CHAPTER 14 Lecture 14

1. Structures on Galois Representations

1.1. Finite/Singular Structures. Let p be a prime number. Fix a global field F and a finite set Σ of places of F containing p and all archimedean places. Let A be as in Lecture 13, Section 3.1 and let W^{∗} be a free A-module of rank n with an A-linear action of G_{F,Σ}, where G_{F,Σ} is the Galois group of the maximal extension of F unramified outside of Σ. We can realize this action as a representation

ρ^{∗} : G_{F,Σ} → Aut_{A}(W^{∗}) ≅ GL_{n}(A),

the last isomorphism given by any choice of any A-basis for W^{∗}.

Since W is free of finite rank over ℤ_{p}, its Cartier dual W = Hom(W^{∗}, μ_{p^{∞}}) is a discrete, torsion, A-divisible A-module with an A-linear G_{F,Σ}-action. We will write

ρ : G_{F,Σ} → Aut_{A}(W)

for the representation of G_{F,Σ} on W.

Let 𝔪 be the maximal ideal of A and let k be the residue field A/𝔪. We make the assumption that the residual representation

\overline{ρ}^{∗} : G_{F,Σ} → Aut_{k}(W^{∗} ⊗_{A} k)

is absolutely irreducible; this just means that the representation

\overline{ρ}^{∗} ⊗_{k} \overline{k} : G_{F,Σ} → Aut\overline{k}(W^{∗} ⊗_{A} \overline{k})

is irreducible. This is in fact equivalent to \overline{ρ} being absolutely irreducible, as is easily seen.

Now, suppose that we are given a finite/singular structure on W. That is, for all places v of F we have chosen subgroups

H_{f}^{1}(G_{v}, W) ⊆ H^{1}(G_{v}, W)

which agree with the standard choices for almost all v. This in turn induces a Cartier dual finite/singular structure on W^{∗}, so that H_{f}^{1}(G_{v}, W) and H_{f}^{1}(G_{v}, W^{∗}) are orthogonal complements under Tate local duality for all v.

Let J be an ideal of A such that A/J is finite, and thus artinian. We will write W [J] for the “kernel” of multiplication by J on W; that is,

W [J] = ∩_{α∈J}W [α],

where W [α] is the kernel of multiplication by α on W. One can use the given finite/singular structures on W to induce structures on W [J] as follows: one simply defines H_{f}^{1}(G_{v}, W [J]) to be the unique subgroup of H^{1}(G_{v}, W) making the diagram

0Last modified April 8, 1998

\xymatrix{H_{f}^{1}(G_{v}, W [J]) \ar[r] \ar@{^{(}->}[d] & H_{f}^{1}(G_{v}, W) \ar@{^{(}->}[d] \\ H^{1}(G_{v}, W [J]) \ar[r] & H^{1}(G_{v}, W)}

cartesian. That is, one sets

H_{f}^{1}(G_{v}, W [J]) = {h ∈ H^{1}(G_{v}, W [J]) | the image of h in H^{1}(G_{v}, W) lies in H_{f}^{1}(G_{v}, W)}.

Of course, we must show that this definition really does give a finite/singular structure on W [J]. This follows from the commutative diagram with exact columns

\xymatrix{0 \ar[d] & 0 \ar[d] \\ H^{1}(𝔤_{v}, W [J]) \ar[d] \ar[r] & H^{1}(𝔤_{v}, W) \ar[d] \\ H^{1}(G_{v}, W [J]) \ar[d] \ar[r] & H^{1}(G_{v}, W) \ar[d] \\ Hom(I_{v}, W [J]) \ar@{^{(}->}[r] & Hom(I_{v}, W)}

Specifically, an easy diagram chase shows that the middle square is cartesian; thus if W is unramified and has the standard finite/singular structure at v (both of which are true for almost all v by assumption), then this identifies our choice of H_{f}^{1}(G_{v}, W [J]) with the standard choice H^{1}(𝔤_{v}, W [J]).

In addition, since W^{∗}/JW^{∗} is Cartier dual to W [J], we get an induced finite/singular structure on W^{∗}/JW^{∗}. We will always assume that all W [J] and W^{∗}/JW^{∗} have finite/singular structures induced from some fixed structure on W in this way.

We will write

ρ_{J} : G_{F,Σ} → Aut_{A/J}(W [J])

and

ρ_{J}^{∗} : G_{F,Σ} → Aut_{A/J}(W^{∗}/JW^{∗})

for the representations induced by ρ and ρ^{∗}.

The Selmer groups of W and W [J] are related in a particularly simple way.

Lemma 1.1. For any ideal J of A of finite index we have

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

and

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

Proof. Let 0 ≠ α ∈ J and consider the exact sequence

0 → W [α] → W ⟶^{α} W → 0.

