CHAPTER 10 Lecture 10

1. Kummer Theory

1.1. The Ring of S-Integers. We present in this section an alternative approach to Kummer theory over global fields. Let F be a number field and let S be a finite set of primes of F. (Later, when we assume that S contains all of the archimedean places, there will be times when only the non-archimedean places are relevant and our notation may not reflect this. We leave these issues to the reader to sort out. Also, we will use both 𝔭 and v to denote primes, depending on the context.) Let O_{F} be the ring of integers of F. We define the ring of S-integers, O_{F,S}, by

O_{F,S} = {x ∈ F | ord_{𝔭}(x) ≥ 0 for all 𝔭 ∉ S}.

Equivalently, O_{F,S} is the subring of F of fractions x/y with x and y integral and y only divisible by primes in S.

Intuitively, then, O_{F,S} is just the localization of O_{F} at an element of F divisible exactly by the primes in S. However, since O_{F} need not be a principal ideal domain, it is not immediately clear that such an element exists. That they do in fact exist follows from the finiteness of the ideal class group. (See [La-ANT, Chapter 5, Section, pp. 100-101].) Specifically, let 𝔭 be a prime in S. Then, since the ideal class group of O_{F} is finite, 𝔭^{n} is principal for some n, say 𝔭^{n} = α_{𝔭}O_{F}. Setting

α = ∏_{𝔭∈S} α_{𝔭},

we see that

O_{F,S} = O_{F} [α].

This description gives an immediate determination of the primes of O_{F,S}: they are simply the primes of O_{F} which are not in S. It also now follows immediately that the localizations and completions of O_{F} and O_{F,S} at primes 𝔭 ∉ S agree. Essentially, then, O_{F,S} has none of the information of O_{F} about 𝔭 ∈ S, but has all of it for 𝔭 ∉ S.

The following theorem is one of two fundamental finiteness results for O_{F,S}.

Theorem 1.1 (Dirichlet Unit Theorem). O_{F,S}^{∗} is a finitely generated abelian group. More precisely, O_{F,S}^{∗}/μ(F^{∗}) is free abelian of rank r_{1} + r_{2} + |S| − 1, where r_{1} is the number of real embeddings of F and r_{2} is the number of pairs of complex conjugate embeddings.

Proof. See [La-ANT, Chapter 5, Section 1, Corollary of p. 105].

0Last modified March 7, 1998

In order to state the other finiteness result, which is just a slight generalization of the finiteness of the ideal class group, we first must define locally free modules over O_{F,S}. We will say that an O_{F,S}-module V is locally free if

V_{v} = V ⊗_{O_{F,S}} (O_{F,S})_{v} = V ⊗_{O_{F,S}} O_{F_{v}}

is free as an O_{F_{v}}-module for all v ∉ S. (This is equivalent to V being free over all of the localizations of O_{F,S}, rather than going all the way to the completions.) We define the Picard group Pic(O_{F,S}) of O_{F,S} to be the set of isomorphism classes of finitely generated O_{F,S}-modules which are locally free of rank 1. Pic(O_{F,S}) is made into a group by tensor product; it is easy to see that the tensor product of two finitely generated locally free of rank 1 O_{F,S}-modules is of the same form. Inverses are given simply by duals,

V^{−1} = Hom_{O_{F,S}}(V, O_{F,S}),

as is easy to check. (For a more geometric treatment of the Picard group, see [Ha, Chapter 2, Section 6, esp. Remark 6.3.2 and pp. 143-146].)

Pic(O_{F,S}) also has an interpretation in terms of ideals. Recall that a fractional ideal of O_{F,S} is a non-zero finitely-generated sub-O_{F,S}-module of F. The fractional ideals are made into a group by multiplication. This group is simply free abelian on the primes of F not in S. A fractional ideal is said to be principal if it is of the form xO_{F,S} for some x ∈ F^{∗}.

Proposition 1.2. There is a canonical isomorphism between Pic(O_{F,S}) and the group of fractional ideals of O_{F,S} modulo principal fractional ideals.

