2-Torsion of the Brauer Group

of an Elliptic Curve:

Generators and Relations

V. Chernousov, V. Guletskiǐ

Received: May 1, 2001

Communicated by Ulf Rehmann

Abstract. In this paper we describe the 2-torsion part of the Brauer group Br E of an elliptic curve E defined over an arbitrary field of characteristic ≠ 2 in terms of generators and relations.

2000 Mathematics Subject Classification: 14H52, 16H05, 16K20 Keywords and Phrases: Elliptic curves, Brauer groups.

1 Introduction

Let E be an elliptic curve defined over a field K of characteristic different from 2 and given by an affine equation

y^{2} = f(x),

where f(x) is a unitary cubic polynomial over K without multiple roots. We will say that E is split, semisplit or non-split if f(x) has 3, 1 or no roots in K respectively. Let Br E be the Brauer group of the curve E. The group Br E plays an important role in arithmetic and algebraic geometry. For example, it can be used to study arithmetical properties of elliptic surfaces and some other algebraic varieties ( cf. [AM72], [CEP71], [CSS98], [S99] ). Another important application is the construction of unirational varieties which are not rational. Let us describe the last point in some more details. We follow the famous paper of Artin and Mumford [AM72] slightly modifying their examples. Let S be a smooth projective surface defined over an algebraically closed field of characteristic ≠ 2, say ℂ for simplicity. Assume that S is a rational elliptic surface defined by a regular map π : S → ℙ^{1} such that the generic fiber E_{ξ} = π^{−1}(ξ) is an elliptic curve. Given a quaternion algebra D = (d_{1}, d_{2}) over the function field L = ℂ(S) of the surface S, whose ramification curve has nonsingular components, one can associate a smooth S-scheme ϕ : V_{D} → S in a natural way, all of whose geometric fibres are isomorphic to ℙ^{1} or to ℙ^{1} ∨ ℙ^{1} (the so-called Brauer-Severi scheme). Let C be the ramification curve of D and let C = C_{1} ∪ … ∪ C_{n} be its decomposition into irreducible components. The remarkable thing about V_{D} is that V_{D} viewed as a variety over ℂ is not rational if all components C_{1}, …, C_{n} are disjoint. Namely, Artin and Mumford [AM72] proved that under these conditions V_{D} has 2-torsion in H^{3}(V_{D}, ℤ). Since the torsion in H^{3} is a birational invariant for complete smooth 3-dimensional varieties, V_{D} is not rational. On the other hand, it turns out that for many quaternion algebras D the variety V_{D} is unirational. To prove it we first remark that if we want to have the ramification curve C of D with disjoint irreducible components it is natural to take D such that C has vertical components (with respect to π) only. It easily follows that all candidates for such D are among quaternion algebras in the Brauer group of the generic fiber E_{ξ}. As we show in this paper, there are lots of non-trivial quaternion algebras in Br E_{ξ}. Taking the appropriate D we may assume that C has ≥ 2 irreducible components. As it was said, this implies that the corresponding V_{D} is not rational. Now let η be a generic point of S. Then V_{η} = ϕ^{−1}(η) is a conic over ℂ(η) = ℂ(S) = L. Consider the extension F/L of degree 4 corresponding to the Kummer map E_{ξ} →^{2} E_{ξ}. It kills D, hence the conic V_{η} has an F -point. In particular V_{η} is rational over F, i.e. the function field F (V_{η}) is isomorphic to F (z) over F, where z is a transcendental variable over F. Furthermore, since

F/L corresponds to the Kummer map, we have F ≃^{ℂ(t)} ℂ(t)(E_{ξ}), hence

F (V_{η}) ≃^{F} F (z) ≃^{ℂ(t)} ℂ(t)(E_{ξ})(z) = ℂ(S)(z)

is a purely transcendental extension of ℂ. Here we used the fact that S is a rational surface. Finally, since ℂ(V_{D}) = L(V_{η}) is a subfield of F (V_{η}), V_{D} viewed as a 3-fold variety over ℂ is unirational.

Our construction shows that if we want to produce an explicit example of an unirational variety which is not rational, one needs to know the structure of 2-torsion of Br E_{ξ}. So it makes sense to get an explicit description of 2-torsion 2Br E of the Brauer group of an elliptic curve E defined over an arbitrary field K. One of the main goals of this paper is to accomplish (to some extent) a description of _{2}Br E in terms of generators and relations. The initial results in this direction were obtained in [Pu98] where a description of quaternion algebras over E is presented and in [GMY97] where an explicit description of generators of _{2}Br E for a split elliptic curve is given. The second-named author [G99] generalized the results of [GMY97] for semisplit elliptic curves. Our paper, in fact, grew out of his preprint [G99] and here we go further and obtain more complete results that concern generators as well as relations for arbitrary elliptic curves. Our arguments are elementary and based only on using standard properties of restriction and corestriction maps for H^{1} with coefficients in certain finite modules. After this paper was released as a preprint [CG00] we learnt of the nice paper [S99] of Skorobogatov where he gave, among other things, a description of generators of the Brauer groups of algebraic varieties X defined over a field K of characteristic 0 satisfying the condition H^{0}(\overline{K}, G_{m}) = \overline{K}[X]^{×} = \overline{K}^{×} where \overline{K} is an algebraical closure of K. In that paper the generators of Br X are given in the form of the cup product of certain torsors over X and cocycles in H^{1} with coefficients in finitely generated submodules of Pic (\overline{X}). The proofs in [S99] are based on the heavy machinery of homological algebra. However, it seems worth while to have elementary constructions and proofs for elliptic curves as well.

