CHAPTER 16 Lecture 16

1. Other Versions of the Main Theorem

1.1. Characters. We briefly review the theory of eigenspaces of group modules; since it will be sufficient for our applications we restrict attention to the simplest case. So let p be a fixed prime number and let G be an abelian group of order dividing p − 1. We will write Ĝ for the group of characters of G. Since G is of order dividing p − 1, the image of any character of G lies in μ_{p−1}. Thus, in this set-up, we can identify Ĝ with Hom(G, ℤ_{p}^{∗}).

Let ε be a character of G and define the ε-idempotent of ℤ_{p}[G] by

e_{ε} = \frac{1}{|G|} ∑_{g∈G} ε^{−1}(g)g ∈ ℤ_{p}[G].

One checks easily that

e_{ε}e_{ε′} = \begin{cases}e_{ε} & ε = ε′ \\ 0 & ε ≠ ε′. \end{cases}

Thus the e_{ε} form a system of idempotents of ℤ_{p}[G], yielding a decomposition

ℤ_{p}[G] = ⨁_{ε∈Ĝ} ℤ_{p}[G]e_{ε}.

Since each ℤ_{p}[G]e_{ε} is non-zero (it contains e_{ε}), it follows from the fact that Ĝ has the same order as G that each ℤ_{p}[G]e_{ε} is free of rank 1 over ℤ_{p}. One also notes that for all g ∈ G,

ge_{ε} = ε(g)e_{ε},

so that G acts on ℤ_{p}[G]e_{ε} via ε.

Now let M be a G-module which is a finite p-group. Then M can be viewed as a ℤ_{p}[G]-module, since each element of M is killed by some power of p. The above decomposition of ℤ_{p}[G] yields a decomposition

M = ⨁_{ε∈Ĝ} Me_{ε}.

In fact, one shows easily from the fact that ge_{ε} = ε(g)e_{ε} that Me_{ε} can also be realized as

Me_{ε} = {m ∈ M | gm = ε(g)m for all g ∈ G}.

We will also write M^{ε} for Me_{ε}, and call it the ε-eigenspace of M.

In the case that G has order 2, we will write M^{+} and M^{−} for the eigenspace of the trivial character and the eigenspace of the non-trivial character respectively.

0Last modified April 9, 1998

1.2. Galois Groups Acting on Cohomology. We now return to the notation of the previous lecture, although we do not yet assume any hypotheses on W. Let us suppose that we are also given a subfield F_{0} of F of index 2, so that F/F_{0} is a quadratic extension. Set

G = Gal(F/F_{0}).

We assume that the G_{F} -action on W is the restriction of a continuous, A-linear action of G_{F_{0}} on W.

Let J be an ideal of A of finite index. We claim that H^{1}(G_{F} , W [J]) has the structure of a G-module. In fact, it has the structure of G_{F_{0}}-modules, where G_{F_{0}} acts on G_{F} by inverse conjugation and on W [J] by the given action. Further, the action of G_{F} on H^{1}(G_{F} , W [J]) in this way is trivial ([Se-LF, Chapter 7, Section 5, Proposition 3]), so that the G_{F_{0}}-action factors through G, as claimed. In the same way we can consider H^{1}(G_{F} , W^{∗}/JW^{∗}) as a G-module. Given this, the discussion of the previous section yields eigenspace decompositions

H^{1}(G_{F} , W [J]) = H^{1}(G_{F} , W [J])^{+} ⊕ H^{1}(G_{F} , W [J])^{−}

and

H^{1}(G_{F} , W^{∗}/JW^{∗}) = H^{1}(G_{F} , W^{∗}/JW^{∗}) ⊕ H^{1}(G_{F} , W^{∗}/JW^{∗})^{−}.

Unfortunately, H_{f}^{1}(G_{F} , W [J]) is not necessarily a G-module. In order to give conditions under which it is, we first must consider the G-action induced on the local cohomology groups. So let v be a place of F. Then the action of τ ∈ G yields a map

H^{1}(G_{τ^{−1}v}, W [J]) = H^{1}(τ^{−1}G_{v}τ, W [J]) → H^{1}(G_{v}, W [J]).

