1. Automorphism groups, Isomorphism sets, and L/K-forms. Recall that if V is an algebraic variety over a field K then Aut_{K}(V) is the group of automorphisms (which are defined over K) of the K-variety V. If L/K is a Galois extension, with Galois group G := Gal(L/K), then Aut_{L}(V) has a natural G-action, i.e., there is a natural homomorphism of G into the group of group-automorphisms of Aut_{L}(V) (defined in the evident way).

By a G-group let us mean a group A together with a G-action in it.

By \underline{Aut}_{L}(V) let us mean the G-group Aut_{L}(V); i.e., it is the group Aut_{L}(V) taken together with its G-action.

If W is another K-variety, let Isom_{K}(W, V) be the set of K-isomorphisms from W to V, noting that it is either empty or else is a set-theoretic torsor with respect to the group Aut_{K}(V) (the operation being composition). There is also a natural G-action on Isom_{K}(W, V) By \underline{Isom}_{L}(W, V) let us mean the G-set Isom_{L}(W, V); i.e., it is the set Isom_{L}(W, V) taken together with its G-action.

So, if we assume Isom_{L}(W, V) to be nonempty, then \underline{Isom}_{L}(W, V) is a torsor with respect to \underline{Aut}_{L}(V) and the G-action on these respect the torsor structure; i.e., we have the formula

γ^{g}α^{g} = (γ · α)^{g},

for all g ∈ G, γ ∈ Isom_{L}(W, V), and α ∈ Aut_{L}(V).

When we have a G-set X, which is a (set-theoretic) torsor over a G-group A such that the formula

ξ^{g}α^{g} = (ξ · α)^{g},

for all ξ ∈ X and α ∈ A, as above, let us say that (the G-set) X is a G-action torsor over (the G-group) A.

In summary, (if Isom_{L}(W, V) is nonempty) the G-set \underline{Isom}_{L}(W, V) is a G-action torsor over (the G-group) \underline{Aut}_{L}(V). We say that W is an L/K-form of V if Isom_{L}(W, V) is non-empty.

Note also that we may systematically insist upon extra structure for our algebraic varieties (e.g. a base point over K, or a reduction of the automorphism group by other means) and get an analogous theory.

2. Sets of isomorphism classes of L/K-forms, G-action torsors. Connection with 1-cohomology.

For simplicity, suppose that L/K is a finite Galois extension, and hence that G = Gal(L/K) is a finite group. We leave setting up the more general theory as an exercise (where L/K is an arbitrary, possibly infinite, Galois extension and its Galois group is given its Krull topology).

Definition 1. Let V be a variety over K and L/K a Galois extension with group G, as above. Let E(L/K, V) denote the pointed set of K-isomorphism classes of L/K-forms of V.

The base point of E(L/K, V) is given by V itself, the trivial L/K-form of V.

Definition 2. Let A be a G-group. By T (G; A) let us mean the pointed set of isomorphism classes of G-action torsors over A.

The base point of T (G; A) is given by A itself, viewed as G-action torsor over itself via its multiplication law.

Definition 3. Let A be a G-group. By Cross.Hom(G; A) let us mean the set of crossed homomorphisms from G to A, meaning the pointed set of maps h : G → A with the property that

h(g_{1} · g_{2}) = h(g_{1})^{g_{2}} · h(g_{2}).

The trivial mapping h(G) = 1 ∈ A is a crossed homomorphism, and provides the base point for the set Cross.Hom(G; A). There is a natural action (not preserving base point) of the group A on the set Cross.Hom(G; A) by the formula,

(α · h)(g) = α^{g} · h(g) · α^{−1},

making Cross.Hom(G; A) naturally an A-set.

Definition 4. By H^{1}(G; A) we mean the pointed set defined as the quotient of Cross.Hom(G; A) with respect the equivalence relation given by its A-action; i.e., H^{1}(G; A) is the orbit-set of Cross.Hom(G; A) under this A-action.

The base point of H^{1}(G; A) is the orbit of the trivial crossed homomorphism.

Exercises :

1. When A is abelian, show that the above definition of H^{1}(G; A) coincides with the group-cohomology definition (the base point corresponding to the origin of the group structure in H^{1}(G; A)).

2. In complete generality, (recalling what we did on Thursday last) show that there is a natural isomorphism of pointed sets

T (G; A) ≅ H^{1}(G; A).

3. In complete generality, (recalling what we did on Thursday last) show that there is a natural injection

E(L/K, V) ↪ T (Gal(L/K); \underline{Aut}_{L}(V)) ≅ H^{1}(Gal(L/K); \underline{Aut}_{L}(V)).

4. Suppose that n ≥ 1 and that K is a field of characteristic not dividing n containing a primitive n-th root of 1. Let G(X) be a polynomial (say with no multiple roots) and consider the plane curve V : Y^{n} = G(X), and its collection of twists V_{D} : DY^{n} = G(X), for D ranging through the nonzero elements of K. The curves V_{D} are L/K-twists of V for L = K(D^{1/n}). Think through the above theory, in the context of these examples.

