Contents

1 Introduction 2

2 Preliminary results 4 2.1 Short exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Local invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Principle homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 The plane of computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Calculus of cocycles 7 3.1 Starting lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Formula for cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Classification of cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4 Direct calculations of cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.5 δ on cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Quaternion and cyclic algebras 13 4.1 Algebras corresponding to cocyles of type I . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 δ for cocycles of type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3 Cross products corresponding to cocycles of type II . . . . . . . . . . . . . . . . . . 14 4.4 Cyclic algebras of degree 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Dihedral algebras 18 5.1 δ for cocycles of type III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.2 Dihedral cross products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.3 End of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Algebras of exponent 2 over an elliptic curve

V.Guletskiǐ

November 15, 1999

1 Introduction

Let k be a field, char(k) ≠ 2, 3, and let X be a smooth projective geometrically integral curve over k. Assume that X has at least one point rational over k. Let k(X) be the field of krational functions on X. The Brauer group Br(X) of X is naturally isomorphic to the unramified Brauer group Br_{nr}(k(X)/k) of k(X)/k (see [Lich69], [Co88]). This allows to identify Br(X) with Br_{nr}(k(X)/k). Denote by J the Jacobian variety of the curve X. In [GMY97] the description of the 2-torsion part in Br(X) in terms of quaternion algebras was made under the condition of k-rationality of the 2-torsion subgroup of J. In [GY98] this result was generated on algebras of an arbitrary exponent n: if all points of order n on J are rational over k, then any element of order n in Br(X) can be presented by a tensor product of cyclic algebras depending on the n-torsion subgroup of J.

Let X = E be an elliptic curve over k. In this paper we consider the problem of a description of 2-torsion part in Br(E) in the case when E has only one k-rational point of order 2 on the Jacobian E. Our arguments based on an explicit computation of 1-cocycles presenting elements of order 2 in the group of homogeneous spaces over E and a construction of a section for some divisor mapping induced by the 2-torsion subgroup of E.

It should be mentioned that in [YM96], [VY96] and [V97] a complete description of the 2-torsion in Br(E) was made in the case of a local field k and all proofs there heavily use the completeness property of the ground field k. In contrast to this we present a different approach which is mainly based on cohomology methods and allows to consider an arbitrary ground field k, char(k) ≠ 2, 3. On the other hand we do not answer the questions about triviality and indices of algebras of exponent 2 in Br(E).

Since the characteristic of the ground field k is not 2 or 3, the elliptic curve E can be defined as the projective closure of a plane curve defined by an equation

y^{2} = F (x) ,

where F (x) is a cubic polynomial over k with the leading coefficient equal to 1 without multiple roots in a separable closure k̄ of k. Let Ē = E(k̄) be the group of geometric points on E and let 2Ē = {O, P, T, S} be the 2-torsion subgroup of Ē. Here O is the infinite point on E playing the rule of zero on the curve E. Let x_{P}, x_{T}, x_{S} be the x-coordinates of the points P, T, S respectively. Then we have the decomposition

F (x) = (x − x_{P})(x − x_{T})(x − x_{S})

over k̄. We say that E is split if all 2-torsion points are rational over k. This is equivalent to the condition that all roots x_{P}, x_{T}, x_{S} of the polynomial F (x) lie in k. We say that E is semisplit if there exists only one k-point in _{2}Ē denoted below by P. In other words, only one root x_{P} of F (x) is k-rational. If all non-trivial points in _{2}Ē are not k-rational, then we say that E is fully non-split.

In the split case it is not hard to compute quaternion algebras generating _{2}Br(E). Recall the result of [GY98] for the group _{2}Br(E). Let E be split. Fix any two non-trivial 2-torsion points on Ē, for example P and T. Then any unramified central simple algebra of exponent 2 over k(E) similar to a tensor product

A ⊗ (b, x − x_{P}) ⊗ (c, x − x_{T}) .

Here A is a numerical algebra over k(E), i.e. an algebra obtained by the scalar extension of some algebra with the center k. The algebras (b, x − x_{P}), (c, x − x_{T}) are quaternion algebras over k(E) with b, c ∈ k^{∗}. Note that (b, x − x_{P}) or (c, x − x_{T}) can be trivial in Br(E). For example, if b is a square in k then (b, x − x_{P}) is zero in Br(E).

If E is fully nonsplit, then the problem of finding of all unramified central simple algebras of exponent 2 over k(E) can be reduced to either a split or semisplit case. Indeed, fully nonsplitness means that F is an irreducible polynomial over k. Then the extension L = k(x_{T}) has degree 3 over k. Let E_{L} = E × Spec(L). The composition

Br(E) ⟶^{res} Br(E_{L}) ⟶^{cor} Br(E)

coincides with the multiplication-by-3 homomorphism on the group Br(E). So keeping in mind that 2 and 3 are relatively prime we get that the corestriction map

cor : _{2}Br(E_{L}) ⟶ _{2}Br(E)

is surjective. Hence one can find algebras in _{2}Br(E) applying cor to algebras from _{2}Br(E_{L}).

Thus we see that the main situation is when _{2}Br(E) has only one non-trivial point P defined over k. In this case

F (x) = (x − x_{P})f(x)  (1)

where f(x) is an irreducible quadratic polynomial over k.

Let G = Gal(k̄/k) be the absolute Galois group of the ground field k, L = k(x_{T}) = k(x_{S}) and H = Gal(k̄/L). Let U be a subgroup of index 2 in H and Z = k̄^{U}. If U is normal in G and G/U is a cyclic group of order 4, then there exists a cyclic algebra (Z · k(E)/k(E), f(x)) over k(E), where the quadratic polynomial f(x) is considered as a rational function on E. If U is not normal in G (this is equivalent to the existing of an element g_{0} ∈ G\H such that U ≠ g_{0}^{−1}Ug_{0}) then one can construct a special central simple algebra B_{U} over k(E) (see Section 5.2 below), which is a cross product defined by a 2-cocycle in an explicit form. We prove the following theorem, which is the main result of this paper.

Theorem 1.0.1 Let k be a field, char(k) ≠ 2, 3, and let E be a semisplit elliptic curve over k. Then any element of order 2 in the quotient group Br(E)/Br(k) is similar to a simple algebra of one of the following three types:

(i)

(a, x − x_{P}) ,

where (a, x − x_{P}) is a quaternion algebra over k(E) with a ∈ k^{∗};

(ii)

(Z · k(E)/k(E), f(x)) ,

where (Z · k(E)/k(E), f(x)) is a cyclic algebra over k(E) generated by a cyclic extension Z/k of degree 4 over k such that L ⊂ Z and the polynomial f is defined by (1) and is viewed as a k-rational function on E;

(iii) an algebra B_{U}, which is defined in Section 5.2 by a 2-cocycle in an explicit form.

Note that the direct computation (see Proposition 5.2.1 below) shows that the algebra B_{U} from (iii) has index at most 4. This agrees with the results of Van den Bergh (see [1], Prop. 1, p. 196) and implies that B_{U} is similar to either a quaternion or biquaternion algebra over k(E). After the paper was finished V. Chernousov inform me that B_{U} can be decomposed in a tensor product of two explicit quaternion algebras determined by f(x) and the subgroup U. The correspoding theorem can be formulated in the following way:

Theorem 1.0.2 Let k be a field, char(k) ≠ 2, 3, and E be a semisplit elliptic curve over k. Without loss of generality the curve E can be defined by the equation y^{2} = (x − p)(x^{2} − a), where p, a ∈ k and a is not a square in k (hence f(x) = x^{2} − a and L = k(\sqrt{a})). Let B_{U} be the algebra from (iii) of Theorem 1.0.1 and b = u + v\sqrt{a} an element in L such that k̄^{U} = L(\sqrt{b}). Then B_{U} is similar to the tensor product

(v, u^{2} − v^{2}a) ⊗ (vx + u, (u^{2} − v^{2}a)(x^{2} − a)) ,

where (v, u^{2} − v^{2}a) is the numerical quaternion algebra defined by the two constants v, u^{2} − v^{2}a ∈ k ⊂ k(E) and (vx + u, (u^{2} − v^{2}a)(x^{2} − a)) is the quaternion algebra defined by the functions vx + u, (u^{2} − v^{2}a)(x^{2} − a) ∈ k(E).

Acknowledgements

This work was supported by the Institute of Mathematics of the Academy of Sciences of Belarus and TMR ERB FMRX CT-97-0107. The author thanks Vladimir Chernousov for useful discussions.

Notations

Symbols ℤ, ℚ denotes integers and rational numbers respectively. If A is an abelian group and n is a natural number, then n : A → A denotes the homomorphism of multiplication by n and nA, A/n are its kernel and cokernel respectively. For any finite set S we denote the number of elements in S by #S. If H is a group, U its normal subgroup and x ∈ H, we denote the left coset xU by \overline{x}. F^{∗} denotes the multiplicative group of a field F. Variety is always smooth projective and geometrically integral scheme over k. For any variety X over k and for any field extension L/k let X(L) be the set of L-points on X and let X̄ = X(k̄) be the set of geometrical points on X. A curve is a variety of dimension one. A point of a curve means closed point. We denote an arbitrary curve by X. An elliptic curve will be always denoted by E.

2 Preliminary results

In this section we recall some well-known facts used below. All of them can be found in [Fadd51],

[Mi81] and [Sch69].