Proof. For a proof in the context of divisors and invertible sheaves, see [Ha, Chapter 2, Section 6, Proposition 6.15]. We give a purely algebraic proof.

There is a natural map from the group of fractional ideals to Pic(O_{F,S}), simply considering a fractional ideal as a locally free O_{F,S}-module. The kernel of the map is easily seen to be the principal fractional ideals, so we are left to show that the map is surjective.

Let V be a locally-free O_{F,S}-module of rank 1. We claim that

V ⊗_{O_{F,S}} F ≅ F.

To see this, choose any prime 𝔭 not in S and let R be the localization of O_{F,S} at 𝔭. Then

V ⊗_{O_{F,S}} F ≅ (V ⊗_{O_{F,S}} R) ⊗_{R} F ≅ R ⊗_{R} F ≅ F,

since V is locally free. (Note that here we only used the fact that V is locally free at one prime.)

Given this isomorphism, we tensor the injection

O_{F,S} ↪ F

with the flat (since it is locally free) O_{F,S}-module V to get an injection

V ↪ V ⊗_{O_{F,S}} F ≅ F,

which realizes V as a submodule of F, and thus, since it is finitely generated, as a fractional ideal.

Corollary 1.3. Pic(O_{F,S}) is finite.

Proof. It is a standard fact that the ideal class group of O_{F} is finite; thus, Pic(O_{F}) is finite. The finiteness for Pic(O_{F,S}) follows immediately, since under the isomorphism of Proposition 1.2 Pic(O_{F,S}) can be identified with a subquotient of Pic(O_{F}).

In fact, one sees easily from the proof of the corollary that by taking S large enough to kill everything in Pic(O_{F}) we get Pic(O_{F,S}) = 0. More precisely, choose representative ideals of Pic(O_{F}) and let S contain every prime divisor of each representative.

1.2. The Kummer map. Let N ≥ 1 be an integer and let F be a number field such that μ_{N}(F̄) ⊆ F^{∗}; we will just write μ_{N} for μ_{N}(F̄). Let S be a set of places of F containing all of the archimedean places and all places of F dividing N. We wish to define a natural map

O_{F,S}^{∗}/O_{F,S}^{∗N} → Hom(G_{F,S}^{ab}, μ_{N}).

Let u be in O_{F,S}^{∗}. Fix a ξ ∈ F̄ such that ξ^{N} = u. (Normally making such a choice is a very bad idea, but we will see that since μ_{N} ⊆ F^{∗} the choice will not matter.) Note that ξ is actually in the maximal abelian extension of F unramified outside of S, since F (ξ)/F is cyclic of order dividing n with discriminant dividing nξ^{n−1}. Since ξ is a unit, F (ξ)/F can only be ramified at places in S.

Now, define φ_{u} ∈ Hom(G_{F,S}^{ab}, μ_{N}) by

φ_{u}(g) = \frac{g(ξ)}{ξ}.

Note first that since the conjugates of ξ are just ζξ for various ζ ∈ μ_{N}, φ_{u} really does have image in μ_{N}. In fact, the map φ_{u} is independent of the choice of N^{th} root of u: the fact that μ_{N} ⊆ L^{∗} implies that any g ∈ G_{F,S}^{ab} acts trivially on μ_{N}, so

\frac{g(ζξ)}{ζξ} = \frac{ζg(ξ)}{ζξ} = \frac{g(ξ)}{ξ}.

Thus we have a canonical map

O_{F,S}^{∗} → Hom(G_{F,S}^{ab}, μ_{N}),

and it is easily seen to be a homomorphism (using the fact that we can use any N^{th} root of u to define φ_{u}). Further, u ∈ O_{F,S}^{∗} defines the trivial homomorphism on G_{F,S}^{ab} if any only if

g(ξ) = ξ

for all g ∈ G_{F,S}^{ab}, where ξ^{N} = u. By Galois theory this happens if and only if ξ ∈ F; thus we have defined a natural injection

O_{F,S}^{∗}/O_{F,S}^{∗N} → Hom(G_{F,S}^{ab}, μ_{N}).

