_{∗} in M_{∗} and integral versions of the comparison isomorphisms involving the M_{λ}.

Let D_{∗} denote the ∗-realization of the (1−k)-Tate twist of the motive associated to ψ. Poincaré duality for X_{1}(N) gives rise to a perfect alternating pairing on M_{∗} with values in D_{∗}, i.e., an isomorphism

μ_{∗} : ∧_{K_{∗}}^{2}M_{∗} → D_{∗},

respecting the various additional structures and comparison isomorphisms. We also have natural O_{∗}-lattices D_{∗} in D_{∗}, but the map ∧_{O_{∗}}^{2}M_{∗} → D_{∗} need not be an isomorphism; the image is ηD_{∗} for a certain ideal η of O_{S}. More generally, if Σ is a finite set of primes contained in S, we analogously define M_{∗}^{Σ}, μ_{∗}^{Σ} and η^{Σ} using level structure which is, roughly speaking, minimal outside Σ.

We let A_{∗} denote the set of trace-zero elements in End_{K_{∗}}M_{∗}. So A_{∗} is threedimensional over K_{∗} and we have comparison isomorphisms, including

I_{∞} : C ⊗ A_{dr} → C ⊗ A_{B},

induced by those for the M_{∗}. Replacing f by a twist does not change this data, so we always assume f has minimal conductor among its twists. The L-function associated to A_{λ} is independent of λ; we denote it L(A, s) and regard it as taking values in K ⊗ C. We know also that L(A, s) extends to an entire function on the complex plane and satisfies the functional equation

Λ(A, s) = ±C^{1−2s}Λ(A, 1 − s),

where C = ±ϵ(A) is a positive integer dividing N and

Λ(A, s) = L(A, s)Γ_{R}(s + 1)Γ_{C}(s + k − 1).

Moreover L(A, 0) and L(A, 1) are invertible elements of R ⊗ K.

We show that the conjecture of Bloch and Kato correctly predicts the values of L(A, 0) and L(A, 1) up to an element of O_{S}^{×}, where S is a certain finite set of primes containing those dividing Nk!. We use the formulation of the conjecture given by Fontaine and Perrin-Riou, which is easily generalized to the setting of K-coefficients.

The fundamental line for A is the K-line defined by

Δ = Hom_{K}(A_{B}^{+}, A_{dr}/Fil^{0}A_{dr})
≅ Hom_{K}(Fil^{k−1}M_{dr}, M_{dr}/Fil^{k−1}M_{dr}).

The Deligne period for A is the natural basis c^{+} for Δ ⊗ R gotten from I_{∞}. One finds that c^{+}L(A, 0)^{−1} is an element δ ∈ Δ characterized by

μ_{dr}(f ∧ δ(f)) = \frac{i^{k−2−η}(k − 2)!ϵ(M) ∏_{p∈P} (1 + p^{−1})}{2ϵ(D)ϵ(A)} · G_{dr}, (1)

where η ∈ {0, 1} has the same parity as k, P is a certain exceptional set of primes dividing N and G_{dr} (for Gauss sum) is the usual basis for D_{dr}. This is proved following Rankin, Shimura and Hida to get a similar formula involving L(A, 1) and then applying the functional equation.

To state the λ-part of the Bloch-Kato conjecture, one chooses a Galois-stable O_{λ}-lattice A_{λ} in A_{λ} ≅ A_{B} ⊗_{K} K_{λ} and an O_{λ}-lattice ω in (A_{dr}/Fil^{0}A_{dr}) ⊗_{K} K_{λ}. The conjecture, which turns out to be independent of these choices, states that

• H_{f}^{1}(G_{Q}, A_{λ}/A_{λ}) is finite;

• δ generates the O_{λ}-submodule

\frac{Fitt_{O_{λ}}H^{0}(G_{Q}, A_{λ}/A_{λ}) · Fitt_{O_{λ}}H^{0}(G_{Q}, A_{λ}^{∗}/A_{λ}^{∗})}{Tam_{ω}^{0}(A_{λ}) · Fitt_{O_{λ}}H_{f}^{1}(G_{Q}, A_{λ}/A_{λ})} · Hom_{O_{λ}}(A_{λ}^{+}, ω)

of Δ ⊗_{K} K_{λ}, where A_{λ}^{∗} = Hom_{Z_{ℓ}}(A_{λ}, Z_{ℓ}(1)), A_{λ}^{∗} = A_{λ}^{∗} ⊗ Q and Tam_{ω}^{0}(A_{λ}) is the Tamagawa ideal of A_{λ} relative to ω.