2.1 Short exact sequences

Let G = Gal(k̄/k) be the absolute Galois group of the ground field k, X a curve of an arbitrary genus defined over k such that X(k) ≠ ∅ and let k̄(X̄) be the field of k̄-rational functions on X̄. One can associate with the curve X the following discrete (left) G-modules: Div(X̄) is the divisor group of X̄; Div^{0}(X̄) is its subgroup of degree zero divisors on X̄; P(X̄) is the subgroup of principles divisors; Pic(X̄) = Div(X̄)/ P(X̄) and Pic^{0}(X̄) = Div^{0}(X̄)/ P(X̄). These G-modules are terms of the following short exact sequences

1 → k̄^{∗} ⟶ k̄(X̄)^{∗} ⟶^{div} P(X̄) → 0 ,  (2)
0 → P(X̄) ⟶ Div(X̄) ⟶ Pic(X̄) → 0 ,  (3)
0 → Pic^{0}(X̄) ⟶ Pic(X̄) ⟶^{deg} ℤ → 0 .  (4)

Here deg is the degree homomorphism and div is the homomorphism sending f ∈ k̄(X̄)^{∗} into the divisor div(f) of zeros and poles of the rational function f. Let J̄ be the group of k̄-points of the Jacobian variety J of the curve X. Since we suppose that X has at least one k-rational point, J̄ and Pic^{0}(X̄) are isomorphic as discrete G-modules. Therefore we have the short exact sequence

0 → P(X̄) ⟶ Div^{0}(X̄) ⟶^{sum} J̄ → 0 .  (5)

2.2 Local invariants

Denote the set of closed points on X (i.e. the set of points of codimension 1 on X) by X^{(1)}. All such points x ∈ X^{(1)} are in one-to-one correspondence with discrete valuations rings of the field k(X) over k. Let k(X)_{x} be the completion of k(X) with respect to the valuation corresponding to a point x ∈ X^{(1)}. Denote by k(x) the residue field of X in x. Then [k(x) : k] < ∞ and one can identify k(x) with a subfield in k̄. Let G_{x} = Gal(k̄/k(x)) for any x ∈ X^{(1)}. Then Br(k(X)_{x}) is isomorphic to the direct sum of the Brauer group Br(k(x)) and the group Hom_{cont}(G_{x}, ℚ/ℤ) of continuous homomorphisms of the profinite group G_{x} into the discrete group ℚ/ℤ. Let

inv_{x} : Br(k(X)) ⟶ Hom_{cont}(G_{x}, ℚ/ℤ)

be the composition of the scalar extension homomorphism

Br(k(X)) ⟶ Br(k(X)_{x})

and the projection

Br(k(X)_{x}) ≅ Br(k(x)) ⊕ Hom_{cont}(G_{x}, ℚ/ℤ) → Hom_{cont}(G_{x}, ℚ/ℤ) .

The homomorphism inv_{x} maps a class of algebras a ∈ Br(k(X)) into its local invariant inv_{x}(a). The order of inv_{x}(a) coincides with the ramification index of algebras from a at the point x ∈ X^{(1)}. Algebras can be ramified only in a finite number of points. Hence one can construct the homomorphism

inv : Br(k(X)) ⟶ ⨁_{x∈X^{(1)}} Hom_{cont}(G_{x}, ℚ/ℤ) .

By definition, its kernel is unramified Brauer group Br_{nr}(k(X)) of k(X)/k. As it was mentioned in the beginning, the group Br_{nr}(k(X)) can be naturally identified with the Brauer group Br(X) of the curve X.

Recall the cohomology description of the homomorphism inv. Below H^{∗}(G,?) is the Galois cohomology functor on the category of (left) discrete G-modules. By Tsen theorem, the group Br(k(X)) is isomorphic to the cohomology group H^{2}(G, k̄(X̄)^{∗}). Let M_{x} be a sumbmodule in Div(X̄) generated by points in X̄ mapping into x ∈ X^{(1)} under the natural projection X̄ → X. The discrete G-module M_{x} is identified with the induced G-module M_{G}^{G_{x}}(ℤ), where ℤ is viewed as a trivial G-module. Then Div(X̄) ≅ ⨁_{x} M_{G}^{G_{x}}(ℤ), where x runs all points in X^{(1)}. Thus we get

H^{2}(G, Div(X̄)) ≅ ⨁_{x} H^{2}(G, M_{G}^{G_{x}}(ℤ)) ≅ ⨁_{x} H^{2}(G_{x}, ℤ) ≅ ⨁_{x} Hom_{cont}(G_{x}, ℚ/ℤ) .

It can be proven that the composition

Br(k(X)) ≅ H^{2}(G, k̄(X̄)^{∗}) ⟶^{div^{∗}} H^{2}(G, Div(X̄)) ≅ ⨁_{x} Hom_{cont}(G_{x}, ℚ/ℤ)

coincides with the homomorphism inv. Here div^{∗} is the homomorphism induced by div on cohomology groups. Therefore,

Br(X) ≅ ker(div^{∗}) .

2.3 Principle homogeneous spaces

Consider the exact sequence

H^{2}(G, k̄^{∗}) → H^{2}(G, k̄(X̄)^{∗}) →^{α} H^{2}(G, P(X̄)) → H^{3}(G, k̄^{∗}) → H^{3}(G, k̄(X̄)^{∗})

induced by (2). Since X(k) ≠ ∅, the left and right homomorphisms are injective for any n (see [Sch69]). Hence we have the short exact sequence

0 → H^{2}(G, k̄^{∗}) ⟶ H^{2}(G, k̄(X̄)^{∗}) ⟶^{α} H^{2}(G, P(X̄)) → 0 .

Taking into account the isomorphisms H^{2}(G, k̄^{∗}) ≅ Br(k) and H^{2}(G, k̄(X̄)^{∗}) ≅ Br(k(X)), one can obtain the exact sequence

0 → Br(k) ⟶^{ι} Br(k(X)) ⟶^{α} H^{2}(G, P(X̄)) → 0 ,

where ι is induced by the scalar extension: if A is a simple algebra with the center k, then ι([A]) = [A ⊗_{k} k(X)].

Now observe that div^{∗} = β ∘ α, where H^{2}(G, P(X̄)) →^{β} H^{2}(G, Div(X̄)) is the homomorphism induced by the embedding P(X̄) ⊂ Div(X̄). Since α is surjective, one has ker(div^{∗}) = α^{−1}(ker(β)). Consider the exact sequence

H^{1}(G, Pic(X̄)) ⟶^{Δ} H^{2}(G, P(X̄)) ⟶^{β} H^{2}(G, Div(X̄))

induced by (3). It follows that ker(div^{∗}) = α^{−1}(im(Δ)). Furthermore, consider the exact sequence

Pic(X̄)^{G} ⟶^{deg} ℤ → H^{1}(G, Pic^{0}(X̄)) → H^{1}(G, Pic(X̄)) → 0

induced by (4). Here Pic(X̄)^{G} is the group of divisor classes invariant under the action of G. The last sequence is ended by zero because H^{1}(G, ℤ) = 0. Since X(k) ≠ ∅, the homomorphism deg is surjective, hence the epimorphism H^{1}(G, Pic^{0}(X̄)) → H^{1}(G, Pic(X̄)) is injective. Thus we have that

H^{1}(G, Pic(X̄)) ≅ H^{1}(G, Pic^{0}(X̄)) ≅ H^{1}(G, J̄) .

Then

ker(div^{∗}) = α^{−1}(im(δ)) ,

where δ : H^{1}(G, J̄) → H^{2}(G, P(X̄)) is the connecting homomorphism induced by (5). Note that the condition H^{1}(G_{x}, ℤ) = 0 implies that H^{1}(G, Div(X̄)) = 0. Then the exact sequence induced by (3) yields that the connecting homomorphism Δ is injective. Hence δ is injective too. Let

δ^{−1} : im(δ) → H^{1}(G, J̄)

be the inverse to the isomorphism δ : H^{1}(G, J̄) → im(δ). Define π as the composition

π : Br(X) ≅ ker(div^{∗}) ⟶^{α} im(δ) ⟶^{δ^{−1}} H^{1}(G, J̄) .  (6)

The definition of π can be illustrated by the following commutative diagram:

\xymatrix{ &  & H^{2}(G, Div^{0}(X̄)) \ar@{^{(}->}[d] \\ H^{1}(G, J̄) \ar@{^{(}->}[r]^{δ} & H^{2}(G, P(X̄)) \ar[ur] \ar[r]_{β} & H^{2}(G, Div(X̄)) \\ ker(inv) \ar[u] \ar@{^{(}->}[r] & H^{2}(G, k̄(X̄)^{∗}) \ar@{->>}[u]_{α} \ar[ur]_{inv} \\ Br(X) \ar[u]^{≅} \ar@{^{(}->}[r] & Br(k(X)) \ar[u]^{≅}}

Now we have the exact sequence

0 → Br(k) ⟶^{ι} Br(X) ⟶^{π} H^{1}(G, J̄) → 0 .  (7)

It splits. Indeed, if x ∈ X(k), then k(x) = k and Br(k(X)_{x}) ≅ Br(k) ⊕ Hom_{cont}(G, ℚ/ℤ). One can check that the composition

Br(k) → Br(k(X)) → Br(k(X)_{x}) → Br(k) ⊕ Hom_{cont}(G, ℚ/ℤ) → Br(k)

is an identical map. Hence ι has a left inverse.

2.4 Torsion

Let n be a natural number. If A ⟶^{ι} B is a homomorphism of abelian groups, then denote by

nA ⟶^{_{n}ι} _{n}B and A/n ⟶^{ι/n} B/n the homomorphisms induced by ι. The sequence (7) induces the exact sequence

0 → _{n}Br(k) ⟶^{_{n}ι} _{n}Br(X) ⟶^{_{n}π} _{n}H^{1}(G, J̄) → Br(k)/n ⟶^{ι/n} Br(X)/n ⟶^{π/n} H^{1}(G, J̄)/n → 0 .

If ι̂ is a left inverse to ι, then ι̂/n is a left inverse to ι/n. Therefore ι/n is injective. Then the last exact sequence yields the short exact sequence

0 → _{n}Br(k) ⟶^{_{n}ι} _{n}Br(X) ⟶^{_{n}π} _{n}H^{1}(G, J̄) → 0 .

We shall use the following simple

Lemma 2.4.1 Let 0 → A ⟶^{ι} B ⟶^{π} C → 0 be an exact sequence. Suppose that the homomorphism ι has a left inverse ι̂: B → C. For any c ∈ C choose b_{c} ∈ π^{−1}(c). Then any element b ∈ B has the unique decomposition b = b_{c} + ι(a), where c = π(b) ∈ C and a = ι^{−1}(b − b_{c}). The map π̂ : C → B, c ↦ b_{c} − ιι̂(b_{c}) is a right inverse homomorphism of π.

2.5 The plane of computations

Now let X = E be an elliptic curve over k. Then J̄ = Ē and there exists the split exact sequence

0 → _{2}Br(k) ⟶^{_{2}ι} _{2}Br(E) ⟶^{_{2}π} _{2}H^{1}(G, Ē) → 0 .

