CHAPTER 8 Lecture 8

1. Notation for Generalized Selmer Groups

1.1. Structures. For convenience we will now set some notation for our generalized Selmer groups. We continue to let F be a number field with the conventions of Lecture 7, Section 1.2. Let M be a finite G_{F} -module. We recall that a structure σ on M is a choice of exact sequences

σ_{v} : 0 → H_{f}^{1}(G_{v}, M) → H^{1}(G_{v}, M) → H_{s}^{1}(G_{v}, M) → 0

for all places v of M, such that the exact sequence σ_{v} is the standard local finite/singular structure for almost all v. We will write M_{σ} for the G_{F} -module M with the structure σ; if there is no chance of confusion we will just write M.

Given two structures σ_{1} and σ_{2} on M, we will say that σ_{1} is more stringent than σ_{2}, written

σ_{1} < σ_{2},

if

H_{f}^{1}(G_{v}, M_{σ_{1}}) ⊆ H_{f}^{1}(G_{v}, M_{σ_{2}})

for all places v. In particular, this implies that

H_{f}^{1}(G_{F} , M_{σ_{1}}) ⊆ H_{f}^{1}(G_{F} , M_{σ_{2}}),

since the conditions to lie in H_{f}^{1}(G_{F} , M_{σ_{1}}) are more restrictive than those to lie in H_{f}^{1}(G_{F} , M_{σ_{2}}). This relation gives a partial ordering on the set of all structures.

Given a structured module M_{σ} we will write σ^{∗} for the structure defined on M^{∗} in Definition 4 of Lecture 7. It follows easily from the definition of this structure on M^{∗} that if σ_{1} < σ_{2}, then σ_{2}^{∗} < σ_{1}^{∗}.

Let S be a finite set of places of F containing all the archimedean places, all the places dividing the order of M and all places where M is ramified. We define the Sstructure on M, written as M_{S}, to be the structure with the standard finite/singular structures for all v ∉ S and with the choices

H_{f}^{1}(G_{v}, M) = H^{1}(G_{v}, M)

for all v ∈ S. (We considered this structure in Lecture 7, Sections 2.2 and 2.3.) So an S-structure is the least stringent structure on M with the standard choices outside of S. In particular, if σ is any structure, then there is some S-structure which is less stringent than σ; simply take S to contain all places where σ is not standard.

0Last modified March 7, 1998

1.2. The Singular Restriction Map. Recall that for every place v we have a canonical restriction map

H^{1}(G_{F} , M) ⟶^{res_{v}} H^{1}(G_{v}, M).

We define the local singular restriction map at v

res_{s,v} : H^{1}(G_{F} , M) → H_{s}^{1}(G_{v}, M)

to be the composition of res_{v} with the quotient map

H^{1}(G_{v}, M) ↠ H_{s}^{1}(G_{v}, M).

We saw in Lecture 7, Section 2.1 that for any h ∈ H^{1}(G_{F} , M) we have res_{v}(h) ∈ H_{f}^{1}(G_{v}, M) for almost all v; in our new notation this says that res_{s,v}(h) = 0 for almost all v. Thus we obtain a map, which we call the singular restriction map,

res_{s} : H^{1}(G_{F} , M) → ⨁_{v} H_{s}^{1}(G_{v}, M),

the direct sum being over all places of F. With this notation we have

H_{f}^{1}(G_{F} , M) = ker res_{s},

so there is an exact sequence

0 → H_{f}^{1}(G_{F} , M) → H^{1}(G_{F} , M) → ⨁_{v} H_{s}^{1}(G_{v}, M).

Recall that by definition of the Kolyvagin group H_{s}^{1}(G_{F} , M) we have an exact sequence

0 → H_{f}^{1}(G_{F} , M) → H^{1}(G_{F} , M) → H_{s}^{1}(G_{F} , M) → 0.

Comparing this to the exact sequence above we see that there is a natural injection

H_{s}^{1}(G_{F} , M) ↪ ⨁_{v} H_{s}^{1}(G_{v}, M).