The last map is surjective since W is A-divisible. The long exact sequence for G_{F} -cohomology yields the exact sequence

H^{0}(G_{F} , W) → H^{1}(G_{F} , W [α]) → H^{1}(G_{F} , W) ⟶^{α} H^{1}(G_{F} , W).

But the first term is 0 since W [𝔪] is irreducible and by Corollary 3.7 of Lecture 13. (This requires the fact that W has rank at least 2.) This gives the first equality in the case that J is principal, and since A is noetherian and W^{G_{F}} = 0, Lemma 1.2 below finishes the proof.

For the second equality, we just use the commutative diagram with exact columns

\xymatrix{0 \ar[d] & 0 \ar[d] \\ H_{f}^{1}(G_{F} , W [J]) \ar[d] \ar[r] & H_{f}^{1}(G_{F} , W)[J] \ar[d] \\ H^{1}(G_{F} , W [J]) \ar[d] \ar[r]^{≅} & H^{1}(G_{F} , W)[J] \ar[d] \\ ⊕_{v}H_{s}^{1}(G_{v}, W [J]) \ar[r] & ⊕_{v}H_{s}^{1}(G_{v}, W)[J]}

A diagram chase and the definition of the induced finite singular structure on W [J] show that the top square is cartesian, and thus that the first horizontal arrow is an isomorphism.

Lemma 1.2. Let M and N be submodules of a G-module W. Suppose that H^{1}(G, M) and H^{1}(G, N) inject into H^{1}(G, W). If also W^{G} = 0, then H^{1}(G, M ∩ N) injects into H^{1}(G, W) and is equal to the intersection of H^{1}(G, M) and H^{1}(G, N).

Proof. Let Z, Z_{M}, Z_{N}, B, B_{M}, B_{N} be the cocycles and coboundaries for W, M and N respectively. Z_{M} and Z_{N} are naturally submodules of Z, and the injectivity assumption means exactly that B_{M} = Z_{M} ∩B and B_{N} = Z_{N} ∩B. (That is, every coboundary w^{g} − w which has image in M or N has a representative with w in M or N respectively.) Thus

H^{1}(G, M)∩H^{1}(G, N) = Z_{M}/(Z_{M} ∩B)∩Z_{N}/(Z_{N} ∩B) = Z_{M} ∩Z_{N}/(Z_{M} ∩Z_{N} ∩B).

Now, clearly Z_{M} ∩Z_{N} is just the cocycles for M ∩N, so it remains to show that Z_{M} ∩ Z_{N} ∩ B are the coboundaries. It is clear that it contains the coboundaries, so suppose that we have w ∈ W such that w^{g} − w ∈ M ∩ N for every g. Then, since B_{M} = Z_{M} ∩ B and B_{N} = Z_{N} ∩ B, we can find m ∈ M and n ∈ N such that

w^{g} − w = m^{g} − m = n^{g} − n

for all g. But then

(m − n)^{g} = m − n,

so m − n ∈ W^{g}. By assumption, then, m − n = 0, so m = n ∈ M ∩ N. This completes the proof.

1.2. The Δ-Hypothesis. We keep the notation and assumptions of the previous section. Since W [J] is finite (since W has finite corank and J has finite index in A), ρ_{J} factors through some finite extension F_{J} of F; that is, ρ_{J} induces an injection

ρ_{J} : Gal(F_{J}/F) ↪ Aut_{A/J}(W [J]).

We will write Δ_{J} for Gal(F_{J}/F).

The natural exact sequence

1 → G_{F_{J}} → G_{F} → Δ_{J} → 1

induces an inflation/restriction sequence

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

In fact, the last term can be though of as Δ_{J}-equivariant homomorphisms if G_{F_{J}}^{ab} is given the correct Δ_{J}-action. Specifically, if δ ∈ Δ_{J} then choose any lifting δ̃ of δ to G_{F}. For any x ∈ G_{F_{J}}^{ab} , we define x^{δ} to be δ̃xδ̃^{−1}; since G_{F_{J}}^{ab} is abelian this does not depend on the choice of lifting of δ. It follows immediately from the relevant definitions that with this action on G_{F_{J}}^{ab} and the usual action on W [J], Hom(G_{F_{J}}^{ab} , W [J])^{Δ_{J}} is just the Δ_{J}-equivariant homomorphisms from G_{F_{J}}^{ab} to W [J].

Now, suppose that H^{1}(Δ_{J}, W [J]) = 0. Then we have an injection

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

which we write as h ↦ φ_{h}. Now, to show that an h ∈ H^{1}(G_{F} , W [J]) is 0 it is enough to show that φ_{h} is 0; to do this it is enough to show that φ_{h} vanishes on a set of Frobenius elements of density 1. Thus the assumption that H^{1}(Δ_{J}, W [J]) = 0 is a very useful one; by the Δ-hypothesis we will mean the assumption that H^{1}(Δ_{J}, W [J]) = 0 for every ideal J of A of finite index.