For every cohomology class c ∈ _{2}H^{1}(G, Ē) we want to choose a Brauer class b_{c} ∈ _{2}Br(E) such that _{2}π(b_{c}) = c. We shall describe classes c ∈ _{2}H^{1}(G, Ē) in terms of explicit cocycles (Section 3). After that we construct central simple algebras representing the Brauer classes b_{c} in Br(E) (Sections 4 and 5). Since by Lemma 2.4.1 any element b ∈ _{2}Br(E) has a unique decomposition b = _{2}ι(a) + b_{c}, where a ∈ _{2}Br(k) and c ∈ _{2}H^{1}(G, Ē), we get that a full description of 2-torsion part of Br(E)/Br(k) is given by central simple algebras representing classes b_{c}.

The Kummer sequence

0 → _{2}Ē ⟶ Ē ⟶^{2} Ē → 0  (8)

yields the exact sequence

0 → E(k)/2 ⟶ H^{1}(G, _{2}Ē) ⟶ _{2}H^{1}(G, Ē) → 0 .  (9)

It follows that any element in _{2}H^{1}(G, Ē) can be presented by a cocycle ϕ : G → _{2}Ē of the group G with coefficients in _{2}Ē. Moreover, the cohomology class cls(ϕ) ∈ _{2}H^{1}(G, Ē) is trivial iff the cohomology class cls(ϕ) ∈ H^{1}(G, _{2}Ē) lies in the image of the embedding E(k)/2 ↪ H^{1}(G, _{2}Ē). This suggests that firstly we need to describe all 1-cocycles of the group G with coefficients in _{2}Ē. It will provide the description of _{2}H^{1}(G, Ē) in terms of explicit 1-cocycles of the group G with coefficients in _{2}Ē.

3 Calculus of cocycles

Let E be a semisplit elliptic curve and let {O, P } be the group k-rational 2-torsion points on E. Then F (x) = (x − x_{P})f(x), where f(x) is an irreducible quadratic polynomial over k. If 2E = {O, P, T, S} is the group of all 2-torsion points on Ē then f(x) = (x − x_{T})(x − x_{S}) over k̄. Let L/k be the minimal subextension in k̄/k such that all 2-torsion points of E are rational over L, i.e. L = k(x_{T}) = k(x_{S}). The group H = Gal(k̄/L) has index 2 in G.

3.1 Starting lemmas

Consider the exact sequence

0 → H^{1}(G/H, _{2}Ē) ⟶^{inf} H^{1}(G, _{2}Ē) ⟶^{inf} H^{1}(H, _{2}Ē)^{G/H} ⟶ H^{2}(G/H, _{2}Ē)

induced by the Hochschild-Serre spectral sequence H^{p}(G/H, H^{q}(H, _{2}Ē)) ⇒ H^{n}(G, _{2}Ē) (see, for example, [Serre]). It is easy to show that H^{1}(G/H, _{2}Ē) = 0 and H^{2}(G/H, _{2}Ē) = 0. So we have the isomorphism

res : H^{1}(G, _{2}Ē) ⟶^{≅} H^{1}(H, _{2}Ē)^{G/H} .  (10)

Recall the action of G/H on H^{1}(H, _{2}Ē). Let ψ ∈ H^{1}(H, _{2}Ē) = Hom(H, _{2}Ē) and g_{0} ∈ G\H. Then (g_{0}ψ)(g) = g_{0}(ψ(g_{0}^{−1}gg_{0})) for any g ∈ G. Note that this action does not depend on the choice of g_{0} and we can consider the action of g_{0} as an action of the unique non trivial element in G/H.

Now fix an isomorphism

2Ē ≅ {O, T } ⊕ {O, S} .

Then

Hom(H, _{2}Ē) ≅ Hom(H, {O, T }) ⊕ Hom(H, {O, S}) .

Let (ψ_{1}, ψ_{2}) be an element in Hom(H, {O, T })⊕Hom(H, {O, S}) and U_{i} = ker(ψ_{i}), i = 1, 2. Then the pair (ψ_{1}, ψ_{2}) lies in (Hom(H, {O, T }) ⊕ Hom(H, {O, S}))^{G/H} if and only if U_{2} = g_{0}^{−1}U_{1}g_{0} for some g_{0} ∈ G\H (this condition does not depend on the choice of an element g_{0} in G\H). Therefore one can identify the set of nontrivial homomorphisms in (Hom(H, {O, T }) ⊕ Hom(H, {O, S}))^{G/H} with the set of subgroups U of index two in H. A such U corresponds to the homomorphism ψ = ψ_{T} + ψ_{S}, where ψ_{T} is the surjective homomorphism H → {O, T } with the kernel U, and ψ_{S} is the surjective homomorphism H → {O, S} with the kernel g_{0}^{−1}Ug_{0}.

Lemma 3.1.1 If ϕ : G → _{2}Ē is a cocycle, then V = ker(ϕ|_{H}) is a normal subgroup in G.

Proof. Isomorphism (10) shows that ϕ |_{H} is invariant under the action of G/H. Then either V = H or V = U ∩ g_{0}^{−1}Ug_{0} for some subgroup U of index 2 in H. In the second case g_{0}^{−1}V g_{0} = g_{0}^{−1}(U ∩ g_{0}^{−1}Ug_{0})g_{0} = g_{0}^{−1}Ug_{0} ∩ g_{0}^{−2}Ug_{0}^{2} = g_{0}^{−1}Ug_{0} ∩ U = V. For any element g ∈ G one can write g = hg_{0} for some h ∈ H. Then g^{−1}V g = g_{0}^{−1}h^{−1}V hg_{0} = g_{0}^{−1}V g_{0} ⊂ V.

Lemma 3.1.2 If ϕ : G → _{2}Ē is a cocycle and ψ = ϕ| H, then ψ(g^{2}) ∈ {O, P } for any g ∈ G\H.

Proof. Since ψ ∈ H^{1}(H, _{2}Ē)^{G/H}, it follows that g_{0}ψ(h) = ψ(g_{0}^{−1}hg_{0}) for any h ∈ H. Substituting h = g^{2} we get

g_{0}ψ(g^{2}) = ψ(g_{0}^{−1}g^{2}g_{0}) = ψ(g_{0}^{−1}g) + ψ(gg_{0})
= ψ(gg_{0}) + ψ(g_{0}^{−1}g)
= ψ(gg_{0}g_{0}^{−1}g)
= ψ(g^{2})
⇔ ψ(g^{2}) ∈ {O, P } .

3.2 Formula for cocycles

Now let g_{0} be some element in G\H. For any Q ∈ _{2}Ē = {O, P, T, S} and for any homomorphism ψ ∈ Hom(H, _{2}Ē) denote by ϕ_{Q,ψ} the map defined by the rule

ϕ_{Q,ψ}(g) = \{ \begin{matrix}ψ(g) , & g ∈ H \\ g_{0}ψ(g_{0}^{−1}g) + Q , & g ∈ G\H . \end{matrix}  (11)

Lemma 3.2.1 Any cocycle ϕ : G → _{2}Ē is of the form ϕ_{Q,ψ} for some Q ∈ _{2}Ē and ψ ∈ Hom(H, _{2}Ē). Conversely, let Q and ψ be as above. Then ∂ϕ_{Q,ψ} = 0 if and only if the following two conditions hold:

(i) g_{0}ψ(h) = ψ(g_{0}^{−1}hg_{0}) for any h ∈ H;

(ii) ψ(g_{0}^{2}) = P for Q ∈ {T, S} and ψ(g_{0}^{2}) = O for Q ∈ {O, P }.

Proof. If ϕ is a 1-cocycle with coefficients in _{2}Ē then for any g ∈ G we have

∂ϕ(g_{0}, g_{0}^{−1}g) = g_{0}ϕ(g_{0}^{−1}g) + ϕ(g) + ϕ(g_{0}) = 0

for some g_{0} ∈ G\H. Therefore ϕ = ϕ_{Q,ψ}, where Q = ϕ(g_{0}) and ψ = ϕ|_{H}.

If h_{1}, h_{2} ∈ H, then it is clear that ∂ϕ_{Q,ψ}(h_{1}, h_{2}) = 0. Let g ∈ G\H and h ∈ H. Then

∂ϕ_{Q,ψ}(g, h) = gψ(h) + g_{0}ψ(g_{0}^{−1}gh) + Q + g_{0}ψ(g_{0}^{−1}g) + Q
= gψ(h) + g_{0}ψ(g_{0}^{−1}g) + g_{0}ψ(h) + g_{0}ψ(g_{0}^{−1}g)
= gψ(h) + g_{0}ψ(h)
= \{ \begin{matrix}2ψ(h) , & ψ(h) ∈ \{O, P \} \\ 2g_{0}ψ(h) , & ψ(h) ∈ \{T, S\} \end{matrix}
= 0

Note that we replace signs ”−” by ”+” because ϕ_{Q,ψ} is a map into a 2-torsion group. If h ∈ H, g ∈ G\H, then for any Q and ψ we have

∂ϕ_{Q,ψ}(h, g) = g_{0}ψ(g_{0}^{−1}g) + Q + g_{0}ψ(g_{0}^{−1}hg) + Q + ψ(h)
= g_{0}ψ(g_{0}^{−1}g) + g_{0}ψ(g_{0}^{−1}hg_{0}) + g_{0}ψ(g_{0}^{−1}g) + ψ(h)
= g_{0}ψ(g_{0}^{−1}hg_{0}) + ψ(h) .

Therefore, if ∂ϕ_{Q,ψ} = 0, then g_{0}ψ(g_{0}^{−1}hg_{0}) + ψ(h) = 0, that is

g_{0}ψ(h) = ψ(g_{0}^{−1}hg_{0}) .  (12)

If g_{1} ∈ G\H, g_{2} ∈ G\H, then for any Q and ψ we have:

∂ϕ_{Q,ψ}(g_{1}, g_{2}) = g_{1}(g_{0}ψ(g_{0}^{−1}g_{2}) + Q) + ψ(g_{1}g_{2}) + g_{0}ψ(g_{0}^{−1}g_{1}) + Q
= ψ(g_{0}^{−1}g_{2}) + ψ(g_{1}g_{2}) + g_{0}ψ(g_{0}^{−1}g_{1}) + g_{1}Q + Q
=^{(12)} ψ(g_{0}^{−1}g_{2}) + ψ(g_{1}g_{2}) + ψ(g_{0}^{−1}g_{0}^{−1}g_{1}g_{0}) + g_{1}Q + Q
= ψ(g_{0}^{−1}g_{2}) + ψ(g_{1}g_{2}) + ψ(g_{0}^{2}) + ψ(g_{1}g_{0}) + g_{1}Q + Q .