3. Algebraic geometric torsors.

Let K be a field and K_{s}/K a separable closure. If V is a variety over K let V̄ denote its pullback to K_{s},; i.e., V̄ is obtained from V by extension of scalars to K_{s}, and if f : V → W is a mapping over K, let f̄: V̄ → W̄ be the mapping induces after extension of scalars to K_{s}. Let Γ be an algebraic group over K, and

μ : Γ × Γ → Γ

the group multiplication mapping.

By an algebraic geometric torsor over Γ let us mean an algebraic variety X over K, together with a mapping (defined over K)

α : Γ × X → X,

(called the Γ-action mapping) such that there exists an isomorphism of K_{s}-varieties u : Γ̄ → X̄ such that ᾱ · (1 × u) = u · μ̄; i.e., such that u identifies the multiplication law on Γ̄ with the Γ̄-action mapping. Call these pairs (X, α) a.g. torsors for Γ, for short.

Remark. When you give an a.g. torsor structure for Γ on an algebraic variety X you are giving a (particular kind of) homomorphism from Γ(K_{s}) to Aut_{K_{s}}(X).

We say that an algebraic geometric torsor is trivial over K if u above can be defined over K; or equivalently, if there is a K-isomorphism between Γ and X bringing α to μ. It is equivalent to ask that X has a K-rational point. If (X, α) is an algebraic geometric torsor for Γ defined over K, and trivial over an extension field L/K, then X is an L/K-form of Γ, but we should be attentive to the difference in structures here. We have a mapping (of isomorphism classes of elements in the sets:

{a. g. torsors for Γ_{/K} becoming trivial over L } ↦ { L/K−forms of the algebraic variety Γ}.

Exercises.

5. Think through the nature of the above mapping. It is not necessarily one:one. Can you find an example?

6. Define the natural mapping (as quickly done in class on Thursday):

{iso.classes of a. g. torsors for Γ_{/K} becoming trivial over L } → H^{1}(Gal(L/K); Γ(L)).

Hint: Choosing a point x ∈ X(L) you can construct a specific 1-cocycle c_{x} (i.e., a crossed homomorphism c_{x} : Gal(L/K) → Γ(L)) by setting c_{x}(g) := that unique γ ∈ Γ(L) such that x^{g} = γ · x. Show that this does the trick.

7. Show that the mapping of Exercise 6. is an injective mapping (of pointed sets).

Definition 5. We shall say that Γ_{/K} has the descent property for torsors if the mapping of Exercise 6. is a bijection for all Galois field extensions L/K.

4. Abelian varieties.

Theorem. Let K be a field, and Γ = A an abelian variety over K.) Then A has the descent property for torsors. We have a canonical identification between the set of isomorphism classes of a.g. torsors for A over K and H^{1}(G_{K}, A(K_{s})).

Remark. This is a serious theorem.

Exercises.

8. Noting that H^{1}(G_{K}, A(K_{s})) has a natural (abelian) group structure, give a nice geometric definition of an addition law for a.g. torsors for A over K such that the natural isomorphism of the above theorem becomes an isomorphism of (abelian) groups.

9. Describe an involution of a.g. torsors over A which realizes the inversion mapping a ↦ −a with respect to the group structure discussed in Exercise 8.

Second definition of the Shafarevich-Tate group.

Definition 6. Let S be a set (any set, not necessarily finite) of places of a number field F, and let A be an abelian variety over F. By Sha_{S}(A/F) let us mean the set of isomorphism classes of a.g. torsors for A defined over F and trivial over F_{v} where v ranges through all places of F not contained in S. If S is empty, we call Sha_{S}(A/F) simply Sha(A/F).

Theorem. There is an exact sequence

0 → Sha_{S}(A/F) → H^{1}(G_{F} , A(F̄)) → ∏_{v∉S} H^{1}(G_{F_{v}}, A(F̄_{v}));

i.e., the second definition is the same as the first definition!

5. Elliptic curves, and curves of genus 1. Let X be a smooth projective curve over K of genus 1. Let E = jac(X) = Pic^{0}(X) denote the jacobian variety of X. Then E is an elliptic curve over K; i.e., it is an abelian variety of dimension 1 defined over K. Discuss Pic^{n}(X) for any n ∈ Z.

Exercises:

10. Let K be a field of characteristic 0, X a smooth projective curve over K of genus 1, and E = jac(X). Give a careful discussion of the structure of the algebraic group \underline{Aut}(X) over K (this will depend upon whether j(E) = 0, j(E) = 1728, or j(E) ≠ 0, 1728). Show that the component of the algebraic group \underline{Aut}(X) containing the identity element, call it \underline{Aut}^{o}(X), is isomorphic to E.