Thus, if v|_{F_{0}} is inert or ramified in F/F_{0}, G does act on H^{1}(G_{v}, W [J]). However, if v|_{F_{0}} splits over F into v and v′, then G acts on the direct sum

H^{1}(G_{v}, W [J]) ⊕ H^{1}(G_{v′}, W [J])

by interchanging the factors. In any event, there is a natural G-action on

⨁_{v} H^{1}(G_{v}, W [J])

and the natural map

res : H^{1}(G_{F} , W [J]) → ∏_{v} H^{1}(G_{v}, W [J])

is G-equivariant, essentially by definition of the action on the local factors.

We make the assumption for the remainder of this section that the G-action respects the finite/singular structure, in the sense that

⨁_{v} H_{f}^{1}(G_{v}, W [J])

is G-stable. The next lemma shows that this is a reasonable condition; the proof is straightforward and we omit it.

Lemma 1.1. Let M be a G_{F_{0}}-module. Let v_{0} be a place of F_{0} which is unramified in F and let v be a place of F above v_{0}. Then H^{1}(𝔤_{v}, M^{I_{v}}) is G-stable. (In the case that v_{0} is inert the meaning of this is clear; if v_{0} splits into v and v′, this means that the G-action interchanges H^{1}(G_{v}, M^{I_{v}}) and H^{1}(G_{v′}, M^{I_{v′}}).)

Now, with this assumption on our finite/singular structures,

⨁_{v} H_{s}^{1}(G_{v}, W [J])

is also a G-module, and G-equivariance of

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

implies that H_{f}^{1}(G_{F} , W [J]) is a G-module as well. In the same way one can impose conditions on the finite/singular structure on W^{∗}/JW^{∗} in order to insure that H_{f}^{1}(G_{F} , W^{∗}/JW^{∗}) is a G-module; we make this assumption as well. We will show in a moment that this assumption is compatible with the usual Cartier dual finite/singular requirements.

Let v be a place of F so that V |_{F_{0}} is either inert or ramified in F_{v_{0}}. Then G acts on H^{1}(G_{v}, W [J]), and we claim that the Tate local pairing

H^{1}(G_{v}, W [J]) ⊗ H^{1}(G_{v}, W^{∗}/JW^{∗}) → ℚ_{p}/ℤ_{p}

is G-invariant. (Equivalently, it is G-equivariant when ℚ_{p}/ℤ_{p} is given the trivial Gaction.) To do this it suffices to check equivariance of cup product, Cartier duality and the invariant map. The first follows from functoriality and the second from an easy direct computation. For the last one must go back to the definition of the invariant map (see [Se-LF, Chapter 13, Section 3]) and see that each step is G-equivariant; we omit the details.

It follows immediately from this invariance and the fact that p is odd that H^{1}(G_{v}, W [J])^{ε} and H^{1}(G_{v}, W^{∗}/JW^{∗})^{−ε} pair to 0, where ε is some choice of sign. Thus Tate local duality restricts to perfect pairings

H^{1}(G_{v}, W [J])^{+} ⊗ H^{1}(G_{v}, W^{∗}/JW^{∗})^{+} → ℚ_{p}/ℤ_{p}

and

H^{1}(G_{v}, W [J])^{−} ⊗ H^{1}(G_{v}, W^{∗}/JW^{∗})^{−} → ℚ_{p}/ℤ_{p}.

These in turn restrict to perfect pairings

H_{f}^{1}(G_{v}, W [J])^{+} ⊗ H_{s}^{1}(G_{v}, W^{∗}/JW^{∗})^{+} → ℚ_{p}/ℤ_{p}

and

H_{f}^{1}(G_{v}, W [J])^{−} ⊗ H_{s}^{1}(G_{v}, W^{∗}/JW^{∗})^{−} → ℚ_{p}/ℤ_{p}.

Among other things these show that, at least when v|_{F_{0}} is inert or ramified, the various requirements on finite/singular structures are compatible. We also note that in the same way one can show that the Tate local pairing is G-equivariant for v|_{F_{0}} which split completely, in the sense that the pairing

(H^{1}(G_{v}, W [J]) ⊗ H^{1}(G_{v}, W^{∗}/JW^{∗})) ⊕ (H^{1}(G_{v′}, W [J]) ⊗ H^{1}(G_{v′}, W^{∗}/JW^{∗}))

→ ℚ_{p}/ℤ_{p} ⊕ ℚ_{p}/ℤ_{p}

is G-equivariant, where the non-trivial element of G acts on ℚ_{p}/ℤ_{p}⊕ℚ_{p}/ℤ_{p} by interchanging the factors. This again shows that the requirements on the finite/singular structures at these places are compatible.