Observe that

ψ(g_{0}^{−1}g_{2}) + ψ(g_{1}g_{2}) + ψ(g_{1}g_{0}) = ψ(g_{1}g_{0}) + ψ(g_{0}^{−1}g_{2}) + ψ(g_{1}g_{2})
= ψ(g_{1}g_{0}g_{0}^{−1}g_{2}) + ψ(g_{1}g_{2})
= ψ(g_{1}g_{2}) + ψ(g_{1}g_{2})
= 2ψ(g_{1}g_{2})
= 0

Hence one can continue:

∂ϕ_{Q,ψ}(g_{1}, g_{2}) = ψ(g_{0}^{−1}g_{2}) + ψ(g_{1}g_{2}) + ψ(g_{0}^{2}) + ψ(g_{1}g_{0}) + g_{1}Q + Q
= ψ(g_{0}^{2}) + g_{1}Q + Q
= ψ(g_{0}^{2}) + g_{0}Q + Q .

Therefore, if ∂ϕ_{Q,ψ} = 0, then

ψ(g_{0}^{2}) = g_{0}Q + Q .

Remark 3.2.1 Clearly, the condition (i) is equivalent to the condition ψ ∈ (Hom(H, _{2}Ē))^{G/H} and can be deduced directly from the isomorphism 10.

3.3 Classification of cocycles

Now one can classify all 1-cocycles ϕ = ϕ_{Q,ψ} of the group G with coefficients in _{2}Ē according to the number of elements in im(ψ).

Denote by ξ the surjective homomorphism G → {O, P } with kernel H. Then ξ = ∂T = ∂S and ξ is a unique non-trivial 1-coboundary of G with coefficients in _{2}Ē. If # im(ψ) = 1, i.e. ψ = 0, then the formula 11 and Lemma 3.2.1 show that either ϕ = 0 or ϕ = ξ. In both cases cls(ϕ) = 0 in H^{1}(G, _{2}Ē).

Let ϕ_{Q,ψ} be a cocycle. Let ψ_{T} and ψ_{S} are the compositions of ψ with the projections of the group _{2}Ē on {O, T } and {O, S} respectively. Let U = ker(ψ_{T}). Then ker(ψ_{S}) = g_{0}^{−1}Ug_{0}. The condition # im(ψ) = 2 is equivalent to the condition g_{0}^{−1}Ug_{0} = U, g_{0} ∈ G\H. In this case ψ is a surjective homomorphism H → {O, P } with kernel U. The group U is normal in G by Lemma 3.1.1. Consider the quotient group G/U. There are two possible cases for G/U: either G/U ≅ ℤ/2 ⊕ ℤ/2 or G/U ≅ ℤ/4. In the first case one can choose g_{0} ∈ G\H such that g_{0}^{2} ∈ U. By Lemma 3.2.1 the point Q lie in {O, P }. But ϕ_{P,ψ} + ϕ_{O,ψ} = ξ = ∂T. Consequently, we can consider only cocycles ϕ_{O,ψ}. If G/U ≅ Z/4 then g_{0}^{2} ∈ H\U. By Lemma 3.2.1 the point Q lies in {T, S}. Adding ξ one can consider only cocycles ϕ_{T,ψ}.

Definition 3.3.1 Let ϕ = ϕ_{Q,ψ} be a cocycle with # im(ψ) = 2 and U = ker(ψ). If G/U ≅ ℤ/2 ⊕ ℤ/2 and Q = O, then we say that ϕ is a cocycle of the type I. If G/U ≅ ℤ/4 and Q = T, then we say that ϕ is a cocycle of the type II.

Now let us consider the case of a cocycle ϕ = ϕ_{Q,ψ} with # im(ψ) = 4. The corresponding homomorphism can be uniquely determined by a subgroup U of index 2 in H such that U ≠ g_{0}^{−1}Ug_{0} for some g_{0} ∈ G\H (note that g_{0}^{−1}Ug_{0} = g′_{0}^{−1}Ug′_{0} for any two elements g_{0}, g′_{0} ∈ G\H). Let U_{O} = U ∩ g_{0}^{−1}Ug_{0}. The condition U ≠ g_{0}^{−1}Ug_{0} implies that the quotient group G/U_{O} is noncommutative of order 8. Moreover, G/U_{O} has 3 elements of order 2 lying in the subgroup H/U_{O}. Therefore G/U_{O} is isomorphic to the dihedral group D_{4} (see, for example, [Hall59], 4.4). But D_{4} has five elements of order 2. Consequently, there exists an element g_{0} in G\H such that g_{0}^{2} lies in U_{O}. By Lemma 3.2.1 the point Q lies {O, P }. Adding ξ if it is necessary one can consider only the case Q = O.

Definition 3.3.2 Cocycles ϕ = ϕ_{T,ψ} with # im(ψ) = 4 will be called cocycles of type III.

In fact we proved the

Proposition 3.3.1 Any element in _{2}H^{1}(G, Ē) can be represented by cocycle of one of three types I, II and III.

3.4 Direct calculations of cocycles

Proposition 3.4.1 Let ϕ = ϕ_{Q,ψ} be a cocycle of the group G with coefficients in _{2}Ē and U = ker(ψ). If ϕ is a cocycle of type I, then ϕ is the surjective homomorphism G → {O, P } with kernel U ∪ g_{0}U. If ϕ = ϕ_{T,ψ} has type II, then

ϕ_{T,ψ}(g) = \{ \begin{matrix}O , if g ∈ U \\ T , if g ∈ g_{0}U \\ P , if g ∈ H\U \\ S , if g ∈ G\(H ∪ g_{0}U) \end{matrix}

Proof. Let ϕ_{O,ψ} be a cocycle of type I. Then G/U ≅ ℤ/2 ⊕ ℤ/2 and im(ψ) = {O, P }. If g ∈ G\(H ∪g_{0}U), then g_{0}^{−1}g ∈ H\U. Using formula (11) one can compute ϕ_{O,ψ}(g) = ψ(g_{0}^{−1}g) = P. Moreover, for any u ∈ U we have ϕ_{O,ψ}(g_{0}u) = ψ(g_{0}^{−1}g_{0}u) = ψ(u) = O.

Let ϕ_{T,ψ} be a cocycle of type II. Fix the isomorphism G/U ≅ ℤ/4 such that \overline{g_{0}} maps into the 1+ℤ. Sometimes it is convenient to identify the group G/U and ℤ/4 by means of this isomorphism. Any g ∈ G can be written in the form

g = g_{0}^{i}u where i ∈ {0, 1, 2, 3} and u ∈ U ,

where i is the order of an element g under the element g_{0}. If the order i of g is even, then g_{0}^{i} ∈ H. Using formula 11 one can obtain that

ϕ_{T,ψ}(g) = ψ(g_{0}^{i}u) = ψ(g_{0}^{i}) = \{ \begin{matrix}O , for i = 0 \\ P , for i = 2 \end{matrix}

If i is odd, then g_{0}^{i} ∈ G\H and hence g = g_{0}^{i}u ∈ G\H. Using the same lemma we have

ϕ_{T,ψ}(g) = ψ(g_{0}^{−1}g) + T = ψ(g_{0}^{−1}g_{0}^{i}u) + T
= ψ(g_{0}^{i−1}u) + T = ψ(g_{0}^{i−1}) + T

= \{ \begin{matrix}T , for i = 1 \\ S , for i = 3 \end{matrix}

Now let us consider cocycles ϕ_{O,ψ} of type III. It follows from the definition that every such cocyle is completely determined by a subgroup U of index 2 in H such that U ≠ g_{0}^{−1}Ug_{0}. Here g_{0} ∈ G\H and g_{0}^{2} ∈ U_{O} = U ∩g_{0}^{−1}Ug_{0}. If ψ_{T} and ψ_{S} are the compositions of ψ with the projections of _{2}Ē on {O, T } and {O, S} respectively, then ψ = ψ_{T} + ψ_{S} and U = ker(ψ_{T}), g_{0}^{−1}Ug_{0} = ker(ψ_{T}) (see Section 3.3). Let

U_{S} = U\(g_{0}^{−1}Ug_{0})
U_{T} = (g_{0}^{−1}Ug_{0})\U
U_{P} = H\(U ∪ g_{0}^{−1}Ug_{0})

be the cosets in H/U_{O}. Then G/U_{O} = {U_{O}, U_{S}, U_{T} , U_{P} , g_{0}U_{O}, g_{0}U_{S}, g_{0}U_{T} , g_{0}U_{P} , }. Schematically one can describe the quotient group G/U_{O} by the diagram

\begin{matrix}U_{S} & U_{P} \\ U_{O} & U_{T} \\ g_{0}U_{S} & g_{0}U_{P} \\ g_{0}U_{O} & g_{0}U_{T} \end{matrix}

Here the top four squares correspond to the group H and all eight squares correspond to the whole group G.

Proposition 3.4.2 Let ϕ = ϕ_{O,ψ} be a cocycle of type III. Let

V_{O} = U_{O} ∪ g_{0}U_{O} ,
V_{S} = U_{S} ∪ g_{0}U_{T} ,
V_{T} = U_{T} ∪ g_{0}U_{S} ,

and

V_{P} = U_{P} ∪ g_{0}U_{P} .