In fact, this is not a particularly restrictive assumption. To give an example, we consider a slightly more general situation: let M be a free R-module of finite rank n for some finite local ring R of residue characteristic p and let Δ be a subgroup of GL_{n}(R). (To relate this to the above situation just take R = A/J, M = W [J] and Δ = Δ_{J}; Δ_{J} is a subgroup of Aut_{A/J}(W [J]) = GL_{n}(R).) We will give one condition under which H^{1}(Δ, M) = 0.

Lemma 1.3 (Sah). Suppose that there is a non-trivial subgroup C of Δ, of order prime to p, which consists entirely of scalar matrices. Then H^{r}(Δ, M) = 0 for all r.

Proof. We have a Hochschild-Serre spectral sequence

H^{p}(Δ/C, H^{q}(C, M)) ⇒ H^{p+q}(Δ, M);

this makes sense since C is in the center of Δ and thus normal. This shows that it will suffice to show that H^{q}(C, M) = 0 for all q. If q > 0 this follows from Lecture 5, Lemma 2.4.

For q = 0 we must show that M^{C} = 0. Now, by hypothesis C ⊆ R^{∗}. So suppose that there is a non-zero m ∈ M such that cm = m for all c ∈ C. Then

(c − 1)m = 0,

so c − 1 is not a unit. Thus c − 1 lies in the maximal ideal of R, so C is a subgroup of the multiplicative group 1 + 𝔪. But this group has order a power of p, while C has order prime to p, which is a contradiction. So M^{C} = 0.

2. Depth and Kolyvagin-Flach Systems

2.1. Depth. We continue to let A and W be as before. Fix a non-unit ideal J of A of finite index. The induced finite/singular structure on W^{∗}/JW^{∗} yields an exact sequence

0 → H_{f}^{1}(G_{F} , W^{∗}/JW^{∗}) → H^{1}(G_{F} , W^{∗}/JW^{∗}) → H_{s}^{1}(G_{F} , W^{∗}/JW^{∗}) → 0.

We will now begin to use the notation of Lecture 8, Section 1.2. Let C be some set of elements of the Kolyvagin group H_{s}^{1}(G_{F} , W^{∗}/JW^{∗}). Fix a place v of F and let C′ be the A/J-submodule of H_{s}^{1}(G_{v}, W^{∗}/JW^{∗}) generated by res_{v,s}(c) for c ∈ C. If I is an ideal of A containing J, we will say that C is of depth I at v if the quotient of H_{s}^{1}(G_{v}, W^{∗}/JW^{∗}) by C′ is annihilated by I.

Let L be a (not necessarily finite) set of places of F disjoint from Σ. If I is some ideal of A containing J, a Kolyvagin-Flach system of classes of depth I for W^{∗}/JW^{∗} for L is an assignment of some subset C^{(v)} of H_{s}^{1}(G_{F} , W^{∗}/JW^{∗}) for each v ∈ L such that Supp c = {v} for all c ∈ C^{(v)} and C^{(v)} is of depth I at v.

2.2. The Main Theorem. We will need two more hypotheses in order to get the main theorem. First, we assume that p ≠ 2; this seems to be a necessary condition and not just the result of a lack of care. Second, we assume that there is an involution τ in the image of

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

which is not a scalar.

In particular, this implies that such a non-scalar involution exists for the representation

ρ_{J} : G_{F,Σ} → Aut_{k}(W [J])

for any ideal J of finite index in A. To see this, first note that

Aut_{A/J}(W [J]) → Aut_{k}(W [𝔪])

is surjective with kernel a p-group. Thus we can lift our given τ to an element of Aut_{A/J}(W [J]) in the image of G_{F,Σ}. It will be of order 2 times a power of p, so taking an appropriate power of this yields the desired non-scalar involution. (It is non-scalar since it still reduces to τ modulo 𝔪, as p is odd.)

Now, fix such a τ for our fixed ideal J. Let L_{τ} be the set of places of F over which there is a place whose associated Frobenius in F_{J} is τ; equivalently, L_{τ} is the set of places of F with Frobenius in F_{J} the conjugacy class of τ. Note that the Tchebatorev density theorem ([La-ANT, Chapter 8, Theorem 10]) implies that L_{τ} has positive density; in particular, it is infinite.

Theorem 2.1. Let A, W and J be as above. Suppose that the residual representation \overline{ρ} is absolutely irreducible; that the Δ-hypothesis holds; and that there exists a non-scalar involution τ in the image of \overline{ρ}. Further suppose that there exists a Kolyvagin-Flach system of classes of depth I for W^{∗}/JW^{∗} for L_{τ}, for some ideal I containing J. Then

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

Note that the statement of this theorem is very much in the spirit of Mazur’s comments in Lecture 8, Section 2.