Given the above pairings, it is now possible to define Kolyvagin-Flach systems for these eigenspaces. We let Σ be a finite set of places of F and L a (not necessarily finite) set of places of F disjoint from Σ. We also let I be some ideal of A containing J and ε some choice of sign.

Definition 8. A Kolyvagin-Flach system of classes of depth I for (W^{∗}/JW^{∗})^{ε} for L is an assignment of some subset C^{(v)} of H_{s}^{1}(G_{F} , W^{∗}/JW^{∗})^{ε} for each v ∈ L such that Supp c = {v} for all c ∈ C^{(v)} and, if C′ is the A-submodule of H_{s}^{1}(G_{v}, W^{∗}/JW^{∗})^{ε} generated by C^{(v)}, then

H_{s}^{1}(G_{v}, W^{∗}/JW^{∗})^{ε}/C′

is annihilated by I.

1.3. The Main Theorem for ± Eigenspaces. We continue with the usual notation. We also continue to assume that \overline{ρ} is absolutely unramified and that H^{1}(Δ_{J}, W [J]) = 0 for all ideals J of finite index in A. We substitute a new condition for the existence of a non-scalar involution.

Specificially, let Δ̃_{J} be the Galois group of F_{J} over F_{0}. There is an exact sequence

1 → Δ_{J} → Δ̃_{J} → G → 1

and a field diagram

[figure: F_{J}; Δ_{J}; Δ̃_{J}; F; G; F_{0}]

We make the hypothesis that there is an involution τ̃ ∈ Δ̃_{J} which projects to the non-trivial element in G and acts on W [𝔪] (and thus on W [J]) as a non-scalar. We define L̃τ̃ to be the set of places v of F such that there is a place w of F_{J} lying above v such that

Fr_{F_{J} /F_{0}}(w) = τ̃ ∈ Δ̃_{J}.

Note that any place v ∈ L̃τ̃ is inert in F/F_{0} since τ̃ projects to the non-trivial element of G = Gal(F/F_{0}).

Theorem 1.2 (Kolyvagin). Let ε be some choice of sign. With the above notation and hypotheses, suppose in addition that there is a Kolyvagin-Flach system of depth I for (W^{∗}/JW^{∗})^{ε} for L̃τ̃. Then

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

Note that the hypothesis of Theorem 1.2 are neither clearly weaker nor stronger than those of Theorem 1.1 of Lecture 15, but the conclusion is weaker. Nevertheless, this version is the appropriate one to apply in many situations.

1.4. Proof of Theorem 1.2. The proof of Theorem 1.2 is quite similar to the proof of Theorem 1.1 of Lecture 15. We will again prove it through two lemmas.

Lemma 1.3. With the hypotheses of Theorem 1.2, if x ∈ H_{f}^{1}(G_{F} , W [J])^{ε} and α ∈ I, then

res_{v}(αx) = 0

for all v ∈ L̃τ̃.

The proof of Lemma 1.3 is almost identical to that of Lemma 1.2 of Lecture 15; we leave the details to the reader.

Lemma 1.4. Continue to suppose the hypotheses of Theorem 1.2. Let X ∈ H_{f}^{1}(G_{F} , W [J])^{ε} be such that

res_{v}(X) = 0

for all v ∈ L̃τ̃. Then X = 0.

Proof. This proof is the same as that of Lemma 1.3 of Lecture 15 at many points. We again begin with the exact sequence

1 → G_{F_{J}} → G_{F} → Δ_{J} → 1,

which, by the inflation-restriction sequence and the Δ-hypothesis, yields an injection

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

In fact, this map is G-equivariant (it is just restriction), so we get an injection

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

Thus it will again 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])^{ε} is trivial.

We again let L/F be the splitting field of φ and set Γ = Gal(L/F_{J}). Thus φ is now thought of as an injection

φ : Γ ↪ W [J].

So Γ is a p-group, and as before it follows that we can lift τ̃ to an involution τ̃′ ∈ Gal(L/F_{0}). Now, choose some g ∈ Γ and let w̃ be some place of L such that

Fr_{L/F_{0}}(w̃) = τ̃′g ∈ Gal(L/F_{0}).

(w̃ exists by the Tchebatorev density theorem, as usual.) We let w, v and v_{0} be the restriction of w̃ to F_{J}, F and F_{0} respectively.