Given any c ∈ H_{s}^{1}(G_{F} , M), this map is simply the map

c ↦ (res_{s,v}(c̃))_{v},

where c̃ is any lifting of c to H^{1}(G_{F} , M). We will write c_{v} for res_{s,v}(c̃) ∈ H_{s}^{1}(G_{v}, M), so that the above map is simply

c ↦ (c_{v})_{v}.

We conclude with a few more definitions. Given any

c = (c_{v})_{v} ∈ ⨁_{v} H_{s}^{1}(G_{v}, M),

define the support of c, Supp(c), to be the set of places v for which c_{v} ≠ 0; our previous discussion shows that Supp(c) is finite. If c is actually in H_{s}^{1}(G_{F} , M), we again write Supp(c) for the set of places for which c_{v} ≠ 0. Also, for any h ∈ H^{1}(G_{F} , M) we will write h_{v} for res_{v}(h) ∈ H^{1}(G_{v}, M). In particular, if h ∈ H_{f}^{1}(G_{F} , M), then h_{v} ∈ H_{f}^{1}(G_{v}, M) for all v

2. A Global Pairing

We continue with the notation of the previous section. In particular, M is a finite structured G_{F} -module with (structured) Cartier dual M^{∗}. For any place v we will write ⟨·, ·⟩_{v} for the pairing

H^{1}(G_{v}, M) ⊗ H^{1}(G_{v}, M^{∗}) → H^{2}(G_{v}, M ⊗ M^{∗}) → H^{2}(G_{v}, μ(F_{v}^{∗})) → ℚ/ℤ

given by cup product, Cartier duality and the invariant map. Thus for nonarchimedean v, ⟨·, ·⟩_{v} is just the Tate pairing; for real v it is the pairing

H^{1}(G_{v}, M) ⊗ H^{1}(G_{v}, M^{∗}) → \frac{1}{2}ℤ/ℤ

of Lecture 7, Proposition 1.1; and for complex v all of the groups are 0, since G_{v} = 0. For any c_{v} ∈ H^{1}(G_{v}, M^{∗}), we will also think of the pairing as giving a map

c_{v} : H^{1}(G_{v}, M) → ℚ/ℤ,

given by c_{v}(h_{v}) = ⟨h_{v}, c_{v}⟩.

We now define a global pairing

H_{f}^{1}(G, M) ⊗ ⨁_{v} H_{s}^{1}(G_{v}, M^{∗}) → ℚ/ℤ

by

h ⊗ (c_{v})_{v} ↦ ∑_{v} ⟨h_{v}, c_{v}⟩_{v} .

(Recall the definition of h_{v} in the previous section.) This is actually well-defined since h_{v} ∈ H_{f}^{1}(G_{v}, M) for all v and by our definition of the structure on M^{∗} the perfect pairings

H^{1}(G_{v}, M) ⊗ H^{1}(G_{v}, M^{∗}) → ℚ/ℤ

yield perfect pairings

H_{f}^{1}(G_{v}, M) ⊗ H_{s}^{1}(G_{v}, M^{∗}) → ℚ/ℤ

for all v. Also, the sum is simply a finite sum

∑_{v} ⟨h_{v}, c_{v}⟩_{v} = ∑_{v∈Supp(c)} ⟨h_{v}, c_{v}⟩_{v}

since (c_{v})_{v} is an element of the direct sum.

We will write the pairing as ⟨h, c⟩; alternately, for any c ∈ ⨁_{v} H_{s}^{1}(G_{v}, M^{∗}), we will also write c for the map

c : H_{f}^{1}(G_{F} , M) → ℚ/ℤ

given by c(h) = ⟨h, c⟩.

This pairing, together with the next proposition, will be our principal tool for bounding the order of Selmer groups.

Proposition 2.1. The subgroup H_{s}^{1}(G_{F} , M^{∗}) of ⨁_{v} H_{s}^{1}(G_{v}, M^{∗}) is orthogonal to all of H_{f}^{1}(G_{F} , M) under the above pairing; that is, the composition

H_{f}^{1}(G_{F} , M) ⊗ H_{s}^{1}(G_{F} , M^{∗}) → H_{f}^{1}(G_{F} , M) ⊗ ⨁_{v} H_{s}^{1}(G_{v}, M^{∗}) → ℚ/ℤ

is zero.