The determination of the cokernel, however, is somewhat more difficult.

1.3. Preparation for Descent. In order to determine the cokernel of the map of the previous section we must introduce the notion of twisting locally free modules. We begin with some easier examples.

Let K be any field and let L be a finite Galois extension, with Galois group G of order d. Then L is a K-vector space of dimension d on which G acts K-linearly. Thus L is a K-representation space for G; in other words L is a K[G]-module. In fact, L is just the regular representation; that is, L is a free K[G]-module of rank 1. This follows immediately from the normal basis theorem ([La-Al]).

In particular, suppose G is abelian and let φ : G → K^{∗} be a character. Define the φ-representation space of L by

L^{φ} = {x ∈ L | g(x) = φ(g)x for all g ∈ G.}.

Identifying L with K[G], we find that

L^{φ} ≅K[G]^{φ}

= {∑_{g∈G} a_{g}g ∈ K[G] | ∑ a_{g}hg = ∑ φ(h)a_{g}g for all h ∈ G}

= {∑ a_{g}g ∈ K[G] | a_{g} = φ(g^{−1})a_{1} for all g ∈ G}

= {a ∑ φ(g^{−1})g} ,

so L^{φ} is one dimensional over K.

Now restrict to the case where K is a local field. Let O_{K} and O_{L} be the rings of integers of K and L respectively. Then O_{L} is a free O_{K}-module of rank d as well as a O_{K}[G]-module. However, it is not always free of rank 1 over O_{K}[G], since the normal basis theorem does not apply to rings of integers. In fact, it can be shown that if L/K is unramified, then there is a form of the normal basis theorem for the ring of integers and O_{L} ≅ O_{K}[G] as O_{K}[G]-modules.

Now suppose that L/K is an arbitrary finite abelian extension of local fields and that V is an O_{K}-module which is free of rank 1. Let φ : G → O_{K}^{∗} be a character of G. (Any character of G to K^{∗} must land in O_{K}^{∗} since its image is just roots of unity, which are integral.) We define the φ-representation space of V by

V^{φ} = (V ⊗_{O_{K}} O_{L})^{φ}
= {v ⊗ x ∈ V ⊗_{O_{K}} O_{L} | v ⊗ g(x) = φ(g)(v ⊗ x) for all g ∈ G}.

We know that V^{φ} is torsion-free, since it is a submodule of V ⊗_{O_{K}} O_{L}, which is free of rank n over O_{K}. Thus, since O_{K} is a principal ideal domain, V^{φ} is free. In fact, it is easy to show that V^{φ} ⊗_{O_{K}} K ≅ K^{φ}, so that the rank must be 1.

1.4. Descent for Global Fields. We now return to the case where F is a global field containing μ_{N} and S is a finite set of places containing the archimedean places and the places dividing N. We will define a map

Hom(G_{F,S}^{ab}, μ_{N}) → Pic(O_{F,S}).

We first work in slightly more generality. Let V be a locally free O_{F,S}-module and let φ be in Hom(G_{F,S}^{ab}, μ_{N}). Since μ_{N} is finite, φ factors through some finite extension L/F, which we can take to be abelian and unramified outside of S:

\xymatrix{G_{F,S}^{ab} \ar[d]^{φ} \ar@{->>}[r] & Gal(L/F) \ar[dl]^{φ} \\ μ_{N}}

Let T be the set of places of L lying above places of S. Then O_{L,T} is an O_{F,S}- module. We define the φ-representation space of V by

V^{φ} = (V ⊗_{O_{F,S}} O_{L,T})^{φ}
= {v ⊗ x ∈ V ⊗_{O_{F,S}} O_{L,T} | v ⊗ g(x) = φ(g)(v ⊗ x) for all g ∈ G}.