Then V_{O} is a (not normal) subgroup in G of index 4. The sets V_{O}, V_{S}, V_{T}, V_{P} are left cosets of the group G by V_{O}. The cocycle ϕ_{O,ψ} can be computed by the formula

ϕ_{O,ψ}(g) = \{ \begin{matrix}O , if g ∈ V_{O} \\ S , if g ∈ V_{S} \\ T , if g ∈ V_{T} \\ P , if g ∈ V_{P} \end{matrix}

Proof. We know that G/U_{O} ≅ D_{4}. Then this proposition can be proved by direct computation with using the formula

ϕ_{O,ψ}(g) = \{ \begin{matrix}ψ(g) , & g ∈ H \\ g_{0}ψ(g_{0}^{−1}g) , & g ∈ G\H . \end{matrix}  (13)

and rules of multiplication in G/U_{O} ≅ D_{4}. For example, let us prove that V_{O} is a subgroup in G and that it is not normal in G. By definition of a cocycle of type III, we have that \overline{g_{0}}^{2} = 1 in G/U_{O}. Hence V_{O} = U_{O} ∪ g_{0}U_{O} is a subgroup in G. Let h_{S} ∈ U_{S}, h_{T} ∈ U_{T} and h_{P} ∈ U_{P}. Then for any u ∈ U_{O} we have:

\overline{h_{S}^{−1}g_{0}uh_{S}} = \overline{h_{S}}\overline{g_{0}}\overline{h_{S}} = \overline{g_{0}}(\overline{g_{0}}^{−1}\overline{h_{S}}\overline{g_{0}})\overline{h_{S}} = \overline{g_{0}}\overline{h_{T}} \overline{h_{S}} = \overline{g_{0}}\overline{h_{P}} = \overline{g_{0}h_{P}} .

In other words, h_{S}^{−1}g_{0}uh_{S} ∈ g_{0}U_{P}. Hence g_{0}^{−1}V_{O}g_{0} ⊄ V_{O} (because by definition of the set V_{O} we have that V_{O} ∩ g_{0}U_{P} = ∅).

We can illustrate the values of cocycle ϕ_{O,ψ} of type III by the diagram

\begin{matrix}S & P \\ O & T \\ T & P \\ O & S \end{matrix}

3.5 δ on cocycles

Now we want to lift the cohomology classes of cocycles ϕ_{Q,ψ} to the group _{2}Br(E). In other words, we want to describe unramified central simple algebras of an exponent 2 such that its classes maps into classes cls(ϕ_{Q,ψ}) ∈ H^{1}(G, Ē) under the homomorphism π.

According to Section 2.3 and formula (6) the first step must be computing δ(cls(ϕ_{Q,ψ})) for cycles of the I and II types. We will determine δ(cls(ϕ_{Q,ψ})) in terms of explicit 2-cocycles of the group G with coefficients in P(Ē).

Consider the following particular case of the short exact sequence (5):

0 → P(Ē) ⟶ Div^{0}(Ē) ⟶^{sum} Ē → 0 .

For any point W ∈ Ē denote by (W) the same point viewed as a divisor on E. Then

sum((W) − (O)) = W .

Let ϕ be a continuous (in sense of the profinite topology on G) 1-cocycle of the group G with coefficients in Ē. Denote by ϕ̂ the continuous map

ϕ̂: G ⟶ Div^{0}(Ē)

such that

ϕ̂(g) = (ϕ(g)) − (O)

for any g ∈ G. Then ∂ϕ̂ is a 2-cocycle of G with coefficients in P(Ē), and

δ(cls(ϕ)) = cls(∂ϕ̂) .

The formula

∂ϕ̂(g_{1}, g_{2}) = g_{1}ϕ̂(g_{2}) − ϕ̂(g_{1}g_{2}) + ϕ̂(g_{2})  (14)

allows to compute ∂ϕ̂ by ϕ̂. We need to do this for cocycles of types I, II and III. First we consider the cases of cocycles of I and II types.

4 Quaternion and cyclic algebras

4.1 Algebras corresponding to cocyles of type I

Proposition 4.1.1 Let ϕ_{O,ψ} be a cocycle of type I, U = ker(ψ) and W = U ∪ g_{0}U the subgroup of index 2 in G (see Proposition 3.4.1). Consider an element a ∈ k\k^{2} such that k(\sqrt{a}) = k̄^{W} is the subfield invariant under the action of W. Then the quaternion algebra

(a, x − x_{P})

with center k(E) is unramified. The class of (a, x − x_{P}) maps into the class cls(ϕ_{O,ψ}) under the homomorphism π : Br(E) → H^{1}(G, Ē).

Proof. Consider the 2-cocycle Φ_{P,a} : G × G → k̄(Ē)^{∗} defined by the formula

Φ_{P,a}(g_{1}, g_{2}) = \{ \begin{matrix}x − x_{P} , if (g_{1}, g_{2}) ∈ (G\W) × (G\W) \\ 1 , in the other case \end{matrix}

The cocycle Φ_{P,a} defines the quaternion algebra (a, x − x_{P}) over k(E). If div : k̄(Ē)^{∗} → P(Ē) is the map sending f ∈ k̄(Ē)^{∗} into the divisor div(f) of zeros and poles of the function f, then ∂ϕ̂_{O,ψ} = div ∘Φ_{P,a}. Taking into account the diagram in Section 2.3, one can see that

inv(cls(Φ_{P,a})) = β(α(cls(Φ_{P,a}))) = β(cls(div ∘Φ_{P,a})) = β(cls(∂ϕ̂_{O,ψ})) = β(δ(cls(ϕ_{O,ψ})) = 0 .

Hence the algebra (a, x − x_{P}) is unramified and π(cls(Φ_{P,a})) = cls(ϕ_{O,ψ}).

4.2 δ for cocycles of type II

Proposition 4.2.1 Let ϕ_{T,ψ} be a cocycle of type II. Then the 2-cocycle ∂ϕ̂_{T,ψ} : G × G → P(Ē) is given in the following table

g_{2} ∈ G\(H ∪ g_{0}U)  0  (S) − (T) + (P) − (O)  2(T) − 2(O)  (T) − (P) + (S) − (O)
g_{2} ∈ g_{0}U  0  (T) − (S) + (P) − (O) (S) − (P) + (T) − (O)  2(S) − 2(O)
g_{2} ∈ H\U  0  2(P) − 2(O)  (P) − (S) + (T) − (O) (P) − (T) + (S) − (O)
g_{2} ∈ U  0  0  0  0
g_{1} ∈ U  g_{1} ∈ H\U  g_{1} ∈ g_{0}U  g_{1} ∈ G\(H ∪ g_{0}U)

Proof. By Proposition 3.4.1 we have the following explicit formula for ϕ̂_{T,ψ}:

ϕ̂_{T,ψ}(g) = \{ \begin{matrix}0 , g ∈ U ⇔ i = 0 \\ (T) − (O) , g ∈ g_{0}U ⇔ i = 1 \\ (P) − (O) , g ∈ H\U ⇔ i = 2 \\ (S) − (O) , g ∈ G\(H ∪ g_{0}U) ⇔ i = 3 \end{matrix}  (15)

Let g_{1}, g_{2} ∈ G and i_{1}, i_{2} are the orders of g_{1}, g_{2} respectively. It is clear that the order i of g_{1}g_{2} is equal to i_{1} + i_{2} modulo 4. The direct computation with the formulas (15) and (14) gives the following explicit form of the cocycle ∂ϕ̂_{T,ψ}:

\begin{matrix}i_{1} & i_{2} & g_{1}ϕ_{T,ψ}(g_{2}) & −ϕ_{T,ψ}(g_{1}g_{2}) & ϕ_{T,ψ}(g_{1}) & sum \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & (T) − (O) & −(T) + (O) & 0 & 0 \\ 0 & 2 & (P) − (O) & −(P) + (O) & 0 & 0 \\ 0 & 3 & (S) − (O) & −(S) + (O) & 0 & 0 \\ 1 & 0 & 0 & −(T) + (O) & (T) − (O) & 0 \\ 1 & 1 & (S) − (O) & −(P) + (O) & (T) − (O) & (S) − (P) + (T) − (O) \\ 1 & 2 & (P) − (O) & −(S) + (O) & (T) − (O) & (P) − (S) + (T) − (O) \\ 1 & 3 & (T) − (O) & 0 & (T) − (O) & 2(T) − 2(O) \\ 2 & 0 & 0 & −(P) + (O) & (P) − (O) & 0 \\ 2 & 1 & (T) − (O) & −(S) + (O) & (P) − (O) & (T) − (S) + (P) − (O) \\ 2 & 2 & (P) − (O) & 0 & (P) − (O) & 2(P) − 2(O) \\ 2 & 3 & (S) − (O) & −(T) + (O) & (P) − (O) & (S) − (T) + (P) − (O) \\ 3 & 0 & 0 & −(S) + (O) & (S) − (O) & 0 \\ 3 & 1 & (S) − (O) & 0 & (S) − (O) & 2(S) − 2(O) \\ 3 & 2 & (P) − (O) & −(T) + (O) & (S) − (O) & (P) − (T) + (S) − (O) \\ 3 & 3 & (T) − (O) & −(P) + (O) & (S) − (O) & (T) − (P) + (S) − (O) \end{matrix}

Rewriting this table in more compact form we obtain the statement of the proposition.

4.3 Cross products corresponding to cocycles of type II

We need some lemma. For short we set

D_{P} = 2(P) − 2(O) = div(x − x_{P}) ,
D_{T} = 2(T) − 2(O) = div(x − x_{T}) ,
D_{S} = 2(T) − 2(O) = div(x − x_{S}) ,

and

D = (P) + (T) + (S) − 3(O) .

Let

Γ = ⟨D, D_{P} , D_{T} , D_{S}⟩

be the subgroup in P(Ē) generated by these principle divisors. Also let

Ω = ⟨y, x − x_{P} , x − x_{T} , x − x_{S}⟩

be the subgroup in k̄(Ē)^{∗} generated by the functions x − x_{P}, x − x_{T}, x − x_{S} and y. It is clear that Γ and Ω are discrete G-submodules in P(Ē) and k̄(Ē)^{∗} respectively. The restriction of the homomorphism div : k̄(Ē)^{∗} → P(Ē) on the submodule Γ is a surjective G-homomorphism

\xymatrix{Ω \ar@{->>}[r]^{div} & Γ} .

Lemma 4.3.1 Let F = F ⟨D, D_{P} , D_{T} , D_{S}⟩ denotes the free abelian group generated by the set {D, D_{P} , D_{T} , D_{S}}. Let τ : F → Ω be the homomorphism defined by τ(D_{P}) = x − x_{P}, τ(D_{T}) = x − x_{T}, τ(D_{S}) = x − x_{S}, τ(D) = y. Let μ : F → Γ be the homomorphism which sends any element of the basis of the group F to itself. Then there exists a unique homomorphism (of discrete Gmodules) γ, which makes the diagram

\xymatrix{F \ar[r]^{μ} \ar[dr]_{τ} & Γ \ar[d]^{γ} \\  & Ω}

commutative. In particular,

div ∘γ = id_{Γ} .

