CHAPTER 15 Lecture 15

1. The Main Theorem

1.1. Statement of the Main Theorem. We briefly recall our running hypotheses. Fix a prime p ≠ 2. Let F be a number field with absolute Galois group G_{F} and let G_{F,Σ} be the maximal quotient of G_{F} unramified outside of a finite set Σ of places of F (containing p and the archimedean places). Let A be a local, finite, faithfully flat ℤ_{p}-algebra with maximal ideal 𝔪 and residue field k. Let W^{∗} be a free A-module of finite rank (at least 2) with a continuous A-linear action of G_{F,Σ}. Let W = Hom(W^{∗}, μ_{p^{∞}}) be its Cartier dual. We let

ρ : G_{F,Σ} → Aut_{A}(W)

be the representation of G_{F,Σ} on W, and for any ideal J of A of finite index we let

ρ_{J} : G_{F,Σ} → Aut_{A/J}(W [J])

be the restriction of ρ to the submodule

W [J] = ∩_{α∈J}W [α].

We make three hypotheses on this data. First, we assume that \overline{ρ} is absolutely irreducible. This implies that W^{G} = 0, by Corollary 3.7 of Lecture 13. (Here we need the fact that W has rank at least 2.) Second, for any ideal J, let F_{J} be the splitting field of ρ_{J}; it is a finite extension of F. Set Δ_{J} = Gal(F_{J}/F). We assume that

H^{1}(Δ_{J}, W [J]) = 0

for all J. Lastly, we suppose that there is a non-scalar involution τ in the image of

\overline{ρ} : G_{F,Σ} → Aut_{k}(W [𝔪]);

we can lift this τ to Aut_{A/J}(W [J]) for any ideal J ⊆ 𝔪 of A of finite index, since p is odd. We fix such a τ.

Theorem 1.1 (Kolyvagin). With the above notation and hypotheses, suppose in addition that there is a Kolyvagin-Flach system of depth I for W^{∗}/JW^{∗} for L_{τ}. Then

H_{f}^{1}(G_{F} , W)[I] = H_{f}^{1}(G_{F} , W)[J].

If one does not assume that H^{1}(Δ_{J}, W [J]) = 0, then one finds that the product of I and any annihilator of H^{1}(Δ_{J}, W [J]) annihilates H_{f}^{1}(G_{F} , W)[J]. This will be clear from the proof below.

We will prove this theorem in two stages. Recall that for any place v of F there is a natural restriction map

res_{v} : H^{1}(G_{F} , W [J]) → H^{1}(G_{v}, W [J]).

0Last modified December 17, 1999

This map sends H_{f}^{1}(G_{F} , W [J]) to H_{f}^{1}(G_{v}, W [J]), so we also obtain a natural map

res_{s,v} : H_{s}^{1}(G_{F} , W [J]) → H_{s}^{1}(G_{v}, W [J]).

Lemma 1.2. With the hypotheses of Theorem 1.1, if x ∈ H_{f}^{1}(G_{F} , W [J]) and α ∈ I, then

res_{v}(αx) = 0

for all v ∈ L_{τ}.

Lemma 1.3. Continue to suppose the hypotheses of Theorem 1.1. Let X ∈ H_{f}^{1}(G_{F} , W [J]) be such that res_{v}(X) = 0 for all v ∈ L_{τ}. Then X = 0.

It is clear that Lemma 1.2 and Lemma 1.3 suffice to prove the theorem. We note also that the proof of Lemma 1.3 will not require the existence of the Kolyvagin-Flach system.

As a corollary, we obtain the following:

Corollary 1.4. In addition to the hypotheses of Theorem 1.1, suppose that 𝔪I ⊇ J. Then

H_{f}^{1}(G_{F} , W)[I] = H_{f}^{1}(G_{F} , W).

Proof. Everything in H_{f}^{1}(G_{F} , W) is killed by some power of 𝔪, so we prove the corollary for each H_{f}^{1}(G_{F} , W)[𝔪^{m}] by induction on m. The result is clear for m = 1, since J ⊆ 𝔪. So suppose that we know the result for m < n. Take x ∈ H_{f}^{1}(G_{F} , W)[𝔪^{n}]. Then 𝔪^{n−1}(𝔪x) = 0, so by the induction hypothesis we have I(𝔪x) = 0. So, since I𝔪 ⊇ J, we have Jx = 0. Now, by Theorem 1.1, we have Ix = 0. This complete the induction.