Proof. Take any h ∈ H_{f}^{1}(G_{F} , M) and c ∈ H_{s}^{1}(G_{F} , M^{∗}), and let c̃ be a lifting of c to H^{1}(G_{F} , M^{∗}). Consider the commutative diagram

[figure: H^{1}(G_{F} , M) ⊗ H^{1}(G_{F} , M^{∗}); ∏_{v} H^{1}(G_{v}, M) ⊗ H^{1}(G_{v}, M^{∗}); H^{2}(G_{F} , M ⊗ M^{∗}); ∏_{v} H^{2}(G_{v}, M ⊗ M^{∗}); H^{2}(G_{F} , μ(F̄^{∗})); ∏_{v} H^{2}(G_{v}, μ(F̄_{v}^{∗})); _{Br_{F}}; _{∏_{v} Br_{F_{v}}}]

where all horizontal maps are restriction and the second column is the product of all of the local pairings ⟨·, ·⟩_{v}. Note also that by Lecture 7, Proposition 3.1, the image of Br_{F} actually lands in ⨁_{v} Br_{F_{v}}. Further, by Lecture 7, Theorem 3.2, Br_{F} is the kernel of the summation map

⨁_{v} Br_{F_{v}} → ℚ/ℤ.

Now, following h ⊗ c̃ clockwise around the diagram and then mapping by summation to ℚ/ℤ gives the global pairing ⟨h, c⟩, by definition. But going counterclockwise shows that the image of h ⊗ c̃ in ⨁_{v} Br_{F_{v}} lies in the image of Br_{F}, and thus maps to 0 under summation. This completes the proof.

The above result is extremely useful in bounding the order of Selmer groups. We quote from Mazur:

Any Kolyvagin element c ∈ H_{s}^{1}(G, M^{∗}) imposes some local condition (at places v in its support) that must be satisfied by all Selmer elements h ∈ H_{f}^{1}(G, M).

For example: If the support of c ≠ 0 is concentrated at exactly one place v_{o}, then any Selmer element h must have the property that h_{v_{0}} ∈ ker(c_{v_{0}}) ⊂ H_{f}^{1}(G_{v_{0}}, M). Here is a way of thinking about the local condition detected by this Kolyvagin element c: It tells us that we may strengthen the f/s structure on the Galois module M, without changing the finite part of its (1-dimensional) cohomology. That is, let us define M′ to be the structured module whose underlying Galois module is M again, whose f/s structure for all places v ≠ v_{0} is equal to the f/s structure on M, but on v_{0} we place a more stringent f/s structure on the G_{v_{0}}-module M by setting:

H_{f}^{1}(G_{v_{0}}, M′) := ker{c_{v_{0}} : H_{f}^{1}(G_{v_{0}}, M) → ℚ/ℤ} ⊂ H_{f}^{1}(G_{v_{0}}, M).

The above discussion gives us that the finite part of the 1dimensional cohomology of the more stringent structure M′ is equal to the finite part of the 1-dimensional cohomology of M. Now this is already a curiously favorable state of affairs, because if we are trying to to show that H_{f}^{1}(G, M) is small, we have (by constructing the Kolyvagin element c) that

H_{f}^{1}(G, M′) = H_{f}^{1}(G, M),

where M′ is more stringent than M, and there is nothing to stop us from trying to do this again inductively, to force the finite cohomology to satisfy more and more stringent local conditions by finding further Kolyvagin elements for these increasingly stringent structures. In the most extreme cases one then shows that H_{f}^{1}(G, M) satisfies such stringent conditions that it is forced to vanish, or else it is generated by specifically constructed cohomology classes. The efficacy of this method is determined by our ability to manufacture Kolyvagin elements with good control over their local components. This approach, combined with the following two further ideas is the underlying philosophy behind Kolyvagin’s method:

1. It is possible, at times, to judiciously enlarge the base field F without losing too much of the Selmer group, but this larger base field gives us an even better chance of constructing Kolyvagin elements.