Proof. Observe that 2D = D_{P} + D_{T} + D_{S}. Show that it is a unique relation in Γ. Indeed, let l, m, n, t ∈ ℤ and D + mD_{P} + nD_{T} + tD_{S} = 0. Then

l(P) + l(T) + l(S) − 3l(O) + 2m(P) − 2m(O) + 2n(T) − 2n(O) + 2t(S) − 2t(O) =
(l + 2m)(P) + (l + 2n)(T) + (l + 2t)(S) − 2(m + n + t)(O) − 3l(O) = 0 ⇒

⇒ \{ \begin{matrix}l + 2m = 0 \\ l + 2n = 0 \\ l + 2t = 0 \\ 2(m + n + t) + 3l = 0 \end{matrix}

Solving this system one can see that m = n = t and l = −2m. Hence the relation lD + mD_{P} + nD_{T} + tD_{S} = 0 is of the form −2mD + mD_{P} + mD_{T} + mD_{S} = 0, that is 2D = D_{P} + D_{T} + D_{S}.

Thus we have that Γ is the quotient group of F = F ⟨D, D_{P} , D_{T} , D_{S}⟩ by the cyclic subgroup generated by the element D_{P} + D_{T} + D_{S} − 2D. Then the existence of γ follows from the equality

τ(D_{P} + D_{T} + D_{S} − 2D) = \frac{(x − x_{P})(x − x_{T})(x − x_{S})}{y^{2}} = \frac{y^{2}}{y^{2}} = 1 .

Definition 4.3.1 Let ϕ_{T,ψ} be a cocycle of type II defined by a normal subgroup U in G such that U ⊂ H and G/U ≅ ℤ/4. Let

Φ_{T,ψ} := γ ∘ ∂ϕ̂_{T,ψ} .

Since γ is a homomorphism of discrete G-modules and ∂ϕ̂_{T,ψ} is a 2-cocycle, the map Φ_{T,ψ} is a 2-cocycle of the group G with coefficients in k̄(Ē)^{∗}. Let

A_{U}

be the central simple algebra over k(E) which is the cross product defined by the cocycle Φ_{T,ψ}.

Proposition 4.3.1 Let ϕ_{T,ψ} be a cocycle of type II and U = ker(ψ). Then the 2-cocyle Φ_{T,ψ} from Definition 4.3.1 can be explicitly given by the following table

g_{2} ∈ G\(H ∪ g_{0}U)  1  \frac{y}{x−x_{T}}  x − x_{T}  \frac{y}{x−x_{P}}

g_{2} ∈ g_{0}U  1  \frac{y}{x−x_{S}}  \frac{y}{x−x_{P}}  x − x_{S}

g_{2} ∈ H\U  1  x − x_{P}  \frac{y}{x−x_{S}}  \frac{y}{x−x_{T}}

g_{2} ∈ U  1  1  1  1

g_{1} ∈ U g_{1} ∈ H\U g_{1} ∈ g_{0}U g_{1} ∈ G\(H ∪ g_{0}U)

The cross product A_{U} is an unramified central simple algebra of exponent 2 over the function field k(E). The class of A_{U} in Br(E) maps into the class cls(ϕ_{T,ψ}) under the homomorphism π : Br(E) → H^{1}(G, Ē).

Proof. To obtain the table it is sufficient to apply the homomorphism γ to the table 4.2.1. Since div ∘γ = id_{Γ}, it follows that div ∘Φ_{T,ψ} = div ∘γ ∘ ∂ϕ̂_{T,ψ} = ∂ϕ̂_{T,ψ}. Hence α(cls(Φ_{T,ψ})) = cls(∂ϕ̂_{T,ψ}) = δ(cls(ϕ_{T,ψ})). Taking into account the commutative diagram from Section 2.3, we see that

inv(cls(Φ_{T,ψ})) = β ∘ α(cls(Φ_{T,ψ})) = β ∘ δ(cls(ϕ_{T,ψ})) = 0 .

Therefore, cls(Φ_{T,ψ}) ∈ ker(inv). This means that A_{U} is an unramified algebra over k(E). The exponent of A_{U} is equal to 2 because 2ϕ_{T,ψ} = 0. The last assertion of the proposition is a direct consequence of the equality δ(cls(ϕ_{T,ψ})) = α(cls(Φ_{T,ψ})) and the definition of the homomorphism π.

4.4 Cyclic algebras of degree 4

Note that the algebra A_{U} splits by a cyclic extension of degree four. To find a corresponding cyclic algebra we need some lemma relating to second cohomology of a cyclic group of order 4. Let C = {1, c, c^{2}, c^{4}} be a such group with a generator c. Let M be a C-module, M^{C} = {m ∈ M| cm = m} and N(M) = {m + cm| m ∈ M}. Then we have the well known isomorphism

v : H^{2}(C, M) ⟶ M^{C}/N(M)

(see, for example, [Mac63], chapter IV, § 7).

Lemma 4.4.1 Let ϕ : C × C → M be a 2-cocycle of C with coefficients in M. Then

v(cls(ϕ)) = ϕ(c^{3}, c) + ϕ(1, 1) + ϕ(c^{2}, c) + ϕ(c, c) + N(A) .

Proof. The proof can be provided by comparison of two standard resolutions of the trivial C-module ℤ for the cyclic group C = {1, c, c^{2}, c^{3}}.

We use Lemma 4.4.1 for computation of cyclic algebras of degree 4 similar to cross products A_{U}.

Proposition 4.4.1 Let ϕ_{T,ψ} be a cocycle of type II, U = ker(ψ), A_{U} be the central simple algebra defined by Φ_{T,ψ} (see Definition 4.3.1). Let Z/k be the cyclic field extension of degree 4 defined by U. Then the algebra A_{U} is similar to the cyclic algebra

(Z · k(E)/k(E), f)

over center k(E). Here f = f(x) is the polynomial from the decomposition (1) considered as an element in k(E).

Proof. Let C = G/U be the cyclic group of order 4 with the generator c = g_{0}U, Z = k̄^{C} and Z(E) be the field of Z-rational functions on the curve E_{Z} = E × Spec(Z). The table from Proposition 4.3.1 shows that the 2-cocycle Φ_{T,ψ} factors through the group C, that is there exists the commutative square

\xymatrix{G × G \ar[r] \ar[d]_{Φ_{T,ψ}} & C × C \ar[d]^{φ_{T,ψ}} \\ k̄(Ē)^{∗} & Z(E)^{∗} \ar[l]}

where the cocyle φ_{T,ψ} : C × C → Z(E)^{∗} can be computed according to the table

c^{3} 1 \frac{y}{x−x_{T}} x − x_{T}  \frac{y}{x−x_{P}}

c 1 \frac{y}{x−x_{S}}  \frac{y}{x−x_{P}} x − x_{S}

c^{2} 1 x − x_{P}  \frac{y}{x−x_{S}}  \frac{y}{x−x_{T}}

1 1  1  1  1

1  c^{2}  c  c^{3}

The cohomology class cls(Φ_{T,ψ}) ∈ H^{2}(G, k̄(Ē)^{∗}) is the image of cls(φ_{T,ψ}) ∈ H^{2}(C, Z(E)^{∗}) under the injective inflation homomorphism inf : H^{2}(C, Z(E)^{∗}) ↪ H^{2}(G, k̄(Ē)^{∗}). In terms of central simple algebras this means that the algebra A_{U} ⊗_{k(E)} Z(E) splits, so the class of algebra A_{U} lies in the relative Brauer group Br(Z(E)/k(E)).

Now observe that for M = Z(E)^{∗} the norm homomorphism N : Z(E)^{∗} → Z(E)^{∗} is the norm N_{Z(E)/k(E)} : Z(E)^{∗} → k(E)^{∗} in the sense of field extensions. According to Lemma 4.4.1 the isomorphism

v : H^{2}(C, Z(E)^{∗}) ⟶^{≅} k(E)^{∗}/N(Z(E)^{∗})

sends φ_{T,ψ} into the class of the rational function

φ_{T,ψ}(c^{3}, c) · φ_{T,ψ}(1, 1) · φ_{T,ψ}(c^{2}, c) · φ_{T,ψ}(c, c) =

= (x − x_{S}) · \frac{y}{x − x_{S}} · \frac{y}{x − x_{P}} =

= \frac{y^{2}}{x − x_{P}} = \frac{(x − x_{P}) · f(x)}{x − x_{P}} = f(x) .

Let (Z · k(E)/k(E), f(x)) be the cyclic algebra over k(E) defined by the cyclic extension Z · k(E)/k(E), the generator c = g_{0}U of the group C = G/U and the rational function f(x) ∈ k(E). Then (Z · k(E)/k(E), f(x)) is the crossed product induced by the 2-cocycle

c^{3} 1 f f f
c 1 1 1 f
c^{2} 1 f 1 f
1 1 1 1 1
1 c^{2} c c^{3}

Using Lemma (4.4.1) once more we see that the isomorphism

v : H^{2}(C, Z(E)^{∗}) →^{≅} k(E)^{∗}/N(Z(E)^{∗})

maps the class of φ′_{T,ψ} into the coset of the rational function

φ′_{T,ψ}(c^{3}, c) · φ′_{T,ψ}(1, 1) · φ′_{T,ψ}(c^{2}, c) · φ′_{T,ψ}(c, c) = f · 1 · 1 · 1 = f .

Thus we have that v(cls(ϕ′_{T,ψ})) = v(cls(ϕ_{T,ψ})), whence cls(ϕ′_{T,ψ}) = cls(ϕ_{T,ψ}). In terms of central simple algebras over k(E) this means that the algebra A_{U} is similar to the algebra (Z · k(E)/k(E), f).