We proceed to describe our results. Let \overline{K} be a separable closure of K and \overline{E} = E(\overline{K}). The starting point of our consideration is the following exact sequence:

0 → Br K ⟶ Br E ⟶^{κ} H^{1}(K, \overline{E}) → 0 .  (1)

Since E(K) ≠ ∅, the homomorphism κ has a section, so that (1) induces the exact sequence

0 → _{2}Br K ⟶ _{2}Br E ⟶^{κ} _{2}H^{1}(K, \overline{E}) → 0 ,

where the subscript 2 means the 2-torsion part. The main result of the paper is formulated in Theorems 3.6, 4.12, 5.2 and 5.3. After some preliminaries given in Section 2 we construct a section for κ in an explicit form. This eventually enables us to give an explicit description of 2Br E in terms of generators and relations. More exactly, let M be the 2-torsion part of \overline{E} and let Γ = Gal (\overline{K}/K). The Kummer sequence

0 → M ⟶ \overline{E} ⟶^{2} \overline{E} → 0,

where the symbol 2 over the arrow means multiplication by 2, yields the exact sequence

0 → E(K)/2 ⟶^{δ} H^{1}(Γ, M) ⟶^{ζ} _{2}H^{1}(Γ, \overline{E}) → 0 .

Here δ : E(K)/2 ↪ H^{1}(Γ, M) is a connecting homomorphism. In Sections 3 through 5 we show that there exists a homomorphism ϵ : H^{1}(Γ, M) → _{2}Br E with the properties

κ ∘ ϵ = ζ,  ϵ (ker(ζ)) = 0.  (2)

The second property implies that ϵ factors through _{2}H^{1}(Γ, \overline{E}), i.e. there is a unique homomorphism ε : _{2}H^{1}(Γ, \overline{E}) → _{2}Br E such that ε ∘ ζ = ϵ, and the first one shows that ε is a required section.

If f(x) = (x − a)(x − b)(x − c) with a, b, c ∈ K, then M ≃ ℤ/2ℤ ⊕ ℤ/2ℤ; hence

H^{1}(Γ, M) ≃ K^{∗}/(K^{∗})^{2} × K^{∗}/(K^{∗})^{2}.

It turns out that the map

ϵ : K^{∗}/(K^{∗})^{2} × K^{∗}/(K^{∗})^{2} → _{2}Br E

which takes a pair (r, s) ∈ K^{∗} × K^{∗} into the product (r, x − b) ⊗ (s, x − c) of quaternion algebras over K(E) satisfies (2). Thus letting I = Im ϵ, we obtain the natural isomorphism _{2}Br E ≃ _{2}Br K⊕I where, by construction, the second summand I is generated by quaternion algebras over K(E) of the form (r, x−b) and (s, x − c) with r, s ∈ K^{∗}. Assume that f(x) does not split over K. We denote the minimal extension of K over which a section ϵ is already constructed by L. Then using standard properties of restriction and corestriction maps we show that for a special map τ : H^{1}(K, M) → H^{1}(L, M) the composition ϵ = cor ∘ ϵ_{L} ∘ τ satisfies (2). As a corollary of our construction, we again obtain the decomposition

2Br E ≃ _{2}Br K ⊕ cor (Im ϵ_{L}) .  (3)

Note that in all cases the degree of L/K is either two or three. This fact enables us to present generators of the second summand in (3) in an explicit form. It turns out that all of them are tensor products of quaternion algebras over K(E) of a very specific form. It follows from the construction that all relations between our generators are given by algebras from (ϵ ∘ δ)(E(K)/2). These algebras are also presented in an explicit form in Theorems 3.6, 4.12, 5.2 and 5.3 and all of them are parametrized by K-points of the elliptic curve E. This result shows that the two problems of an explicit description of the 2-torsion part of Br E (of course, modulo numerical algebras, i.e. algebras from _{2}Br K) and the group E(K)/2 are, in fact, equivalent. So, every time information about E(K)/2 is available we can effectively describe _{2}Br E and vice versa. In the second part of the paper we apply our results to the computation of 2Br E for an elliptic curve E over a local non-dyadic field K. In this case the structure of the group E(K) is well understood. Applying known results we easily construct generators of E(K)/2 in Sections 7 and 8. This, in turn, yields an explicit description of _{2}Br E in the concluding Sections 8 and 9 very quickly. Thus, we reopen a result of Margolin and Yanchevskii [YM96]. It seems that in this part our argument is more natural and shorter (cp. loc. cit.). Finally, we remark that by repeating almost verbatim our argument one can describe in a similar way the 2-torsion part of Br X for a hyperelliptic curve X defined over a field K such that X(K) ≠ ∅. However, in order to keep the volume reasonable we do not consider hyperelliptic curves in the present paper.

If A is an abelian group, A →^{2} A denotes the homomorphism of multiplication by 2 and _{2}A, A/2 are its kernel and cokernel respectively.

|S| denotes the number of elements in a finite set S. Throughout this paper all fields under consideration are of characteristic ≠ 2. For a field K denote by \overline{K} a separable closure of K, K^{∗} its multiplicative group and K^{∗}^{2} the subgroup of squares. By abuse of language, we will write s for a coset sK^{∗}^{2}, whenever there is no danger of confusion. A variety is always a smooth projective and geometrically integral scheme over a field K. For a variety X over K, we write K(X) for the function field of X and X(K) for the set of its K-points. If L/K is a field extension, we put X_{L} = X ×_{Spec K} Spec L. We also write \overline{X} = X ×_{Spec K} Spec \overline{K} and for brevity \overline{K}-points of \overline{X} will be denoted by the same symbol \overline{X}. In the paper we will consider quaternion algebras and their tensor products only. If A is a central simple algebra over a field K then [A] means its class in the Brauer group Br K. If a, b ∈ K^{∗} and (a, b) is a quaternion algebra, then, for short, we write [a, b] instead of [(a, b)]. The group law in a Brauer group we always write additively: if a, b, c, d ∈ F^{∗}, then [(a, b) ⊗ (c, d)] = [a, b] + [c, d]. If Γ is a profinite group, then H^{∗}(Γ, −) is a Galois cohomology functor. Let Λ be a subgroup of finite index in Γ. Then res : H^{∗}(Γ, −) → H^{∗}(Λ, −) and cor : H^{∗}(Λ, −) → H^{∗}(Γ, −) are the restriction and corestriction homomorphisms respectively. In particular, if Γ = Gal(\overline{K}/K) and Λ corresponds to a finite exension F/K then (using the cohomological description of Brauer groups) we have the homomorphism of a scalar extension Br K → Br F and the corestriction homomorphism cor_{F/K} : Br F → Br K. Thus, cor_{F/K}[A] means the value of the homomorphism cor_{F/K} on the class [A] ∈ Br F. If E is an elliptic curve over K, then its Brauer group is naturally isomorphic to the unramified Brauer group Br_{nr}(K(E)/K) (see [Lich69], [Co88]). So we will always identify Br E with Br_{nr}(K(E)/K).