For example,

O_{F,S}^{φ} = {x ∈ O_{L,T} | g(x) = φ(g)x for all g ∈ G}.

For general V, V^{φ} is just

V^{φ} = V ⊗_{O_{F,S}} O_{F,S}^{φ}.

Note first that the O_{F,S}-modules V^{φ} defined by different choices of splitting fields are canonically isomorphic, so that the above definition makes sense. Note also that if V is locally free of rank 1, then so is V^{φ}; this follows from the local computations of the previous section.

Thus, we can define the desired map

Hom(G_{F,S}^{ab}, μ_{N}) → Pic(O_{F,S})

by sending φ to O_{F,S}^{φ}. It is easy to check from the definitions that this is a homomorphism, using the fact that the definition is independent of the choice of splitting field.

Theorem 1.4. There is a natural exact sequence

1 → O_{F,S}^{∗}/O_{F,S}^{∗N} → Hom(G_{F,S}^{ab}, μ_{N}) → Pic(O_{F,S})[N] → 1

where the maps are as defined above.

Proof. We already know that the first map is injective. Next we show that the composition of the maps is zero. So let u be in O_{F,S}^{∗} and let ξ be an N^{th} root. We wish to show that O_{F,S}^{φ_{u}} is isomorphic to O_{F,S}. Set L = F (ξ) (which is indeed abelian and unramified outside of S since μ_{n} ⊆ F^{∗} and S contains the places dividing N) and note that

O_{F,S}^{φ_{u}} = {x ∈ O_{L,T} | g(x) = φ_{u}(g)x for all g ∈ G}

= \{x ∈ O_{L,T} | g(x) = \frac{g(ξ)}{ξ}x for all g ∈ G\}

= \{x ∈ O_{L,T} | g (\frac{x}{ξ}) = \frac{x}{ξ} for all g ∈ G\}.

From this one sees easily that the map

O_{F,S}^{φ} → O_{F,S}

given by multiplication by ξ^{−1} is an isomorphism.

To show that the last map is surjective we use the interpretation of Pic(O_{F,S}) in terms of fractional ideals. So let I be a fractional ideal such that I^{N} is trivial. Thus I^{N} = αO_{F,S} for some α ∈ F^{∗}. Choose an N^{th} root ξ of α and set L = F (ξ). Then L/F is unramified outside of S since α is the N^{th} power of a fractional ideal of O_{F,S}. (This criterion for ramification becomes quite transparent when viewed locally.) Define

φ : G_{F,S}^{ab} → μ_{N}

by

φ(g) = \frac{g(ξ)}{ξ}.

Then φ factors through Gal(L/K). We claim that O_{F,S}^{φ} ≅ I. To see this, first do a computation as above to show that

O_{F,S}^{φ} = ξO_{F,S}.

To check that ξO_{F,S} and I are isomorphic it is enough to check that they agree as fractional ideals of L, since L/K is unramified outside of S. But this is clear, which establishes surjectivity.

Finally, let φ : G_{F,S}^{ab} → μ_{N} be such that O_{F,S}^{φ} is trivial in Pic(O_{F,S})[N]. Thus there is an isomorphism

O_{F,S}^{φ} ≅ O_{F,S}.

Let ξ be the element of O_{F,S}^{φ} corresponding to 1 ∈ O_{F,S} under this isomorphism. One easily checks that φ is given simply by

φ(g) = \frac{g(ξ)}{ξ},

and that O_{F,S}^{φ} = ξO_{F,S}. Now, ξ may not be a unit, but we note that the fractional ideal of O_{F,S} it defines is trivial by hypothesis. Thus we can modify ξ by an element of O_{F,S} to get a unit ξ′ of O_{L,T}. Now (ξ′)^{N} ∈ O_{F,S}^{∗} and ξ′ yields the same homomorphism φ, which completes the proof of exactness.