2. In various different contexts, one can find well-behaved collections of objects (these are the Euler systems of algebraic cycles, Heegner points, circular units, elements in algebraic K-groups, etc.) which produce the desired Kolyvagin elements in cohomology.

3. The Finiteness of the Selmer Group

3.1. Kummer Theory : Perfect Fields. Later in this section we will show that the Selmer group H_{f}^{1}(G_{F} , M) is finite; we review first some Kummer theory which we will need. For the time being, let L be any field; for simplicity we assume that L is perfect. Fix an integer n and consider the exact Kummer sequence

1 → μ_{n}(L̄^{∗}) → L̄^{∗} ⟶^{n} L̄^{∗} → 1.

The long exact sequence of G_{L}-cohomology begins

1 → μ_{n}(L) → L^{∗} ⟶^{n} L^{∗} → H^{1}(G_{L}, μ_{n}(L̄)) → H^{1}(G_{L}, L̄^{∗}).

But, by Hilbert’s Theorem 90, H^{1}(G_{L}, L̄^{∗}) = 0, so we obtain an isomorphism

L^{∗}/L^{∗n} ⟶^{∼} H^{1}(G_{L}, μ_{n}(L̄)).

Recalling the definition of the boundary map in cohomology, this map simply sends any α ∈ L^{∗} to the cocycle

c : G_{L} → μ_{n}(L̄^{∗})

given by

c(σ) = \frac{σ(α^{1/n})}{α^{1/n}};

it is independent of the choice of n^{th} root of α and is constant on the equivalence class of α in L^{∗}/L^{∗n} (up to coboundaries, of course).

Let us now assume that μ_{n} = μ_{n}(L̄^{∗}) ⊆ L; this is the usual situation of Kummer theory. Our above isomorphism becomes

L^{∗}/L^{∗n} ⟶^{∼} Hom(G_{L}, μ_{n}).

Since μ_{n} is abelian of exponent n, we can rewrite our isomorphism as

L^{∗}/L^{∗n} ⟶^{∼} Hom(Gal(L′/L), μ_{n}),

where L′ is the maximal abelian extension of L of exponent n. Taking duals gives

Gal(L′/L) ≅ Hom(L^{∗}/L^{∗n}, μ_{n})

canonically.

Now, let V be a subgroup of L^{∗}/L^{∗n}. Then Hom(V, μ_{n}) is a quotient of Hom(L^{∗}/L^{∗n}, μ_{n}), so it corresponds under our above isomorphism to a quotient of Gal(L′/L); this in turn corresponds to an intermediate field L ⊆ E ⊆ L′. This correspondance sets up a bijection between subgroups of L^{∗}/L^{∗n} and abelian extensions of L of exponent n; if V corresponds to E, then there is an isomorphism

Gal(E/L) ≅ Hom(V, μ_{n}).

Let us attempt to make this correspondance more explicit. To do this, note that Gal(L′/E) is precisely the subgroup of Gal(L′/L) mapping to 0 in Hom(V, μ_{n}); that is, σ ∈ Gal(L′/E) if and only if

\frac{σ(α^{1/n})}{α^{1/n}} = 1

for all α ∈ L^{∗} lifting elements of V. It now follows immediately from Galois theory that

E = L(V^{1/n}),

the notation meaning that E is obtained by adjoining n^{th} roots of lifts of everything in V to L. We state our results of this section as a theorem.

Theorem 3.1. Let L be a perfect field containing μ_{n}(L^{∗}), and let L′ be the maximal abelian extension of L of exponent n. Then

Gal(L′/L) ≅ Hom(L^{∗}/L^{∗n}, μ_{n}).

Furthermore, there is a bijective correspondance between subgroups of L^{∗}/L^{∗n} and intermediate fields between L and L′; if V corresponds to E, then

Gal(E/L) ≅ Hom(V, μ_{n})

and

E = L(V^{1/n}).

