CHAPTER 12 Lecture 12

1. Torsors for Algebraic Groups

1.1. Algebraic Geometric Torsors. Fix a field K. If V is any variety over K we will write \overline{V} for V ×_{Spec K} Spec K_{s}, where K_{s} is the separable closure of K. Let Γ be an algebraic group over K, with multiplication morphism μ:

Γ ×_{Spec K} Γ ⟶^{μ} Γ.

We wish to combine our notion of Γ-torsor from Lecture 5, Section 2.3 with the ideas from Lecture 11. We define an algebraic geometric torsor (AG torsor for short) for Γ to be an algebraic variety X defined over K, together with a morphism

Γ ×_{Spec K} X ⟶^{α} X,

which becomes isomorphic to the standard torsor (Γ, μ) over K_{s}; that is, there is an isomorphism

u : \overline{Γ} → \overline{X}

fitting into a commutative diagram

\xymatrix{\overline{Γ} ×_{Spec K_{s}} \overline{Γ} \ar[r]^{μ} \ar[d]^{1×u} & \overline{Γ} \ar[d] \\ \overline{Γ} ×_{Spec K_{s}} \overline{X} \ar[r]^{α} & \overline{X}}

We will say that X is trivial as an AG torsor for Γ is there is such a u which is definable over K. In fact, this happens precisely when X has a point defined over K; choosing this point to be the identity and using α to define multiplication easily establishes this. Note that in any case u is defined by finite data, so it must actually be defined over some finite extension L of K. Thus any AG torsor becomes trivial over some finite extension of K.

For any extension L/K let us define E(L/K, (Γ, μ)) to be the pointed set of AG torsors for Γ which become trivial over L. Note that there is a natural mapping

E(L/K, (Γ, μ)) → E(L/K, Γ),

where E(L/K, Γ) is simply the set of L/K forms for Γ thought of as an algebraic variety without its group structure. This map is easily seen to be surjective, but it need not be injective; we will see an example of this later in the lecture.

Recall that we constructed an injection

E(L/K, Γ) ↪ H^{1}(Gal(L/K), \underline{Aut}_{L}(Γ)),

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where \underline{Aut}_{L}(Γ) is the group of automorphisms of Γ as an algebraic variety over L. Composing with the forgetful map above this defines a map

E(L/K, (Γ, μ)) → H^{1}(Gal(L/K), \underline{Aut}_{L}(Γ)).

This map is not terribly useful, however, as it is not even an injection. We wish to define a similar map for E(L/K, (Γ, μ)) which is somewhat more germane to our situation.

Proposition 1.1. Let (Γ, μ) be an algebraic group defined over a field K. Then for any L/K there is a natural injection of pointed sets

E(L/K, (Γ, μ)) ⟶^{i} H^{1}(Gal(L/K), Γ(L)),

where Γ(L) is the group of L-valued points of Γ.

It is clear from the proof of the above proposition that our various maps fit into a commutative diagram

\xymatrix{E(L/K, (Γ, μ)) \ar[r] \ar[d]^{i} & E(L/K, Γ) \ar[d]^{ι} \\ H^{1}(Gal(L/K), Γ(L)) \ar[r] & H^{1}(Gal(L/K), \underline{Aut}_{L}(Γ))}

where the bottom map is induced from the natural Gal(L/K)-module injection

Γ(L) → \underline{Aut}_{L}(Γ)

where elements of Γ_{L} are viewed as translations.

We will say that Γ/K satisfies the descent propety for torsors if the map of Proposition 1.1 is an isomorphism.

1.2. Torsors for Abelian Varieties. Let K be a field and let A be an abelian variety defined over K. We have the following fundamental theorem.

Theorem 1.2. An abelian variety A over a field K has the descent property for torsors.

Proof. The proof of this is quite difficult. For the case of elliptic curves, see [Si-AEC, Chapter 10, Theorem 3.6].