Acknowledgements. The authors gratefully acknowledge the support of SFB 343 “Diskrete Strukturen in der Mathematik”, TMR ERB FMRX CT-97-0107 and the hospitality of the University of Bielefeld. We would like also to express our thanks to H. Abels and U. Rehmann for support and encouragement during the preparation of this paper and O. Izhboldin for useful discussions.

2 Preliminaries

Let E be an elliptic curve over a field K defined by an affine equation

y^{2} = f(x),

where f(x) is a unitary cubic polynomial over K without multiple roots. Let O be the infinite point on E. On the set of K-points E(K) there is a natural structure of an abelian group, such that O is a zero element. Throughout the paper we denote the 2-torsion subgroup in \overline{E} by M. Let Γ = Gal(\overline{K}/K) be the absolute Galois group of the ground field K. If

f(x) = (x − a)(x − b)(x − c)

is the decomposition of f(x) over \overline{K}, then

M = { O, (a, 0), (b, 0) (c, 0) }.

We say that E is split if a, b, c ∈ K. In this case M ⊂ E(K); hence M is a trivial Γ-module. We say that E is semisplit if f(x) has one root in K only. If f(x) is irreducible over K, then we say that E is non-split. A starting point of our explicit description of _{2}Br E is the following exact sequence:

0 → Br K ⟶^{ι} Br E ⟶^{κ} H^{1}(Γ, \overline{E}) → 0 .  (4)

Here the maps ι and κ are defined as follows (see details in [Fadd51], [Lich69],

[Mi81] or [Sch69]). Recall that we identify Br E with the unramified Brauer group Br_{nr}(K(E)/K). Then ι is induced by the scalar extension functor: if A is a central simple algebra over K, then ι([A]) = [A ⊗_{K} K(E)]. Next let h ∈ Br E. By Tsen’s theorem (see [P82]), we have Br K(E) ≅ H^{2}(Γ, \overline{K}(E)^{∗}). Hence h can be viewed as an element in H^{2}(Γ, \overline{K}(E)^{∗}). Let Div \overline{E} be the group of divisors on \overline{E} and let P (\overline{E}) be the group of principal divisors on \overline{E}. Let h′ be the image of h under the homomorphism

H^{2}(Γ, \overline{K}(E)^{∗}) ⟶ H^{2}(Γ, P (\overline{E}))

induced by the map \overline{K}(E)^{∗} → P (\overline{E}) that takes a rational function f to its divisor div(f). Since h belongs to the unramified subgroup of Br K(E) ≅ H^{2}(Γ, \overline{K}(E)^{∗}), it follows that h′ lies in the kernel of the homomorphism

H^{2}(Γ, P (\overline{E})) ⟶ H^{2}(Γ, Div(\overline{E}))  (5)

induced by the embedding P (\overline{E}) → Div(\overline{E}). Let Div^{0}(\overline{E}) be the group of degree zero divisors on \overline{E}. Clearly, H^{1}(Γ, ℤ) = 0, so that a natural homomorphism H^{2}(Γ, Div^{0}(\overline{E})) → H^{2}(Γ, Div(\overline{E})) is injective. Therefore, the kernel of (5) coincides with the kernel of

H^{2}(Γ, P (\overline{E})) ⟶ H^{2}(Γ, Div^{0}(\overline{E}))

and the last one coincides with the image of the connecting homomorphism

∂ : H^{1}(Γ, \overline{E}) ⟶ H^{2}(Γ, P (\overline{E}))

induced by the exact sequence

0 → P (\overline{E}) ⟶ Div^{0}(\overline{E}) ⟶ \overline{E} → 0 .

Since E(K) ≠ ∅ and H^{1}(Γ, ℤ) = 0, we easily get

H^{1}(Γ, Div^{0}(\overline{E})) = H^{1}(Γ, Div(\overline{E})) = 1,

so that ∂ is injective. It follows that there exists a unique element h′′ ∈ H^{1}(Γ, \overline{E}) such that ∂(h′′) = h′. Then, by definition, κ(h) = h′′.

We claim that sequence (4) splits. Indeed, if x ∈ E(K) and K(E)_{x} is the completion of K(E) at x, then Br K(E)_{x} ≅ Br K ⊕ Hom_{cont}(Γ, ℚ/ℤ). Let

ς : Br E ⟶ Br K

be the composition

Br E ↪ Br K(E) → Br K(E)_{x} ≅ Br K ⊕ Hom_{cont}(Γ, ℚ/ℤ) → Br K

where the last homomorphism is the projection on the first summand. It is easy to check that the composition ς ∘ ι is an identical map and the claim follows. In view of splitness, (4) induces the exact sequence

0 → _{2}Br K ⟶^{ι} _{2}Br E ⟶^{κ} _{2}H^{1}(Γ, \overline{E}) → 0,  (6)

which also splits. Since _{2}H^{1}(Γ, \overline{E}) can be easily computed, we obtain that for an explicit description of _{2}Br E it suffices to construct a section for κ. To do it, we first consider the Kummer sequence

0 → M ⟶ \overline{E} ⟶^{2} \overline{E} → 0.  (7)

It yields the exact sequence

0 → E(K)/2 ⟶^{δ} H^{1}(Γ, M) ⟶^{ζ} _{2}H^{1}(Γ, \overline{E}) → 0  (8)

where δ : E(K)/2 ↪ H^{1}(Γ, M) is a connecting homomorphism. In the next three sections we will construct a homomorphism ϵ : H^{1}(Γ, M) → _{2}Br E with the properties

κ ∘ ϵ = ζ, ϵ (ker(ζ)) = 0.