2. Cohomology of Abelian Varieties

2.1. Lang’s Theorem. Recall that if A is an abelian variety over a field K, we have an exact sequence of G_{K}-modules

0 → A[N] → A(K̄) ⟶^{N} A(K̄) → 0,

where N is any positive integer and A[N] is the N-torsion in A(K̄). The long exact sequence in G_{K}-cohomology yields an exact sequence

0 → A(K)/NA(K) → H^{1}(G_{K}, A[N]) → H^{1}(G_{K}, A(K̄))[N] → 0.

In order to understand this exact sequence it is important to gain more control over H^{1}(G_{K}, A(K̄)). An important result in this direction is Lang’s theorem.

Theorem 2.1 (Lang). Let A be a smooth, connected, commutative algebraic group over a finite field k. Then

H^{1}(𝔤_{k}, A(\overline{k})) = 0.

Proof. We sketch the proof, the underlying idea being to “think geometrically”. For details see [Car, p. 32] or [PR, Chapter 6, Section 2].

Recall that we have evaluated the cohomology group H^{1}(𝔤_{k}, A(\overline{k})) as

H^{1}(𝔤_{k}, A(\overline{k})) ≅ A(\overline{k})/(Fr −1)A(\overline{k}),

where Fr is the Frobenius automorphism x ↦ x^{q}, q = |k|. So we want to show that

Fr −1 : A(\overline{k}) → A(\overline{k})

is surjective. To do this we want to interpret Fr geometrically and then see that Fr −1 is surjective.

We begin by considering the category of commutative k-algebras. If R is such an algebra, then there is a natural k-linear automorphism of R

Fr : R → R

given by x ↦ x^{q}. Under the natural equivalence of categories between k-algebras and affine schemes over Spec k this yields a Spec k-automorphism Fr for Spec R.

Let X now be an arbitrary scheme over Spec K. Let U = Spec R and V = Spec S be two affine open subsets of X. In order to show that the automorphisms Fr glue to an automorphism of X it will be enough to show that they agree for any W = Spec R_{r} = Spec S_{s}, r ∈ R, s ∈ S. But this is clear, since Fr is given by the same map in both instances. Thus we obtain a natural automorphism Fr on any Spec k-scheme X. (See [Ha, Chapter 4, Section 2, p. 301] for a slightly different approach to defining Fr. Note that in our situation we simply have X_{p} = X.)

The first key point is to observe that Fr induces the zero map on differentials. The essential reason for this is that on the level of differentials, the map induced by Fr involves multiplication by q, which kills any k-algebra.

Now, let A be an algebraic group. Consider the morphism A → A which “sends x ∈ A to x^{−1} Fr(x).” (We leave it to the reader to write this down algebraically in terms of the multiplication, inversion and Frobenius morphisms.) We will simply write this morphism as Fr −1. One now shows that the induced map on differentials is just the sum of the maps induced by Fr and by −1; since Fr induces the zero map and −1 induces isomorphisms, Fr −1 induces isomorphisms on differentials. Since A is smooth, [Ha, Chapter 3, Section 10, Proposition 10.4] now implies that Fr −1 is étale . Thus, by [Mi-EC, Chapter 1, Section 2, Theorem 2.12], it has open image.

Now, further suppose that A is connected. Choose any a ∈ A(\overline{k}) and consider the map ψ_{a} sending x ∈ A(\overline{k}) to x^{−1}a Fr(x). This is just the translation of Fr −1 by a, and thus also has open image. In particular, the image of ψ_{a} intersects that of Fr −1, since any two opens on a connected variety intersect. Thus there must be x_{1}, x_{2} ∈ A(\overline{k}) such that

(Fr −1)(x_{1}) = ψ_{a}(x_{2})

and a little algebra now shows that

(Fr −1)(x_{1}x_{2}^{−1}) = a.

Thus Fr −1 is surjective, as claimed.

Corollary 2.2. If A is an abelian variety over a finite field k, then

A(k)/NA(k) ≅ H^{1}(𝔤_{k}, A[N])

for any integer N.

Proof. This follows immediately from Theorem 2.1 and the Kummer sequence.