5 Dihedral algebras

Now let us calculate unramified central simple algebras over k(E) corresponding to cocycles ϕ_{O,ψ} : G → _{2}Ē of type III. Cocycles of type III was defined in Section 3.3 and calculated in Section 3.4. We shall use the notations of these sections. In particular, U is a subgroup in H such that [H : U] = 2 and U ≠ g_{0}^{−1}Ug_{0}. If ψ_{T} : H → {O, T } is the surjective homomorphism with kernel U and ψ_{S} : H → {O, T } is the surjective homomorphism with kernel g_{0}^{−1}Ug_{0}, then ψ = ψ_{T} + ψ_{S}.

5.1 δ for cocycles of type III

Proposition 5.1.1 Let ϕ_{O,ψ} be a cocycle of type III defined by a subgroup U of index 2 in H such that g_{0}^{−1}Ug_{0} ≠ U, where g_{0}^{2} ∈ U_{O} = U ∩ g_{0}^{−1}Ug_{0} (see Sections 3.3 and 3.4). Let ϕ̂_{O,ψ} : G → Div^{0}(Ē) be the map such that ϕ̂_{O,ψ}(g) = (Q) − (O) if and only if ϕ_{O,ψ}(g) = Q ∈ _{2}Ē. Then the 2-cocycle ∂ϕ̂_{O,ψ} : G × G → P(Ē) can be calculated by the formula

ϕ_{O,ψ}(g_{1}, g_{2}) = \{ \begin{matrix}0 , (g_{1}, g_{2}) ∈ (V_{O} × G) ∪ (G × V_{O}) \\ div(x − x_{S}) , (g_{1}, g_{2}) ∈ (U_{S} × V_{S}) ∪ (g_{0}U_{T} × V_{T}) \\ div(x − x_{T}) , (g_{1}, g_{2}) ∈ (U_{T} × V_{T}) ∪ (g_{0}U_{S} × V_{S}) \\ div(x − x_{P}) , (g_{1}, g_{2}) ∈ V_{P} × V_{P} \\ div(\frac{y}{x−x_{S}}) , (g_{1}, g_{2}) ∈ (V_{T} × V_{P}) ∪ (U_{P} × V_{T}) ∪ (g_{0}U_{P} × V_{S}) \\ div(\frac{y}{x−x_{T}} ) , (g_{1}, g_{2}) ∈ (V_{S} × V_{P}) ∪ (U_{P} × V_{S}) ∪ (g_{0}U_{P} × V_{T}) \\ div(\frac{y}{x−x_{P}} ) , (g_{1}, g_{2}) ∈ ((U_{S} ∪ g_{0}U_{S}) × V_{T}) ∪ ((U_{T} ∪ g_{0}U_{T}) × V_{S}) \end{matrix}

Proof. For the proof we use the formula

∂ϕ̂_{O,ψ}(g_{1}, g_{2}) = g_{1}ϕ̂_{O,ψ}(g_{2}) − ϕ̂_{O,ψ}(g_{1}g_{2}) + ϕ̂_{O,ψ}(g_{1})

and the rule of multiplication in G/U_{O} ≅ D_{4}. All calculations are in the following eight tables:

\begin{matrix}g_{1} & g_{2} & g_{1}ϕ_{O,ψ}(g_{2}) & −ϕ_{O,ψ}(g_{1}g_{2}) & ϕ_{O,ψ}(g_{1}) & sum \\ U_{O} & U_{O} & 0 & 0 & 0 & 0 \\ U_{O} & U_{S} & (S) − (O) & −(S) + (O) & 0 & 0 \\ U_{O} & U_{T} & (T) − (O) & −(T) + (O) & 0 & 0 \\ U_{O} & U_{P} & (P) − (O) & −(P) + (O) & 0 & 0 \\ U_{O} & g_{0}U_{O} & 0 & 0 & 0 & 0 \\ U_{O} & g_{0}U_{S} & (T) − (O) & −(T) + (O) & 0 & 0 \\ U_{O} & g_{0}U_{T} & (S) − (O) & −(S) + (O) & 0 & 0 \\ U_{O} & g_{0}U_{P} & (P) − (O) & −(P) + (O) & 0 & 0 \end{matrix}

\begin{matrix}g_{1} & g_{2} & g_{1}ϕ_{O,ψ}(g_{2}) & −ϕ_{O,ψ}(g_{1}g_{2}) & ϕ_{O,ψ}(g_{1}) & sum \\ U_{S} & U_{O} & 0 & −(S) + (O) & (S) − (O) & 0 \\ U_{S} & U_{S} & (S) − (O) & 0 & (S) − (O) & 2(S) − 2(O) \\ U_{S} & U_{T} & (T) − (O) & −(P) + (O) & (S) − (O) & (T) − (P) + (S) − (O) \\ U_{S} & U_{P} & (P) − (O) & −(T) + (O) & (S) − (O) & (P) − (T) + (S) − (O) \\ U_{S} & g_{0}U_{O} & 0 & −(S) + (O) & (S) − (O) & 0 \\ U_{S} & g_{0}U_{S} & (T) − (O) & −(P) + (O) & (S) − (O) & (T) − (P) + (S) − (O) \\ U_{S} & g_{0}U_{T} & (S) − (O) & 0 & (S) − (O) & 2(S) − 2(O) \\ U_{S} & g_{0}U_{P} & (P) − (O) & −(T) + (O) & (S) − (O) & (P) − (T) + (S) − (O) \end{matrix}

\begin{matrix}g_{1} & g_{2} & g_{1}ϕ_{O,ψ}(g_{2}) & −ϕ_{O,ψ}(g_{1}g_{2}) & ϕ_{O,ψ}(g_{1}) & sum \\ U_{T} & U_{O} & 0 & −(T) + (O) & (T) − (O) & 0 \\ U_{T} & U_{S} & (S) − (O) & −(P) + (O) & (T) − (O) & (S) − (P) + (T) − (O) \\ U_{T} & U_{T} & (T) − (O) & 0 & (T) − (O) & 2(T) − 2(O) \\ U_{T} & U_{P} & (P) − (O) & −(S) + (O) & (T) − (O) & (P) − (S) + (T) − (O) \\ U_{T} & g_{0}U_{O} & 0 & −(T) + (O) & (T) − (O) & 0 \\ U_{T} & g_{0}U_{S} & (T) − (O) & 0 & (T) − (O) & 2(T) − 2(O) \\ U_{T} & g_{0}U_{T} & (S) − (O) & −(P) + (O) & (T) − (O) & (S) − (P) + (T) − (O) \\ U_{T} & g_{0}U_{P} & (P) − (O) & −(S) + (O) & (T) − (O) & (P) − (S) + (T) − (O) \end{matrix}

\begin{matrix}g_{1} & g_{2} & g_{1}ϕ_{O,ψ}(g_{2}) & −ϕ_{O,ψ}(g_{1}g_{2}) & ϕ_{O,ψ}(g_{1}) & sum \\ U_{P} & U_{O} & 0 & −(P) + (O) & (P) − (O) & 0 \\ U_{P} & U_{S} & (S) − (O) & −(T) + (O) & (P) − (O) & (S) − (T) + (P) − (O) \\ U_{P} & U_{T} & (T) − (O) & −(S) + (O) & (P) − (O) & (T) − (S) + (P) − (O) \\ U_{P} & U_{P} & (P) − (O) & 0 & (P) − (O) & 2(P) − 2(O) \\ U_{P} & g_{0}U_{O} & 0 & −(P) + (O) & (P) − (O) & 0 \\ U_{P} & g_{0}U_{S} & (T) − (O) & −(S) + (O) & (P) − (O) & (T) − (S) + (P) − (O) \\ U_{P} & g_{0}U_{T} & (S) − (O) & −(T) + (O) & (P) − (O) & (S) − (T) + (P) − (O) \\ U_{P} & g_{0}U_{P} & (P) − (O) & 0 & (P) − (O) & 2(P) − 2(O) \end{matrix}

\begin{matrix}g_{1} & g_{2} & g_{1}ϕ_{O,ψ}(g_{2}) & −ϕ_{O,ψ}(g_{1}g_{2}) & ϕ_{O,ψ}(g_{1}) & sum \\ g_{0}U_{O} & U_{O} & 0 & 0 & 0 & 0 \\ g_{0}U_{O} & U_{S} & (T) − (O) & −(T) + (O) & 0 & 0 \\ g_{0}U_{O} & U_{T} & (S) − (O) & −(S) + (O) & 0 & 0 \\ g_{0}U_{O} & U_{P} & (P) − (O) & −(P) + (O) & 0 & 0 \\ g_{0}U_{O} & g_{0}U_{O} & 0 & 0 & 0 & 0 \\ g_{0}U_{O} & g_{0}U_{S} & (S) − (O) & −(S) + (O) & 0 & 0 \\ g_{0}U_{O} & g_{0}U_{T} & (T) − (O) & −(T) + (O) & 0 & 0 \\ g_{0}U_{O} & g_{0}U_{P} & (P) − (O) & −(P) + (O) & 0 & 0 \end{matrix}

\begin{matrix}g_{1} & g_{2} & g_{1}ϕ_{O,ψ}(g_{2}) & −ϕ_{O,ψ}(g_{1}g_{2}) & ϕ_{O,ψ}(g_{1}) & sum \\ g_{0}U_{S} & U_{O} & 0 & −(T) + (O) & (T) − (O) & 0 \\ g_{0}U_{S} & U_{S} & (T) − (O) & 0 & (T) − (O) & 2(T) − 2(O) \\ g_{0}U_{S} & U_{T} & (S) − (O) & −(P) + (O) & (T) − (O) & (S) − (P) + (T) − (O) \\ g_{0}U_{S} & U_{P} & (P) − (O) & −(S) + (O) & (T) − (O) & (P) − (S) + (T) − (O) \\ g_{0}U_{S} & g_{0}U_{O} & 0 & −(T) + (O) & (T) − (O) & 0 \\ g_{0}U_{S} & g_{0}U_{S} & (S) − (O) & −(P) + (O) & (T) − (O) & (S) − (P) + (T) − (O) \\ g_{0}U_{S} & g_{0}U_{T} & (T) − (O) & 0 & (T) − (O) & 2(T) − 2(O) \\ g_{0}U_{S} & g_{0}U_{P} & (P) − (O) & −(S) + (O) & (T) − (O) & (P) − (S) + (T) − (O) \end{matrix}