The second property implies that ϵ induces a unique homomorphism ε : 2H^{1}(Γ, \overline{E}) → _{2}Br E such that ε∘ζ = ϵ. Then it follows that κ∘ε∘ζ = κ∘ϵ = ζ. Since ζ is surjective, we conclude that κ ∘ ε = 1, i.e. ε is a required section for κ. Letting I = Im ε, we have _{2}Br E ≅ I ⊕ Im ι ≅ I ⊕ _{2}Br K. As we see in Sections 3, 4 and 5, elements in I are tensor product of quaternion algebras over K(E) of a very specific form. So our construction eventually gives a simple system of generators of _{2}Br E modulo numerical algebras (i.e. algebras from Im ι) and according to the construction of the maps ϵ and ε all relations between the generators are given by algebras from ϵ(ker(ζ)). Thus, to find all relations explicitly, we first have to describe the subset Im δ ⊂ H^{1}(Γ, M) and then apply ϵ to its elements. Since the structure of the group H^{1}(Γ, M) (and hence the construction of ϵ) depends on splitting properties of the polynomial f(x), to realize our program we consider split, semisplit and non-split cases in the next three sections separately.

3 Split elliptic case

Let E be a split elliptic curve. Then M is a trivial Γ-module; hence we have

H^{1}(Γ, M) = Hom (Γ, M) .

Fix two non-zero points in M, say (b, 0) and (c, 0). Considering them as generators of M we have an isomorphism

M ≅ ℤ/2 ⊕ ℤ/2 .

It induces the isomorphism

H^{1}(Γ, M) = Hom(G, M) ≅ K^{∗}/K^{∗}^{2} ⊕ K^{∗}/K^{∗}^{2} .

Consider a map

ϵ_{b} : K^{∗}/K^{∗}^{2} ⟶ _{2}Br E

which takes s ∈ K^{∗} into the class [s, x − b]. Here and below, for an element r ∈ K the polynomial x − r is considered as a rational function on E. Clearly, the quaternion algebra (s, x − b) is unramified and ϵ_{b} is a homomorphism. Analogously, consider a homomorphism

ϵ_{c} : K^{∗}/K^{∗}^{2} ⟶ _{2}Br E

which takes s ∈ K^{∗} into the class [s, x − c]. Let now

ϵ = ϵ_{b} ⊕ ϵ_{c} : K^{∗}/K^{∗}^{2} ⊕ K^{∗}/K^{∗}^{2} = Hom(Γ, M) ⟶ _{2}Br E .  (9)

Using the description of κ given in Section 2 it is easy to show that κ ∘ ϵ = ζ.

Lemma 3.1 κ ∘ ϵ = ζ.

Proof. Let P be a non zero point in M. For any s ∈ K^{∗}\K^{∗}^{2} let ϕ_{P,s} be a homomorphism from Γ into M, such that ϕ_{P,s}(g) = P if g ∉ U_{s} = Gal(\overline{K}/K(\sqrt{s})) and ϕ_{s,b}(g) = O otherwise. The group H^{1}(Γ, M) = Hom(Γ, M) is generated by the homomorphisms of type ϕ_{P,s}. Therefore it is sufficient to show that (κ ∘ ϵ)(ϕ_{P,s}) = ζ(ϕ_{P,s}) for any P and s. Let Φ_{P,s} be a homomorphism from Γ into Div^{0}(\overline{E}), such that Φ_{P,s}(g) = (P) − (O) if g ∉ U_{s} and Φ_{s,b}(g) = 0 otherwise. Let dΦ_{P,s} : Γ × Γ → Div^{0}(\overline{E}) be a codifferential of Φ_{P,s}, that is

(dΦ_{P,s})(g_{1}, g_{2}) = g_{1}Φ_{P,s}(q_{2}) − Φ_{P,s}(g_{1}g_{2}) + Φ_{P,s}(g_{2})

for any g_{1}, g_{2} ∈ Γ. Then dΦ_{P,s} takes its values in P(E) and ∂(cls(ϕ_{P,s})) = cls(dΦ_{P,s}) where cls denotes a cohomology class of a cocycle. Using the above formula for dΦ_{P,s} it is easy to compute that dΦ_{P,s}(g_{1}, g_{2}) = 2(P) − 2(O) if g_{1} and g_{2} lie in Γ\U_{s} and dΦ_{P,s}(g_{1}, g_{2}) = 0 otherwise. Let x(P) be the x-coordinate of P and let ψ_{P,s} : Γ × Γ → \overline{K}(E)^{∗} be a map, such that Ψ_{P,s}(g_{1}, g_{2}) = x − x(P) if g_{1} and g_{2} lie in Γ\U_{s} and dΦ_{P,s}(g_{1}, g_{2}) = 1 otherwise. Then we see that the composition of Ψ_{P,s} with the natural homomorphism div : \overline{K}(E)^{∗} → P(\overline{E}) coincides with the homomorphism dΦ_{P,s}. Therefore ∂(cls(ϕ_{P,s})) = η(cls(Ψ_{P,s})). Since Ψ_{P,s} is a cocycle of the unramified quaternion algebra (s, x − x(P)), we see that κ([s, x − x(P)]) = cls(ϕ_{P,s}). But [s, x − x(P)] is equal to ϵ(ϕ_{P,s}). So we have κ(ϵ(ϕ_{P,s})) = cls(ϕ_{P,s}) = ζ(ϕ_{P,s}).