1.2. Proof of Lemma 1.2. Fix v ∈ L_{τ}. Since W [J] and W^{∗}/JW^{∗} are Cartier dual, Tate local duality (Lecture 6, Theorem 2.2) yields a perfect cup product pairing

H_{f}^{1}(G_{v}, W [J]) ⊗ H_{s}^{1}(G_{v}, W^{∗}/JW^{∗}) → ℚ/ℤ.

Since v ∈ L_{τ}, we have a set C^{(v)} of Kolyvagin-Flach classes such that res_{s,v′}(c) = 0 for c ∈ C^{(v)} and v ≠ v′, and if C′ is the A-submodule of H_{s}^{1}(G_{v}, W^{∗}/JW^{∗}) generated by C^{(v)}, then

H_{s}^{1}(G_{v}, W^{∗}/JW^{∗})/C′

is annihilated by I.

Next recall that under our global pairing (see Lecture 8, Section 2)

H_{f}^{1}(G_{F} , W [J]) ⊗ ⊕_{v}H_{s}^{1}(G_{v}, W^{∗}/JW^{∗}) → ℚ/ℤ,

H_{f}^{1}(G_{F} , W [J]) and H_{s}^{1}(G_{F} , W^{∗}/JW^{∗}) are orthogonal. Since res_{s,v′}(c) = 0 for c ∈ C^{(v)} and v′ ≠ v, the definition of the pairing and the above orgthogonality imply that

res_{v}(x) ∪ res_{s,v}(c) = 0

for all c ∈ C^{(v)}. Thus, res_{v}(x) is contained in the orthogonal complement of C′ in H_{f}^{1}(G_{v}, W [J]) under Tate local duality. Since Tate local duality is a perfect pairing and I annihilates

H_{s}^{1}(G_{v}, W^{∗}/JW^{∗})/C′,

it follows immediately that I annihilates res_{v}(x) ∈ H_{f}^{1}(G_{v}, W [J]), as desired.

1.3. Proof of Lemma 1.3. Consider the exact sequence

1 → G_{F_{J}} → G_{F} → Δ_{J} → 1.

This yields an inflation-restriction sequence

0 → H^{1}(Δ_{J}, W [J]) → H^{1}(G_{F} , W [J]) → Hom_{Δ_{J}} (G_{F_{J}}^{ab} , W [J]).

(See Lecture 14, Section 1.2 for a discussion of the Δ_{J} action on G_{F_{J}}^{ab} .) By hypothesis, H^{1}(Δ_{J}, W [J]) = 0, so it will be enough to show that the image

φ : G_{F_{J}}^{ab} → W [J]

of X in Hom_{Δ_{J}} (G_{F_{J}}^{ab} , W [J]) is trivial.

Let L be the fixed field of ker φ, and set Γ = Gal(L/F_{J}).

\xymatrix{G_{F_{J}}^{ab} \ar[rr]^{φ} \ar@{->>}[dr] &  & W [J] \\  & Γ \ar@{^{(}->}[ur]^{φ}}

There is an exact sequence

1 → Γ → Gal(L/F) → Δ_{J} → 1.

Since Γ injects into W [J], it is a finite abelian p-group.

Let τ be our fixed non-scalar involution in Δ_{J}. Since Γ has odd order we can lift τ to a non-scalar involution τ̃ ∈ Gal(L/F). Let g be any element of Γ and consider τ̃g ∈ Gal(L/F). By the Tchebatorev density theorem ([La-ANT, Chapter 8, Theorem 10]) there exists an unramified place w̃ of L such that

Fr_{L/F} (w̃) = τ̃g.

Set w = w̃|_{F_{J}} and v = w|_{F}.

[figure: L; Γ; Gal(L/F); w̃; F_{J}; Δ_{J}; w; F; v]

By standard properties of Frobenius, we have

Fr_{F_{J} /F} (w) = Fr_{L/F} (w̃)|_{F_{J}} = τ̃g|_{F_{J}} = τ

(so in particular v ∈ L_{τ}), and

Fr_{L/F_{J}} (w̃) = (Fr_{L/F} (w̃))^{deg(w/v)} = (τ̃g)^{2}.