\begin{matrix}g_{1} & g_{2} & g_{1}ϕ_{O,ψ}(g_{2}) & −ϕ_{O,ψ}(g_{1}g_{2}) & ϕ_{O,ψ}(g_{1}) & sum \\ g_{0}U_{T} & U_{O} & 0 & −(S) + (O) & (S) − (O) & 0 \\ g_{0}U_{T} & U_{S} & (T) − (O) & −(P) + (O) & (S) − (O) & (T) − (P) + (S) − (O) \\ g_{0}U_{T} & U_{T} & (S) − (O) & 0 & (S) − (O) & 2(S) − 2(O) \\ g_{0}U_{T} & U_{P} & (P) − (O) & −(T) + (O) & (S) − (O) & (P) − (T) + (S) − (O) \\ g_{0}U_{T} & g_{0}U_{O} & 0 & −(S) + (O) & (S) − (O) & 0 \\ g_{0}U_{T} & g_{0}U_{S} & (S) − (O) & 0 & (S) − (O) & 2(S) − 2(O) \\ g_{0}U_{T} & g_{0}U_{T} & (T) − (O) & −(P) + (O) & (S) − (O) & (T) − (P) + (S) − (O) \\ g_{0}U_{T} & g_{0}U_{P} & (P) − (O) & −(T) + (O) & (S) − (O) & (P) − (T) + (S) − (O) \end{matrix}

\begin{matrix}g_{1} & g_{2} & g_{1}ϕ_{O,ψ}(g_{2}) & −ϕ_{O,ψ}(g_{1}g_{2}) & ϕ_{O,ψ}(g_{1}) & sum \\ g_{0}U_{P} & U_{O} & 0 & −(P) + (O) & (P) − (O) & 0 \\ g_{0}U_{P} & U_{S} & (T) − (O) & −(S) + (O) & (P) − (O) & (T) − (S) + (P) − (O) \\ g_{0}U_{P} & U_{T} & (S) − (O) & −(T) + (O) & (P) − (O) & (S) − (T) + (P) − (O) \\ g_{0}U_{P} & U_{P} & (P) − (O) & 0 & (P) − (O) & 2(P) − 2(O) \\ g_{0}U_{P} & g_{0}U_{O} & 0 & −(P) + (O) & (P) − (O) & 0 \\ g_{0}U_{P} & g_{0}U_{S} & (S) − (O) & −(T) + (O) & (P) − (O) & (S) − (T) + (P) − (O) \\ g_{0}U_{P} & g_{0}U_{T} & (T) − (O) & −(S) + (O) & (P) − (O) & (T) − (S) + (P) − (O) \\ g_{0}U_{P} & g_{0}U_{P} & (P) − (O) & 0 & (P) − (O) & 2(P) − 2(O) \end{matrix}

Now one can rewrite the results of these calculations in more compact way. This yields the third assertion of the proposition.

5.2 Dihedral cross products

Now observe that the image of the cocycle ∂ϕ̂_{O,ψ} lies in Γ (see Section 4.3 for the definition of Γ). Hence one can obtain the cocycle

Φ_{O,ψ} = γ ∘ ∂ϕ̂_{O,ψ} : G × G ⟶ Ω ⊂ k̄(Ē)^{∗} .

According Proposition 5.1.1 the cocycle Φ_{O,ψ} can be calculated by the formula

Φ_{O,ψ}(g_{1}, g_{2}) = \{ \begin{matrix}1 , (g_{1}, g_{2}) ∈ (V_{O} × G) ∪ (G × V_{O}) \\ x − x_{S} , (g_{1}, g_{2}) ∈ (U_{S} × V_{S}) ∪ (g_{0}U_{T} × V_{T}) \\ x − x_{T} , (g_{1}, g_{2}) ∈ (U_{T} × V_{T}) ∪ (g_{0}U_{S} × V_{S}) \\ x − x_{P} , (g_{1}, g_{2}) ∈ V_{P} × V_{P} \\ \frac{y}{x−x_{S}} , (g_{1}, g_{2}) ∈ (V_{T} × V_{P}) ∪ (U_{P} × V_{T}) ∪ (g_{0}U_{P} × V_{S}) \\ \frac{y}{x−x_{T}} , (g_{1}, g_{2}) ∈ (V_{S} × V_{P}) ∪ (U_{P} × V_{S}) ∪ (g_{0}U_{P} × V_{T}) \\ \frac{y}{x−x_{P}} , (g_{1}, g_{2}) ∈ ((U_{S} ∪ g_{0}U_{S}) × V_{T}) ∪ ((U_{T} ∪ g_{0}U_{T}) × V_{S}) \end{matrix}

Definition 5.2.1 Denote by B_{U} the cross product defined by the cocycle Φ_{O,ψ} (constructed by the cocycle ϕ_{O,ψ} of type III). Then B_{U} is a central simple algebra with center k(E). We shall call B_{U} a dihedral cross product because it can be splited by the normal extension k̄^{U_{O}} with Galois group G/U_{O} ≅ D_{4}.

Proposition 5.2.1 The dihedral cross product B_{U} is an unramified central simple algebra over k(E) of exponent 2 and index at most 4. The class of B_{U} in Br(E) maps under the homomorphism π into the cohomology class of the cocycle ϕ_{O,ψ}.

Proof. Taking into account the considerations in Section 2.3 one can calculate

inv(cls(Φ_{O,ψ})) = β(α(cls(Φ_{O,ψ})))
= β(cls(div ∘Φ_{O,ψ}))
= β(cls(∂ϕ̂_{O,ψ}))
= β(δ(cls(ϕ_{O,ψ}))
= 0 .

Hence B_{U} is unramified and π([B_{U}]) = cls(ϕ_{O,ψ}).

Denote by ρ the homomorphism H^{2}(G, Γ) → H^{2}(G, k̄(Ē)^{∗}) induced by the composition Γ →^{γ} Ω ⊂ k̄(Ē)^{∗}. Then

2 cls(Φ_{O,ψ}) = 2 cls(γ ∘ ∂ϕ̂_{O,ψ})
= 2ρ cls(∂ϕ̂_{O,ψ})
= 2ρδ(cls(ϕ_{O,ψ}))
= ρδ(cls(2ϕ_{O,ψ}))
= 0 .

Hence B_{U} has exponent 2.

Since the restriction Φ_{O,ψ}|_{V_{O}×V_{O}} is zero (see explicit formula for Φ_{O,ψ} below), the algebra B_{U} splits by the field k̄^{V_{O}}. But the subgroup V_{O} has index 4 in G by Proposition 5.1.1. Consequently, the index of B_{U} is at most 4.

5.3 End of the proof

Let c ∈ _{2}H^{1}(G, Ē). According to Proposition 3.3.1 the class c can be presented by a cocycle ϕ of one of three types I, II or III. Let U = ker(ψ), where ψ = ϕ|_{H}. If ϕ is a cocycle of type I, then π([(a, x − x_{P})]) = c, where a ∈ k such that k̄^{U} = k(\sqrt{a}) (see Proposition 4.1.1). So one can put b_{c} = [(a, x − x_{P})] ∈ _{2}π^{−1}(c) (see Section 2.5). If ϕ is a cocycle of type II, then π([(Z, f)]) = c, where Z = k̄^{U} (see Propositions 4.3.1 and 4.4.1) such that k̄^{U} = k(\sqrt{a}). So one can take b_{c} = [(a, x−x_{P})]. At last, if ϕ is a cocycle of type III, then let b_{c} = [B_{U}] (see Proposition 5.2.1). Now applying Lemma 2.4.1 we finish the proof of Theorem 1.0.1.

References

[Lich69] S.Lichtenbaum. Duality Theorems for Curves over P -adic Fields. Invent. Math. 7, 120 - 136 (1969).

[Co88] J.-L.Colliot-Thélène (with the collaboration of J.-J.Sansuc). The rationality problem for fields of invariants under linear algebraic groups (with special regard to the Brauer group). Unpublished Lecture Notes from the 9th ELAM, Santiago de Chile, 1988.

[GMY97] V.I.Guletskiǐ, G.L.Margolin, V.I.Yanchevskiǐ. Presentation of two-torsion part in Brauer groups of curves by quaternion algebras. (In Russian). Dokl. NANB. 41 (1997), no. 6, 4 - 8.

[GY98] V.I.Guletskiǐ, V.I.Yanchevskiǐ. Presentation of torsion in Brauer groups of curves by cyclic algebras.(In Russian). Dokl. AN Belarusi, 42 (1998), no. 2, 52-55.

[YM96] V.I.Yanchevskiǐ, G.L.Margolin. The Brauer groups and the torsion of local elliptic curves. St.Peterburgsburg Math. J. Vol.7 (1996), no.3.

[VY96] G.B.Vasiuk, V.I.Yanchevskiǐ. On the 2-torsion part of Brauer group of local elliptic curves with good reduction defined over the field of 2-adic numbers.(In Russian). Dokl. AN Belarusi, 40 (1996), no.6, 31-33.

[V97] G.B.Vasiuk. On the 2-torsion part of Brauer group of 2-adic elliptic curves in the nonsplit case of bad reduction.(In Russian). Dokl. AN Belarusi, 41 (1997), no.6, 13-16.

[1] M. van den Bergh The algebraic index of a division algebra. Lect. Notes in Math. 1197 (1985) 190-206.

[Fadd51] D.K.Faddeev. Simple algebras over a function field in one variable. Proc. Steklov Inst. 38 (1951) 321 - 344; English transl. in AMS Transl. 3 (1956) 15 - 38.

[Mi81] J.S.Milne. Comparison of the Brauer group with the Tate-Šafarevič group. J. Fac. Science, Univ. Tokyo, Sec. IA. 28 (1981) 735-743 .

[Sch69] W.Scharlau. Über bie Brauer-Gruppe eines algebraischen Funktionen-körpers in einer Variabein. J. für die reine und angew. Math. Bd. 239/240 (1969), S. 1-6.

[Serre] J.-P.Serre. Cohomologie Galoisienne. Springer-Verlag, 1964.

[Hall59] M.Hall. The theory of groups. N.Y. MacMillan, 1959.

[Mac63] S.Maclane. Homology. Die grundlehren der mathematischen wissenschaften. Springer-Verlag, 1963.

[Pie82] R.S.Pierce. Associative algebras. Graduate Texts in Mathematics 88, Springer-Verlag, 1982.