According to our plan we also need to make sure that ϵ(Im δ) = 0. The description of Im(δ) in the split case is well known. However for the reader’s convenience we describe this image in details. To ease notation, for a point (u, v) ∈ E(K) the coset (u, v) + 2E(K) will be denoted by the same symbol (u, v). We start with a simple lemma which gives a formula for dividing a point (u, v) ∈ E(K) in the group \overline{E} by 2. Let

r = \sqrt{u − a} , s = \sqrt{u − b} , t = \sqrt{u − c} and w = r + s − t .

Let also

p = \frac{1}{2}(w^{2} − (r^{2} + s^{2} + t^{2})) + u = rs − rt − st + u and q = w(p − u) + v .

Lemma 3.2 We have (p, q) ∈ \overline{E} and 2(p, q) = (u, v).

Proof. This is a straightforward calculation (see also the proof of Theorem 4.1 on page 38 in [Hu87]) and we omit the details to the reader.

Proposition 3.3 Let (u, v) ∈ E(K). Then

δ(u, v) = \{ \begin{matrix}(u − c, u − b) & if u ≠ b and u ≠ c, \\ (b − c, (b − c)(b − a)) & if u = b, \\ ((c − a)(c − b), c − b) & if u = c, \\ (1, 1) & if u = ∞. \end{matrix}

Proof. If u = b, then u ≠ a and u ≠ c and, analogously, if u = c, then u ≠ a and u ≠ b. Therefore, by the symmetry argument, it suffices to prove the statement in the case u ≠ b and u ≠ c. Moreover, we consider only “a generic case” where u − b and u − c generate a subgroup in K^{∗}/K^{∗}^{2} of order 4, i.e. u − b and u − c are nontrivial and different modulo squares. The other cases can be handled in a similar way. We keep the notation of Lemma 3.2. Since 2(p, q) = (u, v), the cocycle δ(u, v) corresponds to the homomorphism ϕ_{(u,v)} : Γ → M that takes γ to the point (p, q)^{γ} −(p, q). Let U = Gal(\overline{K}/K(s)) and V = Gal(\overline{K}/K(t)). We fix arbitrary automorphisms

σ ∈ U\V and τ ∈ V \U .

Let ψ_{(u,v)} ∈ Hom (Γ, M) be the homomorphism corresponding to the pair (u − c, u − b). Clearly, ϕ_{(u,v)}(γ) = ψ_{(u,v)}(γ) = 0 for all γ ∈ Gal(\overline{K}/K(s, t)) and ψ_{(u,v)}(σ) = b, ψ_{(u,v)}(τ) = c. So it suffices to show that the abscissas of the points (p, q)^{σ} − (p, q) and (p, q)^{τ} − (p, q) are b and c respectively. Note that, by construction, we have

σ(r) = −r , σ(s) = s and σ(t) = −t .

Then it easily follows that (p, q)^{σ} ≠ ±(p, q). Denoting by m the abscissa of the point (p, q)^{σ} − (p, q) and taking into account the group law algorithm given on p. 58 in [Sil85], we have

\begin{aligned}m & = & (\frac{q+σ(q)}{σ(p)−p})^{2} + a + b + c − σ(p) − p \\ & = & (\frac{q+σ(q)}{σ(p)−p})^{2} + 3u − r^{2} − s^{2} − t^{2} − σ(p) − p . \end{aligned}

Since q = w(p − u) + v and p = rs − rt − st + u , we can write

q + σ(q) = w(p − u) + v + σ(w)σ(p − u) + v
= w(p − u) + σ(w)σ(p − u) + 2v
= (r + s − t)(rs − rt − st) + (−r + s + t)(−rs − rt + st) + 2rst
= 2r^{2}s − 4rst + 2st^{2}
= 2s(r − t)^{2} ,

and

σ(p) − p = −rs − rt + st − rs + rt + st = 2s(t − r) .

Thus, we obtain

m = (\frac{(2s(r−t)^{2}}{2s(t−r)} )^{2} + 3u − r^{2} − s^{2} − t^{2} + 2rt − 2u
= −s^{2} + u
= b .

The equality (p, q)^{τ} − (p, q) = (c, 0) is proved in exactly the same fashion.

Proposition 3.4 ϵ(Im δ) = 0.

Proof. Let (u, v) ∈ E(K). Since κ ∘ ϵ = ζ, we have (κ ∘ ϵ) (δ(u, v)) = 0, i.e. the algebra ϵ(δ(u, v)) is numerical. We claim that this algebra is trivial. Indeed, we may assume that (u, v) is a point in E(K) such that u−b ≠ 0 and u−c ≠ 0. Then the evaluation of the algebra

ϵ(δ(u, v)) = [u − c, x − b] + [u − b, x − c]

at the point (u, v) yields

[u − c, u − b] + [u − b, u − c] = 2[u − c, u − b] = 0 .

This implies that the algebra ϵ(δ(u, v)) is itself trivial, as required.

Summarizing the above results, we obtain the following Proposition 3.5 Let E/K be a split elliptic curve over K, char K ≠ 2. Let κ : _{2}Br E → _{2}H^{1}(Γ, \overline{E}) be the homomorphism described in Section 2 and let ζ : H^{1}(Γ, M) → _{2}H^{1}(Γ, \overline{E}) be the homomorphism induced by the embedding M ⊂ \overline{E}. Let also

ϵ : H^{1}(Γ, M) ⟶ _{2}Br E

be the homomorphism defined by (9). Then

(i) κ ∘ ϵ = ζ .

(ii) There exists a unique homomorphism

ε : _{2}H^{1}(Γ, \overline{E}) ⟶ _{2}Br E

such that ε ∘ ζ = ϵ and κ ∘ ε = 1_{_{2}H^{1}(Γ,\overline{E})} is an identical map.