It follows from this and Proposition 1.1 that for any finite L/K we have an identification

E(L/K, (A, μ)) ≅ H^{1}(Gal(L/K), A(L)).

Since A(L) is abelian, H^{1}(G_{K}, A(L)) is an abelian group. The induced group structure on E(L/K, (A, μ)) is in fact just the Baer sum; see Lecture 5, Section 2.2. However, it is fairly difficult to see that the necessary quotient operations are really defined.

It is much easier to describe the involution induced on E(L/K, (A, μ)) by inversion in H^{1}(G_{K}, A(L)). This is done as follows: let ι : A → A be the inversion morphism on A and let (X, α) be an algebraic geometric torsor for A. Then the inverse of X, say X^{∼}, is simply X considered as a torsor with A-action α ∘ (ι × 1). Note, however, that X^{∼} and X give exactly the same element of E(L/K, A), since ι is defined over K. Thus this gives an example of a case in which the forgetful map

E(L/K, (A, μ)) → E(L/K, A)

is not injective. More generally elements of Aut_{K}(Γ) which are not translations can be used to manufacture such examples.

1.3. The Shafarevich-Tate Group. Let F be a number field. We will now relate torsors to the Shafarevich-Tate group � (A/F) defined in Lecture 9, Section 2.1. More generally, we let S be any finite set of places of F and we define the pointed set

� _{S}(A/F)

to be the set of isomorphism classes of AG torsors for A which are trivial when viewed over the local fields F_{v} for v ∉ S. Thus a non-trivial element in � _{S}(A/F) is a torsor for A which has an F_{v}-rational point for every v ∉ S but has no F -rational point.

Using Theorem 1.2 we have the following immediate cohomological interpretation of � _{S}(A/F).

Proposition 1.3. There is an exact sequence

0 → � _{S}(A/F) → H^{1}(G_{F} , A(F̄)) ⟶^{res} ∏_{v∉S} H^{1}(G_{v}, A(F̄_{v})).

In particular, � _{∅}(A/F) can be identified with the Shafarevich-Tate group � (A/F).

Proof. All that is necessary is, for every L/F, to identify the map

E(L/F, (A, μ)) → E(L_{w}/F_{v}, (A ×_{Spec F} Spec F_{v}))

with the restriction map in cohomology, but this follows easily from the defintions of the isomorphisms of Theorem 1.2.

Proposition 1.3 gives a geometric interpretation of the Shafarevich-Tate group; however, it is still extremely difficult (both in theory and in practice) to compute it. As an example of the results of one such computation we quote the following theorem which follows from careful consideration of the work of Kolyvagin and Rubin.

Theorem 1.4. Let E be the elliptic curve over ℚ defined by the equation

x^{3} + y^{3} + 60z^{3} = 0,

with origin (−1, 1, 0). Then

� (E/ℚ) ≅ ℤ/3ℤ × ℤ/3ℤ.

The non-trivial elements of � (E/ℚ) are represented by

Σ_{2} : 3x^{3} + 4y^{3} + 5z^{3} = 0;
Σ_{3} : 12x^{3} + y^{3} + 5z^{3} = 0;
Σ_{4} : 15x^{3} + 4y^{3} + z^{3} = 0;
Σ_{5} : 3x^{3} + 20y^{3} + z^{3} = 0;

together with the inverses Σ_{i}^{∼}.

The proof of this theorem is quite difficult; we will eventually prove most of it in this course.

The group � (A/F) can be interpreted as an “obstruction” to the Hasse principle. Recall that the Hasse-Minkowski theorem (see [Se-CA, Chapter 4]) states that a quadric curve has a point in ℚ if and only if it has a point in each ℚ_{p} and in ℝ. Unfortunately, this results fails for curves of higher genus; this is reflected in the existence of non-trivial elements of � (A/F). The curve Σ_{2} is a famous example of Selmer, which he used to demonstrate the explicit failure of the Hasse principle for cubic curves.