3.2. Kummer Theory : Local and Global Fields. We now take K to be a local field with uniformizer π. We continue to assume that K contains the n^{th} roots of unity. Let V ⊆ K^{∗}/K^{∗n} and E/K correspond under the correspondance of Theorem 3.1. We wish to relate the ramification in E/K to V.

This is fairly easy given that E = K(V^{1/n}). Let α be a lift of some element of V and let β be an n^{th} root of α. E is the compositum of K(β) for all such β, so it will be ramified over K if and only if K(β)/K is ramified for some α.

Now, write α = uπ^{m} for some unit u. Suppose first that n divides m. Then K(β) = K(β′), where β′ is an n^{th} root of u. Now, β′ satisfies the polynomial

x^{n} − u,

although it may not be minimal. In any event, a standard theorem of algebraic number theory (see [Se-LF, Chapter 3, Section 5, Corollary 1 and Section 6, Corollary 2]) says that K(β′)/K is ramified only if nu^{n−1} is not a unit in K(β′); this occurs only if π divides n. (However, even if π divides n we can not be sure that the extension is ramified.)

On the other hand, if n does not divide m, then K(β) must contain some nontrivial root of π; this immediately implies that K(β)/K is ramified. We rephrase our results so far as a proposition.

Proposition 3.2. Let K be a local field with uniformizer π. Suppose that K contains the n^{th} roots of unity. Let E/K be an abelian extension of exponent n corresponding to a subgroup V ⊆ K^{∗}/K^{∗n}. If there is an α ∈ V such that

v(α) ≠ 0 (mod n),

where v is the normalized valuation on K, then E/K is ramified. If instead

v(α) ≡ 0 (mod n)

for all α ∈ V and v(n) = 0, then E/K is unramified.

Now let F be a global field. Considering ramification locally immediately yields the following theorem.

Theorem 3.3. Let F be a number field containing the n^{th} roots of unity. Let E/F be an abelian extension of exponent n corresponding to a subgroup V ⊆ F^{∗}/F^{∗n}. Then E/F is ramified at a non-archimedean place v if there is an α ∈ V with

v(α) ≠ 0 (mod n);

E/F is unramified at all other v except possibly those dividing n.

Corollary 3.4. Let F be a number field and let S be a finite set of places of F. Let L be the maximal abelian extension of F of exponent n, unramified outside of S. Then L/F is finite.

Proof. Let F′ = F (μ_{n}(F̄^{∗})). Then the corresponding extension of F′ contains that of F, so we can assume that F contains the n^{th} roots of unity. In the same way we can assume that F is totally imaginary, so that the archimedean places are irrelevant.

There is a natural map

F^{∗} → ⨁_{v non-arch} ℤ,

given by the valuation maps at all non-archimedean places, with kernel U, the units of the ring of integers of F. This map descends to an exact sequence

0 → U/U^{n} → F^{∗}/F^{∗n} → ⨁_{v} ℤ/nℤ.

(The kernel is a priori U/(F^{∗n} ∩U), but any element of F^{∗} which has a power which is a unit is itself a unit.) If V is the subgroup of F^{∗}/F^{∗n} corresponding to L, then by Theorem 3.3 the image of V in each ℤ/nℤ factor for v ∉ S is zero; thus we have an exact sequence

0 → V ∩ U/U^{n} → V → ⨁_{v∈S} ℤ/nℤ.

By Dirichlet’s Theorem, U is finitely generated, so U/U^{n} is finite. Since S is also finite, V must be finite. This implies that L/F is finite.

3.3. The Finiteness of Selmer Groups.

Theorem 3.5. Let F be a number field and let M be a finite structured G_{F} - module. Then H_{f}^{1}(G_{F} , M) is finite.