2.2. Abelian Schemes over Local Fields. Let S be a scheme. An abelian scheme over S is a proper, smooth group scheme over S with connected geometric fibers. (As usual, see [Sh, Section 2] for the relevant diagrams.) In particular, for every S scheme X, the set

A(X) =^{def} Hom_{S}(X, A)

of X-valued points of A is a group; in fact, it turns out that the group structure must be commutative. (See [Mi-AV, Section 20, Corollary 20.2]. If T = Spec R is affine we will often just write A(R) for A(T).) Note first that if T is an S-scheme, then

A(T) = (A ×_{S} T)(T),

as follows immediately from the definition of the fiber product. Also, in the case that S is the spectrum of a field K, then an abelian scheme over S is the same as an abelian variety over S; thus any such abelian scheme is necessarily projective, and under any projective embedding it is easy to check that the above notion of Spec K-valued points corresponds with the usual notion of points in projective space.

Let K be a local field with ring of integers O_{K} and residue field k. Let A be abelian variety over Spec K. We will say that A has good reduction if it extends to an abelian scheme over Spec O_{K}; that is, if there is some abelian scheme A′ over Spec O_{K} such that

A′ ×_{Spec O_{K}} Spec K ≅ A.

A fundamental result for abelian varieties with good reduction is that all Spec O_{K}- valued points extend to Spec K-valued points. Intuitively the idea is that any point in projective space over K can be “scaled” to have at least one coordinate integral. We give a “coordinate-free” proof instead.

Proposition 2.3. Let A be an abelian scheme over O_{K}. Then the natural map A(O_{K}) → A(K) is a bijection.

Proof. The valuative criterion for properness ([Ha, Chapter 2, Theorem 4.7]) tells us that given any diagram

\xymatrix{Spec K \ar[r] \ar[d] & A \ar[d] \\ Spec O_{K} \ar[r] & Spec O_{K}}

there is a unique map Spec O_{K} → A making the diagram commute. Phrased differently, the natural map A(O_{K}) → A(K) is an isomorphism, as required.

We can use the above result to gain more control over the structure of the K-valued points of an abelian variety. So let A be an abelian scheme over Spec O_{K} and let 𝔪 be the maximal ideal of O_{K}. Then we have the equality

A(O_{K}) = \underleftarrow{lim}_{ν} A(O_{K}/𝔪^{ν}).

(In fact, this is true for any Spec O_{K}-scheme X. In the affine case it follows immediately from the universal property of inverse limits and the general case follows from the affine case.) Thus A(O_{K}) is a profinite group; in fact, it is a p-adic Lie group. Define a subgroup A_{1} of A(O_{K}) to be the kernel of the natural reduction map

A(O_{K}) → A(k),

so that there is an exact sequences

0 → A_{1} → A(O_{K}) → A(k) → 0.

(The last map is surjective since O_{K} → k is formally smooth.) It is a standard fact for abelian varieties that A_{1} can be expressed as the 𝔪-valued points of a certain formal group. (See [Si-AEC, Chapter 4, Section 1] for the case of elliptic curves and [Co2] for the general case.)

Now, suppose that N is prime to the characteristic of k. Then the map

A_{1} ⟶^{N} A_{1}

is an isomorphism by [Si-AEC, Chapter 4, Proposition 2.3], so there is an exact commutative diagram

\xymatrix{0 \ar[r] & A_{1} \ar[r] \ar[d]^{N} & A(O_{K}) \ar[r] \ar[d]^{N} & A(k) \ar[r] \ar[d]^{N} & 0 \\ 0 \ar[r] & A_{1} \ar[r] \ar[d] & A(O_{K}) \ar[r] \ar[d] & A(k) \ar[r] \ar[d] & 0 \\  & 0 \ar[r] & A(O_{K})/NA(O_{K}) \ar[r] & A(k)/NA(k) \ar[r] & 0}

Combined with Theorem 2.1 and Proposition 2.3 this gives us isomorphisms

A(K)/NA(K) ≅ A(O_{K})/NA(O_{K}) ≅ A(k)/NA(k) ≅ H^{1}(𝔤_{k}, A[N]).

