CHAPTER 9 Lecture 9

1. Abelian Varieties

We will refer to the two articles [Ro] and [Mi-AV] in [CS] for most of the results in this section. Both articles refer to [Mu] for most proofs, but since we feel that attempting to read [Mu] is a much more demanding activity than reading the summaries in [CS] we will simply refer to those and let them refer the reader to the appropriate part of [Mu].

1.1. Complex Tori. A complex torus T is a compact, connected commutative Lie group over ℂ. Let g be its dimension, and let 𝔱 be the tangent space to T at the origin; we have 𝔱 ≅ ℂ^{g}. The exponential map

exp : 𝔱 → T

in this situation is surjective, and since T is compact and connected the kernel Λ must be a a discrete subgroup of 𝔱 ≅ ℂ^{g} of maximal rank; that is, Λ is a lattice. In this way we get a complex analytic group isomorphism

T ≅ 𝔱/Λ ≅ ℂ^{g}/Λ.

(See [Ro, Section 1] for details.)

Let

m : T → T

be the multiplication by m map; it is a homomorphism since T is commutative. In fact, the identification T ≅ ℂ^{g}/Λ makes it clear that it is surjective, so we have an exact sequence

0 → T [m] → T ⟶^{m} T → 0,

where T [m] is the m-torsion in T. Considering T once again as ℂ^{g}/Λ, we find that

T [m] ≅ \frac{1}{m}Λ/Λ ≅ (ℤ/mℤ)^{2g},

the last isomorphism being non-canonical.

1.2. Abelian Varieties. An abelian variety over a field L is a proper, geometrically connected variety A defined over L together with a point O ∈ A(L), a multiplication morphism

μ : A × A → A

and an inversion morphism

ι : A → A,

both defined over L, making A into a group object in the category of proper varieties over L, with O as the identity element. (All this means is that the appropriate diagrams expressing associativity, inverses and the identity all commute. See [Sh, Section 2] for the diagrams.)

0Last modified March 7, 1998

There are some rather remarkable consequences of this definition. First, A is nonsingular, since any variety is nonsingular on a non-empty open set and μ can be used to translate this set over all of A. Second, A is actually projective; see [Mi-AV, Section 7]. Lastly, the group law must be commutative; see [Mi-AV, Section 2]. Thus for any field L′ ⊇ L the L′-valued points A(L′) form an abelian group.

Now, suppose that L is a subfield of ℂ. Then A(ℂ) is a compact, connected (since we assumed that A was geometrically connected), commutative Lie group over ℂ; that is, A(ℂ) is a complex torus. (Not every complex torus can be given the structure of an algebraic variety; see [Ro, Section 3].) In particular, we have an exact sequence

0 → A(ℂ)[m] → A(ℂ) ⟶^{m} A(ℂ) → 0

as above. Recall that A(ℂ)[m] ≅ (ℤ/mℤ)^{2g}, where g is the dimension of A.

Lemma 1.1. Every point of A(ℂ)[m] actually lies in A(L′) for some finite extension L′ of L.

Proof. Let P be an m-torsion point over ℂ. Since μ and thus m : A → A is defined over L,

m(P^{σ}) = m(P)^{σ} = 0

for any σ ∈ Gal(ℂ/L). In particular, all of the conjugate points to P are also in A(ℂ)[m]. Thus P has only finitely many conjugates over L, which implies that it is defined over a finite extension of L.

In particular, A(ℂ)[m] = A(L̄)[m]. This is not exactly what one would like; the optimal situation would be to define a projective algebraic group A[m] over L by

A[m] = A ×_{A} 0,

where the map A → A is multiplication by m and 0 is the trivial abelian variety over L. (Intuitively this is the correct notion, since a point in A ×_{A} 0 is an ordered pair (a, 0) where a maps to 0 under multiplication by m.) One then would want to show that the morphism A ⟶^{m} A is surjective with finite kernel (such a map is called an isogeny), so that A[m] is a finite projective algebraic group over L. (Recall that a map f : A → B of abelian varieties over L is surjective if it satisfies any of the following equivalent conditions:

1. f is surjective as a map of topological spaces;

2. f : A(L̄) → B(L̄) is surjective as a map of abelian groups;

3. the induced map between the étale sheaves defined by A and B is surjective.) This program can actually be carried out over an arbitrary field L; one finds

that multiplication by m is an isogeny of degree m^{2g} and that it is étale if and only if the characteristic of L does not divide m. In particular, the proof of Lemma 1.1 generalizes to show that A[m] has all points in L̄, and that

A[m](L̄) ≅ (ℤ/mℤ)^{2g},

assuming that the characteristic of L does not divide m. See [Mi-AV, Section 8] for more details.