[figure: L; Γ; w̃; F_{J}; Δ_{J}; Δ̃_{J}; w; F; G; v; F_{0}; v_{0}]

We now have

Fr_{F_{J} /F_{0}}(w) = Fr_{L/F_{0}}(w̃)|_{F_{J}} = τ̃′g|_{F_{J}} = τ̃ ∈ Δ̃_{J},

so v ∈ L̃τ̃. Also,

Fr_{L/F_{J}} (w̃) = (Fr_{L/F_{0}}(w̃))^{deg(w/v_{0})} = (τ̃′g)^{2},

since τ̃ = Fr_{F_{J} /F_{0}}(w) has order 2. Furthermore, exactly as in the proof of Lemma 1.3 of Lecture 15, we have

φ(Fr_{L/F_{J}} (w̃)) = 0,

so that

φ(τ̃′gτ̃′g) = 0.

Thus,

φ(τ̃g) + φ(g) = 0.

Now, φ is Δ_{J}-equivariant, but not necessarily Δ̃_{J}-equivariant. However, it does lie in

Hom_{Δ_{J}} (G_{F_{J}}^{ab} , W [J])^{ε}.

From this it follows easily, since τ̃ projects non-trivially to G, that

φ(τ̃g) = ετ̃φ(g).

Thus φ(g) lies in the subspace of W [J] on which τ̃ acts by −ε and the A-module H generated by φ(Γ) lies in W [J]^{−ε}. (Here by W [J]^{−ε} we mean the −ε eigenspace for τ̃.) H is also Δ_{J}-stable, since φ is Δ_{J}-equivariant, so from here one proceeds as before, using the fact that τ̃ acts as a non-scalar on W [𝔪] to conclude that W [𝔪]^{−ε} ≠ W [𝔪].

1.5. A Generalization of the Main Theorem. In this section we give one last version of the main theorem. We keep the usual notation and continue to assume that \overline{ρ} is absolutely irreducible and that H^{1}(Δ_{J}, W [J]) = 0 for all ideals J of finite index in A. We no longer assume that p ≠ 2, and instead of assuming that we have a non-scalar involution τ, we simply assume that there is some τ ∈ Δ_{J} such that

W [𝔪]/(τ − 1)W [𝔪] ≠ 0.

Theorem 1.5 (Rubin). 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].

Proof. The proof is quite similar to that of Theorem 1.1 of Lecture 15. Lemma 1.2 of Lecture 15 goes through with no changes. For Lecture 15, Lemma 1.3, we proceed as follows. As usual, it suffices to show that the homomorphism φ : G_{F_{J}} → W [J] is trivial. However, now we choose also a representative cocycle c : G_{F} → W [J] for the cohomology class X. Let L be some finite extension of F through which c factors; then φ necessarily factors through Gal(L/F_{J}) = Γ. That is, we now have maps

c : Gal(L/F) → W [J]

and

φ : Γ → W [J].

Note that φ need no longer be injective on Γ.

Choose any g ∈ Γ and some lifting τ̃ of τ to Gal(L/F). By the Tchebatorev density theorem we can find some place w̃ of L such that

Fr_{L/F} (w̃) = τ̃g.

Let w and v be the restriction of w̃ to F_{J} and F respectively. As usual we have

Fr_{F_{J} /F} (w) = τ,

so that v ∈ L_{τ}.

Now, c|_{Gal(Lw̃/F_{v})} is a coboundary, since res_{v}(X) = 0. Thus

c(Fr_{L/F} (w̃)) ∈ (Fr_{L/F} (w̃) − 1)W [J];

that is,

c(τ̃g) ∈ (τ̃g − 1)W [J] = (τ − 1)W [J],

since τ̃ acts on W [J] as τ and g acts trivially on W [J]. Since c is a cocycle,

c(τ̃g) = c(τ̃) + τ̃c(g) = c(τ̃) + τc(g) ∈ (τ − 1)W [J].

Furthermore, this is true for all g ∈ Γ. Taking g = 1 we find that

c(τ̃) ∈ (τ − 1)W [J].

Thus

τc(g) ∈ (τ − 1)W [J].

Since also

(τ − 1)c(g) ∈ (τ − 1)W [J],

we conclude that

c(g) ∈ (τ − 1)W [J].

Thus the image of c, and therefore the image of φ, lies in (τ − 1)W [J]. Letting H be the A-submodule of (τ − 1)W [J] generated by φ(Γ), the proof continues in the usual way, using the fact that

(τ − 1)W [𝔪] ≠ W [𝔪]

to show that H = 0.