Now consider the above commutative diagram with K replaced by arbitrary finite extensions L. If L is taken large enough to contain all of A[N](K̄), then the kernels of the diagram yield an isomorphism

A[N](K̄) ≅ A[N](\overline{k}).

In fact, this isomorphism is G_{K}-equivariant, as is easy to check.

2.3. Local Abelian Variety Structures. Let A be an abelian variety over a local field K and suppose that A has good reduction. Let N be an integer prime to the residue characteristic of K. We wish to compare the two exact sequences

0 → A(K)/NA(K) → H^{1}(G_{K}, A[N]) → H^{1}(G_{K}, A(K̄))[N] → 0

and

0 → H^{1}(𝔤_{k}, A[N]) → H^{1}(G_{K}, A[N]) → H^{1}(I_{K}, A[N])^{𝔤_{k}} → 0.

In the previous section we defined an isomorphism between the first terms in the exact sequences. We claim that this isomorphism lies in a commutative diagram

[figure: 0; A(K)/NA(K); H^{1}(G_{K}, M); H^{1}(G_{K}, A(K̄))[N]; 0; 0; H^{1}(𝔤_{k}, A[N]); H^{1}(G_{K}, M); H^{1}(I_{K}, A[N])^{𝔤_{k}}; 0]

The proof of this is quite easy if A has been embedded in projective space over K in such a way that it “reduces” to an abelian variety over k; the main step becomes identifying the map A(K) → A(k) with the usual “scaling and reduction” map. (See [Si-AEC, Chapter 7, Section 2].) We instead again present a “coordinatefree” proof.

Since A has good reduction it extends to an abelian scheme over Spec O_{K} which we will also denote by A. Note that the group of K̄-valued points A(K̄) is naturally a G_{K}-module even when thought of as Hom_{Spec O_{K}} (Spec K̄, A), since any g ∈ G_{K} induces a Spec K-automorphism

Spec K̄ ⟶^{g} Spec K̄.

Any choice of projective embedding of A will identify this G_{K}-module with the usual G_{K}-module of K-valued points. Similarly A(k) has a natural structure of 𝔤_{k}-module. Considering A[N] as a group scheme over Spec O_{K} we can realize the G_{K}-module A[N](K̄) as the K̄-valued points of the group scheme A[N].

The key to the coordinate-free proof of the commutativity of the above diagram is the following commutative diagram which comes immediately from the fact that A is a group scheme over Spec O_{K} (here L is any finite extension of K, O_{L} is its ring of integers and ℓ is its residue field):

\xymatrix{A(L) × A(L) \ar[r] & A(L) \\ A(O_{L}) × A(O_{L}) \ar[r] \ar[u]^{≅} \ar[d] & A(O_{L}) \ar[u]^{≅} \ar[d] \\ A(ℓ) × A(ℓ) \ar[r] & A(ℓ)}

Here the vertical maps are the natural maps induced by Spec L → Spec O_{L} and Spec ℓ → Spec O_{L} and the horizontal maps are multiplication; the indicated maps are isomorphisms by Proposition 2.3. Passing to the limit over L and the induced multiplication-by-N diagram, we obtain a commutative diagram

\xymatrix{A(K̄) \ar[r]^{N} \ar[d] & A(K̄) \ar[d] \\ A(\overline{k}) \ar[r]^{N} & A(\overline{k})}

Now, note that in order to establish the desired commutativity it will be enough to establish it for the diagram

[figure: A(K); H^{1}(G_{K}, A[N](K̄)); A(k); H^{1}(𝔤_{k}, A[N](\overline{k})); inf; H^{1}(G_{K}, A[N](K̄))]

since the map of cokernels is simply the one induced by the first two vertical maps. (Here the map A(K) → A(k) is the reduction map and the other two unlabeled maps are the boundary maps from the Kummer exact sequence.) We recall the definition of the Kummer maps: given P ∈ A(K), pick a Q ∈ A(K̄) such that N · Q = P. Then the image of P in H^{1}(G_{K}, A[N]) is the cocycle which sends any g ∈ G_{K} to Q^{g} − Q ∈ A[N]. The definition is similar for A(k).