The usefulness of our above results is that we have obtained a well-behaved Gal(L̄/L)-module A[m](L̄); in fact, by Lemma 1.1 it is actually discrete.

2. Selmer Groups of Abelian Varieties

2.1. Definitions. Let F be a global field, let A be an abelian variety defined over F and let m be an integer. Since multiplication by m is surjective, we have an exact sequence

0 → A[m](F̄) → A(F̄) ⟶^{m} A(F̄) → 0.

We saw in the previous section that A[m](F̄) is a finite, discrete G_{F} -module, so we can apply all of the theory we have developed to it. We will write it (only slightly ambiguously) just as A[m].

Consider the portion of the long exact cohomology sequence

A(F) ⟶^{m} A(F) → H^{1}(G_{F} , A[m]) → H^{1}(G_{F} , A) ⟶^{m} H^{1}(G_{F} , A)

We can reduce this to a short exact sequence

0 → A(F)/mA(F) → H^{1}(G_{F} , A[m]) → H^{1}(G_{F} , A)[m] → 0.

If we are interested in the arithmetic of A(F), this exact sequence is a good start; we have expressed a quotient of A(F) as a subgroup of a cohomology group. Unfortunately, both of the cohomology groups in the exact sequence are quite large. We will use local considerations to replace them by much smaller subgroups.

To do this, we consider the composition

H^{1}(G_{F} , A[m]) → H^{1}(G_{F} , A(F̄)) → ∏_{v} H^{1}(G_{v}, A(F̄_{v})),

where the last map is the product of the restriction maps over all places v of F. The groups we are looking for are the kernel of the second map and the kernel of the composition.

Definition 5. The m^{th} Selmer group S_{m}(A/F) is defined by

S_{m}(A/F) = ker (H^{1}(G_{F} , A[m]) → ∏_{v} H^{1}(G_{v}, A(F̄_{v}))) ,

the map being the composition above. The Shafarevich-Tate group � (A/F) is defined by

� (A/F) = ker (H^{1}(G_{F} , A(F̄)) → ∏_{v} H^{1}(G_{v}, A(F̄_{v}))) .

By their definitions, S_{m}(A/F) and � (A/F) fit into an exact commutative diagram

[figure: 0; 0; 0; 0; A(F)/mA(F); S_{m}(A/F); � (A/F); m; � (A/F); 0; A(F)/mA(F); H^{1}(G_{F} , A[m]); H^{1}(G_{F} , A(F̄)); m; H^{1}(G_{F} , A(F̄)); ∏_{v} H^{1}(G_{v}, A(F̄_{v})); m; ∏_{v} H^{1}(G_{v}, A(F̄))]

Exactness of the top row follows from a simple diagram chase. Extracting that row, we obtain the fundamental exact sequence

0 → A(F)/mA(F) → S_{m}(A/F) → � (A/F)[m] → 0.

2.2. The Abelian Variety Structure on A[m]. The next step is to endow A[m] with a suitable finite/singular structure (in the sense of Lecture 7, Section 1.2) so that we can apply our general theory. In fact, by choosing all of the local structures correctly we will identify H_{f}^{1}(G_{F} , A[m]) with the m^{th} Selmer group S_{m}(A/F); Theorem 3.3 of Lecture 8 will then have immediate important arithmetic consequences.

To define this structure, we want to extend the bottom row of our above commutative diagram by using the local versions

0 → A(F_{v})/mA(F_{v}) → H^{1}(G_{v}, A[m]) → H^{1}(G_{v}, A)[m] → 0

of our original exact sequence

0 → A(F)/mA(F) → H^{1}(G_{F} , A[m]) → H^{1}(G_{F} , A)[m] → 0.