Proof. The equality κ ∘ ϵ = ζ is proved in Lemma 3.1. Since ζ is the cokernel of the homomorphism δ and, by Proposition 3.4, ϵ(Im δ) = 0, there exists a unique homomorphism ε : _{2}H^{1}(Γ, \overline{E}) → _{2}Br E, such that ε ∘ ζ = ϵ. Since κ ∘ ε ∘ ζ = κ ∘ ϵ = ζ, we obtain that κ ∘ ε = 1_{_{2}H^{1}(Γ,\overline{E})} because ζ is an epimorphism.

Reformulating the results of Proposition 3.5 in terms of central simple algebras and using Proposition 3.3, we obtain

Theorem 3.6 Let E/K be a split elliptic curve defined by an affine equation

y^{2} = (x − a)(x − b)(x − c),

where a, b, c ∈ K and char K ≠ 2. Let ϵ : _{2}H^{1}(Γ, \overline{E}) → _{2}Br E be the section for the homomorphism κ : _{2}Br E → _{2}H^{1}(Γ, \overline{E}) constructed in Proposition 3.5 and let I = Im ε. Then

2Br E = _{2}Br K ⊕ I

and every element in I is represented by a biquaternion algebra

(r, x − b) ⊗ (s, x − c)

with r, s ∈ K^{∗}. Conversely, every algebra of such a type is unramified over E. An algebra A = (r, x − b) ⊗ (s, x − c) is trivial in I = Im (ε) if and only if A is similar to an algebra of one of the three following types:

(i) an algebra

(u − c, x − b) ⊗ (u − b, x − c),

where u is the abscissa of a point in E(K) such that u − b ≠ 0 and u − c ≠ 0;

(ii) an algebra

(b − c, x − b) ⊗ ((b − c)(b − a), x − c) ;

(iii) an algebra

((c − a)(c − b), x − b) ⊗ (c − b, x − c) .

4 Semisplit elliptic case

Let E be a semisplit elliptic curve given by an affine equation

y^{2} = (x − w)(x^{2} − d),

where w, d ∈ K, char K ≠ 2 and d is not a square in K^{∗}. Let L = K(\sqrt{d}), Γ = Gal (\overline{K}/K) and Λ = Gal (\overline{K}/L). Clearly, Λ is a subgroup of index two in Γ and

M ≅ M_{Γ}^{Λ}(ℤ/2),

where M_{Γ}^{Λ}(ℤ/2) is an induced Γ-module. Therefore, by Shapiro’s lemma (see, for example, [Serre64]), we have

H^{1}(Γ, M) = H^{1}(Γ, M_{Γ}^{Λ}(ℤ/2)) ≅ H^{1}(Λ, ℤ/2) ≅ L^{∗}/L^{∗}^{2} .

Let us consider the split elliptic curve E_{L} = E ×_{K} L over L. Fixing its points (b, 0), (c, 0), where b = \sqrt{d}, c = −\sqrt{d}, we get the isomorphisms over L

M ≅ ℤ/2 ⊕ ℤ/2, H^{1}(Λ, M) ≅ L^{∗}/L^{∗}^{2} ⊕ L^{∗}/L^{∗}^{2} .

Under these identifications the restriction map is given by the formula

res : H^{1}(Γ, M) → H^{1}(Λ, M), l ∈ L^{∗}/L^{∗}^{2} → (l^{σ}, l) ∈ L^{∗}/L^{∗}^{2} ⊕ L^{∗}/L^{∗}^{2} (10)

where σ is the nontrivial automorphism L/K. We denote the homomorphisms constructed in the previous section for the split curve E_{L} by the same symbols but equipped with the subscript L. Thus, we have the homomorphisms

ϵ_{L} : H^{1}(Λ, M) ⟶ _{2}Br (E_{L}) ,
ζ_{L} : H^{1}(Λ, M) ⟶ _{2}H^{1}(Λ, \overline{E})

and

ε_{L} : _{2}H^{1}(Λ, \overline{E}) ⟶ _{2}Br (E_{L}) .

Let

H^{1}(Γ, M) ≅ L^{∗}/L^{∗}^{2} ⟶^{τ} L^{∗}/L^{∗}^{2} ⊕ L^{∗}/L^{∗}^{2} ≅ H^{1}(Λ, M)

be the homomorphism which takes l into the pair (1, l). We define the homomorphism

ϵ : H^{1}(Γ, M) ⟶ _{2}Br E

by means of the following commutative diagram

\xymatrix{H^{1}(Λ, M) \ar[r]^{ϵ_{L}} & _{2}Br (E_{L}) \ar[d]^{cor} \\ H^{1}(Γ, M) \ar[u]^{τ} \ar[r]^{ϵ} & _{2}Br E}

Documenta Mathematica · Quadratic Forms LSU 2001 · 85–120

Proposition 4.1 Let E/K be a semisplit elliptic curve. Let ζ : H^{1}(Γ, M) → 2H^{1}(Γ, \overline{E}) be the homomorphism induced by the embedding M ⊂ \overline{E} and let ϵ be the above homomorphism. Then there exists a homomorphism

ε : _{2}H^{1}(Γ, \overline{E}) ⟶ _{2}Br E

such that κ ∘ ε = 1_{_{2}H^{1}(Γ,\overline{E})} (i.e. ε is a section for the homomorphism κ) and ε ∘ ζ = ϵ .

Proof. The proof is based on a diagram chase. We divide it into a sequence of simple observations.

Lemma 4.2 The restriction homomorphism

H^{1}(Γ, M) ⟶^{res} H^{1}(Λ, M)

is injective.

Proof. This easily follows from (10).

Lemma 4.3 The composition

H^{1}(Γ, M) ⟶^{τ} H^{1}(Λ, M) ⟶^{cor} H^{1}(Γ, M)

coincides with the identical map 1_{H^{1}(Γ,M)}.