11. Show that the set of isomorphism classes of a.g. torsors for E over K is equal to the set of isomorphism classes of couples (X, ι) where X is a smooth projective curve over K, and ι : E ≅ \underline{Aut}^{o}(X), is an isomorphism of algebraic groups over K.

12. If the E-torsor (X, ι) corresponds to the cohomology class ξ ∈ H^{1}(G_{K}; E(K̄)) compare (X, ι) to the E-torsor Pic^{1}(X), and in terms of this comparision and ξ, give a formula for the cohomology class ∈ H^{1}(G_{K}; E(K̄)) corresponding to the E-torsor Pic^{n}(X) (for any n).

6. Finding elements of Sha(E/F) in projective space. Let X be a smooth projective curve of genus 1 over a number field F, and let E denote its jacobian. We identify X with the E-torsor X_{1} := Pic^{1}(X). Let ξ ∈ H^{1}(G_{K}; E(K̄)) be the cohomology class corresponding to X_{1} and let n be the order of ξ. Then X_{n} = Pic^{n}(X) has an F - rational point. Does X also possess a divisor of degree n defined over F? Let us analyze this question, by considering the exact sequence of G_{F} -modules,

0 → F̄(X)^{∗}/F̄^{∗} → Z[X(F̄)] → Pic(X)(F̄) → 0,

and passing to G_{F} -cohomology, we have:

Z[X(F̄)]^{G_{F}} → Pic(X)(F) → H^{1}(G_{F} , F̄(X)^{∗}/F̄^{∗}).

To analyze the right-hand group, we make use of Hilbert’s Theorem 90, and compute H^{1}(G_{F} , F̄(X)^{∗}/F̄^{∗}) from the exact sequence of G_{F} -modules

0 → F̄^{∗} → 0 → F̄(X)^{∗} → 0 → F̄(X)^{∗}/F̄^{∗} → 0,

giving

H^{1}(G_{F} , F̄(X)^{∗}/F̄^{∗}) = Ker {Br_{F} → Br_{F (X)} }.

Now, if ξ is an element of Sha(E/F)[n], we have:

Theorem* .

Ker {Br_{F} → Br_{F (X)} } = 0.

Proof. Since Br_{F} ⊂ ⨁_{v} Br_{F_{v}}, it suffices to show that Br_{F_{v}} → Br_{F_{v}(X)} is injective for all places v of F. Let

X_{v} = X ×_{Spec(F)} Spec(F_{v})

and using the theory in A. Grothendieck’s “Le groupe de Brauer III. Exemples et Complements”, we may factor Br_{F_{v}} → Br_{F_{v}(X)} through

Br_{F_{v}} → Br_{X_{v}} → Br_{F_{v}(X)}

and, by Prop. 2.1 of loc. cit. (since X_{v} is a noetherian regular integral scheme of dimension 1) the second mapping in the diagram above is injective. Since X_{v} has an F_{v}-rational point, the first mapping has a right-inverse, and therefore is also injective.

Corollary 1. If X is an everywhere locally trivial E-torsor corresponding to an element of order n in Sha(E/F), then there exists a divisor D on X of degree n and which is defined over F.

Under the hypotheses (and conclusion) of Corollary 1, suppose that n ≥ 3, and consider the invertible coherent sheaf O_{X}(D) of regular functions locally defined on X with poles bounded by D. Letting P^{n−1} denote the projective space (over F) dual to the projectivization of the vector space of global sections, H^{0}(X, O_{X}(D)), denote by ϕ : X ↪ P^{n−1} the natural imbedding derived from the linear system containing D, which identifies X with a curve of genus 1 over F, of degree n, in projective (n − 1)-space.

Corollary 2. If X is an E-torsor corresponding to an element of order n in Sha(E/F), then X is isomorphic, over F, to a “normally imbedded” curve of genus 1, of degree n in projective (n − 1)-space.

Idle question*. Fix n ≥ 3. Consider the Chow variety, call it C_{n} of all (smooth) projective “normally imbedded” curve of genus 1, of degree n in projective (n − 1)-space. Then C_{n} is a quasi-projective variety over Q, with a standard (quasi-) projective imbedding. In terms of this standard imbedding, let

H : C_{n}(Q̄) → R^{≥0}

denote the classical (Weil) height function. Given an elliptic curve E over a number field F, by the height, h(σ) of an element σ ∈ Sha(E/F)[n] ) let us mean the minimum of the heights (in the sense of the H just mentioned) of all the points in C_{n}(F) which correspond to normally imbedded curves of genus 1, of degree n in projective (n − 1)-space which are isomorphic over F to E-torsors representing the element σ. Of course, h(σ) = h(−σ). Can we find a (good) upper bound for the values of h on Sha(E/F)[n] (depending, of course, on E, F, and n; or perhaps just on the absolute norm of the conductor of E, the absolute degree of F, and n) ?