(Note that the A[m] notation is not really ambiguous here, since we have chosen an embedding F̄ ⊆ F̄_{v}; Lemma 1.1 then shows that A[m](F̄_{v}) = A[m](F̄).) We thus obtain a commutative diagram

[figure: 0; 0; 0; 0; A(F)/mA(F); S_{m}(A/F); � (A/F); m; � (A/F); 0; A(F)/mA(F); H^{1}(G_{F} , A[m]); H^{1}(G_{F} , A(F̄)); m; H^{1}(G_{F} , A(F̄)); 0; ∏_{v} A(F_{v})/mA(F_{v}); ∏_{v} H^{1}(G_{v}, A[m]); ∏_{v} H^{1}(G_{v}, A(F̄_{v})); m; ∏_{v} H^{1}(G_{v}, A(F̄_{v}))]

with exact rows, although only the last two columns are exact. After a little bit of comtemplation of the above diagram and the definition of S_{m}(A/F) it becomes clear what the appropriate finite/singular structure should be.

Definition 6. We define the abelian variety structure on A[m] to be the finite/singular structure σ defined by the local exact sequences

σ_{v} : 0 → A(F_{v})/mA(F_{v}) → H^{1}(G_{v}, A[m]) → H^{1}(G_{v}, A)[m] → 0;

that is, we define

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

and

H_{s}^{1}(G_{v}, A[m]) = im(H^{1}(G_{v}, A[m]) → H^{1}(G_{v}, A)).

The exact sequence

0 → A(F_{v})/mA(F_{v}) → H^{1}(G_{v}, A[m]) → H^{1}(G_{v}, A)

shows that these definitions are compatible. It is not immediately clear, however, that σ is really a finite/singular structure. We will prove this in the next lecture; for the time being we derive some consequences.

Proposition 2.1. If F is a number field, A/F is an abelian variety and A[m] is endowed with the abelian variety structure σ, then

H_{f}^{1}(G_{F} , M_{σ}) = S_{m}(A/F).

Proof. This is just an easy diagram/definition chase in the above diagram.

Corollary 2.2. The m^{th} Selmer group S_{m}(A/F) is finite.

Proof. This follows immediately from Proposition 2.1 and Theorem 3.3 of Lecture 8.

Corollary 2.3 (The Weak Mordell-Weil Theorem). If A is abelian variety and F is a number field, then A(F)/mA(F) is finite for any integer m.

The full Mordell-Weil theorem states that A(F) is a finitely generated abelian group; to derive this from Corollary 2.3 one must use the theory of height functions. (See [Si-AEC, Chapter 8, Sections 5 and 6] or [Si-H].)

Corollary 2.4. The m-torsion in � (A/F) is finite for any m.

S_{m}(A/F) itself is effectively computable in many cases; see [Si-AEC, Chapter 10, Section 1] for the case of elliptic curves with 2-torsion. More recent work of Schaffer and Grant has given good algorithms for computing it for abelian varieties of dimension 1 or 2, independent of any other hypotheses.

To relate S_{m}(A/F) to A(F)/mA(F) requires knowledge of � (A/F), however, and this group has proved much more resistant to computation. Note that Corollary 2.4 does not imply that � (A/F) itself is finite. That � (A/F) is finite is a standard conjecture; it was not known for a single abelian variety over a number field until the work of Karl Rubin in 1986. He showed that it was finite for elliptic curves over ℚ with complex multiplication and analytic rank 0 or 1. (See [Ru-TS].) It is now known, using the work of Kolyvagin, Gross-Zagier, Murty-Murty and Bump-Friedberg-Hoffstein, for all modular elliptic curves over ℚ of analytic rank 0 or 1. (See [Ko], [GZ], [MM], [BFH].)

Note that knowledge of A(F)_{tors} and A(F)/mA(F) for a single m > 1 determines the abelian group structure of A(F). Alternately, knowledge of A(F)/mA(F) for a few values of m would suffice. Nevertheless, we know very little about the rank of A(F) in general. For example, although it is relatively easy to construct infinite families of abelian varieties over ℚ of rank r and dimension g such that

\frac{r}{g} ≥ 1

for all sufficiently large g, it is not known how to extend these constructions to give

\frac{r}{g} ≥ 1 + ε

for any ε.