Proof. By Lemma 4.2, the homomorphism res : H^{1}(Γ, M) → H^{1}(Λ, M) is injective. Therefore, it is sufficient to prove that res ∘ cor ∘τ = res. Let l ∈ L^{∗}. Using (10) we have

(res ∘ cor ∘τ)(l) = (res ∘ cor)(1, l) = (1, l) + (1, l)^{σ} =
(1, l) + (l^{σ}, 1) = (l^{σ}, l) = res(l) .

Lemma 4.4 κ ∘ ϵ = ζ .

Proof. The commutative diagram

\xymatrix{H^{1}(Λ, M) \ar[r]^{ζ_{L}} \ar[d]^{cor} & _{2}H^{1}(Λ, \overline{E}) \ar[d]^{cor} & _{2}Br (E_{L}) \ar[l]_{κ_{L}} \ar[d]^{cor} \\ H^{1}(Γ, M) \ar[r]^{ζ} & _{2}H^{1}(Γ, \overline{E}) & _{2}Br E \ar[l]_{κ}}

and Lemma 4.3 imply

κ ∘ ϵ = κ ∘ cor ∘ ϵ_{L} ∘ τ = cor ∘ κ_{L} ∘ ε_{L} ∘ ζ_{L} ∘ τ = cor ∘ ζ_{L} ∘ τ = ζ ∘ cor ∘ τ = ζ .

Documenta Mathematica · Quadratic Forms LSU 2001 · 85–120

Lemma 4.5 cor ∘ ζ_{L} ∘ τ = ζ .

Proof. Clearly, we have cor ∘ ζ_{L} = ζ ∘ cor. Multiplying from the right hand by τ we obtain that cor ∘ ζ_{L} ∘τ = ζ ∘cor ∘ τ = ζ (the last equality holds by Lemma 4.3).

Lemma 4.6 ϵ (Im δ) ⊂ Im ι.

Proof. By Lemma 4.4, we have κ ∘ ϵ = ζ, hence

ϵ (Im δ) = ϵ (ker ζ) ⊂ ker κ = Im ι .

Lemma 4.7 Im ϵ ∩ Im ι = 0.

Proof. Our computations are illustrated by the following commutative diagram

\xymatrix{ & _{2}H^{1}(Λ, \overline{E}) \ar[dr]^{ε_{L}} \\ H^{1}(Λ, M) \ar[ur]^{ζ_{L}} \ar[rr]^{ϵ_{L}} &  & _{2}Br E_{L} \ar[d]^{cor} \ar[r]^{ς_{L}} & _{2}Br L \ar[d]^{cor} \ar[l]^{ι_{L}} \\ H^{1}(Γ, M) \ar[u]^{τ} \ar[rr]^{ϵ} &  & _{2}Br E \ar[r]^{ς} & _{2}Br K \ar[l]^{ι}}

Let b ∈ _{2}Br E be such that b = ϵ(h) = ι(a) for some h ∈ H^{1}(Γ, M) and some a ∈ _{2}Br K. Let c = ζ_{L}(τ(h)). Then

a = (ς ∘ ι)(a) = ς(b) = (ς ∘ cor ∘ ε_{L})(c) = (cor ∘ ς_{L} ∘ ε_{L})(c) = 0,

because ς_{L} ∘ ε_{L} = 0.

Lemma 4.8 ϵ (Im δ) = 0.

Proof. By Lemmas 4.6 and 4.7, we have ϵ (Im δ) ⊂ Im ϵ ∩ Im ι = 0.

We are now in the position to finish the proof of Proposition 4.1. Since ϵ (Im δ) = ϵ (ker ζ) = 0, it follows that there exists a unique homomorphism ε : _{2}H^{1}(Γ, \overline{E}) → _{2}Br E such that ϵ = ε ∘ ζ. Furthermore,

κ ∘ ε ∘ ζ = κ ∘ ϵ = κ ∘ cor ∘ ϵ_{L} ∘ τ = κ ∘ cor ∘ ε_{L} ∘ ζ_{L} ∘ τ =

cor ∘ κ_{L} ∘ ε_{L} ∘ ζ_{L} ∘ τ = cor ∘ ζ_{L} ∘ τ = ζ ∘ cor ∘ τ = ζ .

Since ζ is an epimorphism, it follows that κ ∘ ε = 1_{_{2}H^{1}(Γ,\overline{E})}. Proposition 4.1 is proved.

To reformulate the results of Proposition 4.1 in terms of central simple algebras we need three well-known lemmas which describe images of quaternion algebras under corestriction homomorphisms.

Lemma 4.9 Let F be a field and let P be a finite separable extension of F. Then for elements a ∈ F and b ∈ P we have

cor_{P/F} [a, b] = [a, N_{P/F} (b)]

in the Brauer group Br F.

Proof. This is a well-known fact (see, for instance, [Serre79], p. 209).

Lemma 4.10 Let F be a field and let P be a quadratic extension of F. Suppose that P = F (\sqrt{s}), where s ∈ F. Then for elements a, b ∈ F with the property a + b ≠ 0 we have

cor_{P/F} [a + \sqrt{s}, b − \sqrt{s}] = [a + b, (a^{2} − s)(b^{2} − s)] .

Proof. Let

t = \frac{a + \sqrt{s}}{a + b} and l = \frac{b − \sqrt{s}}{a + b} .

Then t + l = 1, whence [t, l] = [t, 1 − t] = 0 in Br P. Substituting t and l, we have

0 = [t, l] = [\frac{a + \sqrt{s}}{a + b}, \frac{b − \sqrt{s}}{a + b}] =

= [a + \sqrt{s}, b − \sqrt{s}] + [a + b, b − \sqrt{s}] + [a + \sqrt{s}, a + b] + [a + b, a + b].

Taking cor_{P/F} and using Lemma 4.9 we obtain that

0 = cor_{P/F} [a + \sqrt{s}, b − \sqrt{s}] + [a + b, b^{2} − s] + [a^{2} − s, a + b] + [a + b, (a + b)^{2}] .