Here by deg(w/v) we mean the degree of the local field extension F_{J,w}/F_{v}; it is 2 since τ has order 2 and w/v is unramified.

We claim that since res_{v}(x) = 0, we have

φ(Fr_{L/F_{J}} (w̃)) = 0.

To see this, begin with the commutative diagram

\xymatrix{X ∈ H^{1}(G_{F} , W [J]) \ar[r]^{^{res^{v}}} \ar[d] & H^{1}(G_{v}, W [J]) \ar[d] \\ φ ∈ Hom(G_{F_{J}} , W [J]) \ar[r] & Hom(G_{F_{J} ,w}, W [J])}

Since φ factors through Γ = Gal(L/F_{J}), φ|_{G_{F_{J} ,w}} factors through Gal(Lw̃/F_{J,w}), which is generated by Fr_{L/F_{J}} (w̃) since w̃/w is unramified. But res_{v}(X) = 0, so φ|_{G_{F_{J} ,w}} = 0; thus φ(Fr_{L/F_{J}} (w̃)) = 0 as claimed.

Combining all of this we have

φ(τ̃gτ̃g) = 0.

But τ̃^{2} = 1, so τ̃gτ̃g = τ̃gτ̃^{−1}g. By the definition of the Δ_{J}-action on Γ, this is none other than ^{τ} g · g. Thus

φ(^{τ} g) + φ(g) = 0,

so by Δ_{J}-equivariance

τφ(g) = −φ(g) ∈ W [J].

This is true for all g ∈ Γ, so if we let H be the A-submodule of W [J] generated by φ(Γ), we have

H ⊆ W [J]^{−}.

Here W [J]^{−} is the minus eigenspace for τ acting on W [J]:

W [J]^{−} = {z ∈ W [J] | τz = −z}.

Note that W [J]^{−} ≠ W [J] since τ is non-scalar. Since φ is Δ_{J}-equivariant and Δ_{J} acts A-linearly, H is Δ_{J}-stable. But if we consider H[𝔪] ⊆ W [𝔪]^{−}, we must have H[𝔪] = 0, since W [𝔪] is irreducible as a G_{F,Σ}-representation space and W [𝔪]^{−} ≠ W [𝔪]. But if H[𝔪] = 0, then H = 0, by Corollary 3.7 of Lecture 13. Thus φ = 0 as desired.

2. The Main Theorem for Rank 1 Modules

The end of the proof of Lemma 1.3, and thus the proof of Theorem 1.1, relied very heavily on the fact that W^{G} = 0. Thus need not be true in the case that W has rank 1, even if \overline{ρ} is absolutely irreducible. In fact, the version of the theorem which is useful for rank 1 A-modules is somewhat simpler than the main theorem; essentially, one is able to take τ = 1. We state the result for all W even though it is only useful in practice for W of rank 1.

We keep all of the definitions of the first paragraph of Section 15.1, except that we now allow W to have rank 1. We need not assume that \overline{ρ} is absolutely irreducible or that there is a non-scalar involution τ. The only extra hypothesis we need is the Δ-hypothesis

H^{1}(Δ_{J}, W [J]) = 0.

We let L = L_{1} be the set of unramified places of F with trivial Frobenius in F_{J}.

Theorem 2.1. With the above notation and hypotheses, suppose in addition that there is a Kolyvagin-Flach system of depth I for W^{∗}/JW^{∗} for L. Then

H_{f}^{1}(G_{F} , W)[I] = H_{f}^{1}(G_{F} , W)[J].

Proof. The proof is actually somewhat simpler than the proof of Theorem 1.1. Lemma 1.2 goes through with no changes; Lemma 1.3 in fact simplifies greatly, as one finds that

Fr_{L/F_{J}} (w̃) = g,

and thus φ(g) = 0 immediately by Lemma 1.2.

This theorem is only useful in the case where W has rank 1; in the higher rank cases there are no Kolyvagin-Flach systems for trivial τ.