The two main problems are to compute the local contribution at ℓ to the Tamagawa ideal, Tam_{ℓ,ω}(A_{λ}), and the length of H_{f}^{1}(G_{Q}, A_{λ}/A_{λ}). We restrict our attention to primes λ not in S, and let A_{∗} denote the set of elements of End_{O_{∗}}(M_{∗}) of trace zero. One can then use existing machinery to show that Tam_{ℓ,ω}(A_{λ}) = O_{λ}, where ω = (A_{dr}/Fil^{0}A_{dr}) ⊗_{O_{dr}} O_{λ}. The proof requires only that λ not divide Nk! and uses the integral version of Faltings’ comparison theorem. For primes between k and 2k, the argument is slightly delicate since the filtration length of A_{dr} is greater than ℓ − 2.

The computation of the H_{f}^{1} relies on the methods of Wiles and Taylor-Wiles, but now we have to impose another hypothesis on λ:

(2) ρ̄ : G_{Q} → Aut_{O_{λ}/λ}(M_{λ}/λM_{λ}) has absolutely irreducible restriction to G_{L}, where L = Q(\sqrt{(−1)^{(ℓ−1)/2}ℓ}).

We give an axiomatic formulation of their method of the following nature: Suppose

ℜ = {ρ : G_{Q} → Aut_{K_{ρ}}V_{ρ}}

is a set of liftings of ρ̄, where for each ρ in ℜ, K_{ρ} is a finite extension of K_{λ} in K̄_{λ} and the restriction of ρ to G_{Q_{ℓ}} is crystalline of Hodge-Tate type (0, k − 1). For any finite set of primes Σ not containing ℓ, let ℜ^{Σ} denote the set of ρ ∈ ℜ such that ρ is minimally ramified outside Σ. Suppose that for each ρ ∈ ℜ^{Σ} we are given an isomorphism

μ_{ρ}^{Σ} : ∧_{K_{ρ}}^{2}V_{ρ} → D_{λ} ⊗_{K_{λ}} K_{ρ}.

We assume that for fixed ρ and varying Σ, the ratios of the μ_{ρ}^{Σ} are determined by Euler factors. We assume also that ⊕_{ρ∈ℜ}(V_{ρ} ⊗_{K_{ρ}} K̄_{λ}) contains a lattice V having a certain self-duality property with respect to the pairings. We show that if the numbers #ℜ^{Σ} satisfy a certain numerical criterion, then

• every lifting of ρ̄ which is crystalline of Hodge-Tate type (0, k−1) is isomorphic to an element of ℜ; • if ρ ∈ ℜ^{Σ} and K_{ρ} = K_{λ}, then

Fitt_{O_{λ}}H_{Σ}^{1}(G_{Q}, W_{ρ} ⊗_{O_{λ}} (K_{λ}/O_{λ})) = Fitt_{O_{λ}}D_{λ}/μ_{ρ}^{Σ}(∧_{O_{λ}}^{2}V_{ρ}),

where V_{ρ} = V ∩ V_{ρ} and W_{ρ} is the set of elements of End_{O_{λ}}V_{ρ} of trace zero. We then verify that the set ℜ of modular liftings of ρ̄ satisfies these hypothe-

ses. We thus obtain the following result in the direction of the Fontaine-Mazur conjecture:

Theorem 1. Suppose ρ : G_{Q} → Aut_{K_{λ}}V_{λ} is continuous, two-dimensional, unramified outside a finite set of primes and has crystalline restriction to G_{Q_{ℓ}} of Hodge-Tate type (0, d) with 0 < d < ℓ − 1. If ρ̄ is modular and has absolutely irreducible restriction to G_{L}, then ρ is modular.

We also find that Fitt_{O_{λ}}H_{Σ}^{1}(G_{Q}, A_{λ}/A_{λ}) = η^{Σ}O_{λ}. We can then compute the length of H_{f}^{1}(G_{Q}, A_{λ}/A_{λ}) as well as that of H_{f}^{1}(G_{Q}, A_{λ}^{∗}/A_{λ}^{∗}). Note that under our hypotheses H^{0}(G_{Q}, A_{λ}/A_{λ}) and H^{0}(G_{Q}, A_{λ}^{∗}/A_{λ}^{∗}) both vanish. Applying equation (1) and its analogue for L(A, 1), we conclude:

Theorem 2. Let f be a newform of weight k ≥ 2 and level N with coefficients in a number field K. Let S denote the set of primes λ such that λ divides Nk! or (2) fails. For λ not in S, the λ-part of the Bloch-Kato conjecture holds for L(A, 0) and L(A, 1).

Finally we remark that the set S is finite. In fact, if λ does not divide Nk! and (2) fails, then either ℓ = 2k − 1 or M_{λ}/λM_{λ} defines a reducible representation of G_{Q}, and reducibility for infinitely many λ would violate the Ramanujan conjecture.