Proof. Let S be a finite set of places of F containing all places where M does not have the standard finite/singular structure; in particular, S contains all archimedean places, all places dividing the order of M, and all places where M is wildly ramified. Let us also take S to contain all places where M is at all ramified. Now, let M_{S} be the same G_{F} -module as M with the S-structure (see Section 1.1). Then we have

H_{f}^{1}(G_{F} , M) ⊆ H_{f}^{1}(G_{F} , M_{S}),

so it will be enough to prove finiteness for M_{S}.

Next, let L be a finite Galois extension of F which splits M; that is, such that G_{L} acts trivially on M. (It is easy to see that such an L exists; see Lecture 1, Section 1.1.) If L/F is ramified outside of S, then enlarge S to contain all places where L/F is ramified. Let T be the set of places of L lying above places of F in S, and let M_{T} be M considered as a G_{L}-module with the T -structure. Consider, for all places v of F and all places w of L lying over v, the maps

H^{1}(G_{F_{v}}, M_{S}) ⟶^{res} H^{1}(G_{L_{w}}, M_{T}).

We claim that under these maps the image of H_{f}^{1}(G_{F_{v}}, M_{S}) lies in H_{f}^{1}(G_{L_{w}}, M_{T}). If v ∈ S (so w ∈ T) this is immediate; if v ∉ S, then this follows from the commutative diagram

\xymatrix{H^{1}(𝔤_{v}, M_{S}) \ar[r]^{res} \ar[d]^{inf} & H^{1}(𝔤_{w}, M_{T}) \ar[d]^{inf} \\ H^{1}(G_{F_{v}}, M_{S}) \ar[r]^{res} & H^{1}(G_{L_{w}}, M_{T})}

This implies that we have a natural map

⨁_{v∉S} H_{s}^{1}(G_{F_{v}}, M_{S}) → ⨁_{w∉T} H_{s}^{1}(G_{L_{w}}, M_{T})

sitting in a commutative diagram

\xymatrix{H^{1}(G_{F} , M_{S}) \ar[r] \ar[d] & ⨁_{v} H_{s}^{1}(G_{F_{v}}, M_{S}) \ar[d] \\ H^{1}(G_{L}, M_{T}) \ar[r] & ⨁_{w} H_{s}^{1}(G_{L_{w}}, M_{T})}

We now get induced maps on the kernels of the horizontal maps, which are just the Selmer groups:

H_{f}^{1}(G_{F} , M_{S}) → H_{f}^{1}(G_{L}, M_{T}).

Let ker be the kernel of this map; it sits in an exact commutative diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & ker \ar[r] \ar[d] & H^{1}(Gal(L/F), M) \ar[d] \\ 0 \ar[r] & H_{f}^{1}(G_{F} , M_{S}) \ar[r] \ar[d] & H^{1}(G_{F} , M_{S}) \ar[r] \ar[d] & ⨁_{v} H_{s}^{1}(G_{F_{v}}, M_{S}) \ar[d] \\ 0 \ar[r] & H_{f}^{1}(G_{L}, M_{T}) \ar[r] & H^{1}(G_{L}, M_{T}) \ar[r] & ⨁_{w} H_{s}^{1}(G_{L_{w}}, M_{T})}

It is clear from the cocycle interpretation that H^{1}(Gal(L/F), M) is finite; thus ker is also finite. It follows that it will be enough to prove that H_{f}^{1}(G_{L}, M_{T}) is finite.

We are now in the situation of Lecture 7, Section 2.2: we have

H_{f}^{1}(G_{L}, M_{T}) = Hom(G_{L,T}^{ab} , M)

where G_{L,T}^{ab} is the Galois group of the maximal abelian extension of L unramified outside of T. Decomposing M as a product of cyclic groups, we see that to show this it will be enough to show that

Hom(G_{L,T}^{ab} , ℤ/nℤ)

is finite for any n.

Now we have

Hom(G_{L,T}^{ab} , ℤ/nℤ) = Hom(Gal(L′/L), ℤ/nℤ),

where L′ is the maximal abelian extension of L of exponent n, unramified outside of T. By Corollary 3.4 we know that L′ is a finite extension of L, and thus that

Hom(Gal(L′/L), ℤ/nℤ)

is finite. This completes the proof.