So let P be in A(K) and pick Q ∈ A(K̄) as above. By the commutative diagram for multiplication by N, we find that N · Q̄ = P̄, where P̄ and Q̄ are the images of the respective points in A(k). Thus P maps to the G_{K}-cocycle

g ↦ Q^{g} − Q

and P̄ maps to the 𝔤_{k}-cocycle

g ↦ Q̄^{g} − Q̄ = \overline{Q^{g} − Q}.

To complete the proof we now simply have to use the fact that A[N](K̄) ≅ A[N](\overline{k}) as G_{K}-modules. This shows that the two cocycles become the same after inflation, so the diagram really does commute.

2.4. Global Abelian Variety Structures. Now, let A be an abelian variety defined over a global field F. Let N be an integer and let M = A[N] be the G_{F} - module of N-torsion points. A fundamental fact about such an A is that it has good reduction at almost all places v of F; that is, the base change

A_{v} = A ×_{Spec F} Spec F_{v}

has good reduction for almost all v.

We sketch the proof. For a few more details see [Mi-AV, Section 20, Remark 20.9], or for more many more details and a “coordinate-free” approach see [Co]. We need to show that there is some finite set S of places such that A extends to an abelian scheme over Spec O_{F,S}. Now, recall that any abelian variety is projective. Fix an embedding of A into projective space. Here A is simply the zero set of some finite set of homogeneous polynomials. Since all that matters are the zeros of the polynomials, we can scale them so as to assume that all of the coefficients are integral. Let A′ be the projective scheme the equations define over Spec O_{F}. A′ is automatically proper and is easily seen to be flat, and its generic fiber is A. However, A′ need not yet be smooth, a group scheme or have geometrically connected fibers.

By [Ha, Chapter 3, Section 10, Theorem 10.2], to get smoothness it is enough to remove all primes which have nonsingular fibers lying over them. A fiber is nonsingular if and only if the Jacobian determinant vanishes modulo the corresponding prime. Since the fibers are all defined by the same equations and by assumption the Jacobian does not vanish in K, it can only vanish for finitely many fibers. Thus we need only remove a finite number of primes to get smoothness.

Next consider the multiplication and inversion morphisms of A. These are given by ratios of homogeneous polynomials with coefficients which we can take to be in O_{K}, and thus will be regular for almost all primes. Thus by removing a finite number of primes we can get multiplication and inversion morphisms for A′ over O_{F,S}, where S is the (finite set) of discarded primes.

Lastly, then, we must consider the geometrically connected fiber condition. Here one shows that the existence of the (group) identity section shows that connected implies geometrically connected, so that it will be enough to establish that the fibers are connected. Here one again uses the fact that the generic fiber is connected to show that almost all of the fibers are connected. See [ST, Section 1, Lemma 3] for the details.

Finally, then, we have constructed an abelian scheme A′ over Spec O_{F,S} (for some finite set of primes S) with generic fiber A. Base changing A′ to the appropriate completion for v ∉ S shows that A has good reduction for v ∉ S, and thus that A has good reduction almost everywhere.

Recall now that we defined the abelian variety structure on M by setting

H_{f}^{1}(G_{v}, M) = im(A(F_{v})/NA(F_{v}) ↪ H^{1}(G_{v}, A[N]))

under the Kummer map. We now immediately see that this really is a finite/singular structure; this follows from the computations of the previous section, which apply whenever A has good reduction at v and v does not divide N. In particular, this completes the proofs of all of the results of Lecture 9, Section 2.2.

Note, however, that the abelian variety structure on M depends on the particular abelian variety A and the isomorphism A[N] ≅ M and thus is not an invariant of M as a G_{K}-module.