Therefore,

cor_{P/F} [a + \sqrt{s}, b − \sqrt{s}] = [a + b, b^{2} − s] + [a^{2} − s, a + b] .

Lemma 4.11 Let F be a field and let P = F (\sqrt{s}) be a quadratic extension of F. Let u_{1}, v_{1}, u_{2}, v_{2} ∈ F be such that v_{1} ≠ 0, v_{2} ≠ 0 and v_{1}u_{2} ≠ u_{1}v_{2}. Then

cor_{P/F} [u_{1} + v_{1}\sqrt{s}, u_{2} + v_{2}\sqrt{s}] =

[v_{1}, u_{1}^{2} − v_{1}^{2}s] + [−v_{2}, u_{2}^{2} − v_{2}^{2}s] + [v_{1}u_{2} − u_{1}v_{2}, (u_{1}^{2} − v_{1}^{2}s)(u_{2}^{2} − v_{2}^{2}s)] .

Proof. Let

a = \frac{u_{1}}{v_{1}} and b = −\frac{u_{2}}{v_{2}} .

Then

[u_{1} + v_{1}\sqrt{s}, u_{2} + v_{2}\sqrt{s}] = [v_{1}(a + \sqrt{s}), −v_{2}(b − \sqrt{s})] =
= [v_{1}, −v_{2}] + [a + \sqrt{s}, b − \sqrt{s}] + [v_{1}, b − \sqrt{s}] + [a + \sqrt{s}, −v_{2}] .

Lemmas 4.10 and 4.9 give

cor_{D/F} [u_{1} + v_{1}\sqrt{s}, u_{2} + v_{2}\sqrt{s}] =
[a + b, (a^{2} − s)(b^{2} − s)] + [v_{1}, b^{2} − s] + [−v_{2}, a^{2} − s]

and it remains to substitute a = u_{1}/v_{1}, b = −u_{2}/v_{2}.

Theorem 4.12 Let E be a semisplit elliptic curve over K, char K ≠ 2, given by an affine equation y^{2} = (x − w)(x^{2} − d), where w, d ∈ K and d is not a square in K. Let ε : _{2}H^{1}(Γ, \overline{E}) → _{2}Br E be the section for the homomorphism κ : _{2}Br E → _{2}H^{1}(Γ, \overline{E}) constructed in Proposition 4.1 and let I = Im ε. Then

2Br E ≅ _{2}Br K ⊕ I

and every element in I is represented by either a quaternion algebra

(r, x − w),

where r ∈ K^{∗}, or a biquaternion algebra

(t, r^{2} − t^{2}d) ⊗ (tx + r, (r^{2} − t^{2}d)(x^{2} − d))

where r, t ∈ K and t ≠ 0. Conversely, every algebra of the above types is unramified over E. It is trivial in I if and only if it is similar to a quaternion algebra

(x + u, (u − w)(x − w)),

where u is the abscissa of a point in E(K).

Proof. The first statement is trivial because ε is a section for the homomorphism κ. To prove the second one we have to compute ϵ(h) in terms of quaternion algebras for all h ∈ H^{1}(Γ, M). By definition, ϵ = cor ∘ ϵ_{L} ∘ τ, where L = K(\sqrt{d}). Recall that we identify L^{∗}/L^{∗}^{2} ≅ H^{1}(Γ, M) and L^{∗}/L^{∗}^{2}⊕L^{∗}/L^{∗}^{2} ≅ H^{1}(Λ, M) and that τ : L^{∗}/L^{∗}^{2} → L^{∗}/L^{∗}^{2} ⊕ L^{∗}/L^{∗}^{2} takes l ∈ L^{∗}/L^{∗}^{2} into (1, l). Let l ∈ L^{∗}. Then we have

(cor ∘ ϵ_{L} ∘ τ) (l) = (cor ∘ ϵ_{L}) (1, l) = cor_{L(E)/K(E)} [l, x − \sqrt{d}] .

Let l = r + t\sqrt{d}. If t = 0, then, by Lemma 4.9, we have

cor_{L(E)/K(E)} [r, x − \sqrt{d}] = [r, x^{2} − d] = [r, x − w] .

If t ≠ 0, then, by Lemma 4.11, we have

cor_{L(E)/K(E)}[r + t\sqrt{d}, x − \sqrt{d}] =
[t, r^{2} − t^{2}d] + [1, x^{2} − d] + [tx + r, (r^{2} − t^{2}d)(x^{2} − d)] =
[t, r^{2} − t^{2}d] + [tx + r, (r^{2} − t^{2}d)(x^{2} − d)] .

It remains to find out when an algebra b ∈ I = Im ϵ is trivial. Let b = ϵ (l). By Proposition 4.1, we have ϵ = ε ∘ ζ and ker ε = 0. So b is trivial if and only if l ∈ ker ζ = Im δ. Let (u, v) ∈ E(K) and l = δ(u, v). The commutative square

\xymatrix{E(L)/2 \ar@{^{(}->}[r]^{δ_{L}} & L^{∗}/L^{∗}^{2} ⊕ L^{∗}/L^{∗}^{2} \\ E(K)/2 \ar[u]^{res} \ar@{^{(}->}[r]^{δ} & L^{∗}/L^{∗}^{2} \ar[u]^{res}}

shows that

(l^{σ}, l) = res (l) = (res ∘ δ)(u, v) = (δ_{L} ∘ res) (u, v) = δ_{L}(u, v),

where σ is a unique nontrivial automorphism L/K. Proposition 3.3 gives

δ_{L}(u, v) = (u + \sqrt{d}, u − \sqrt{d}).

Thus, l = u − \sqrt{d} and finally we get

(ϵ ∘ δ) (u, v) = (cor_{L/K} ∘ ϵ_{L} ∘ τ) (l)
= (cor_{L/K} ∘ ϵ_{L}) (1, l)
= cor_{L/K} [u − \sqrt{d}, x + \sqrt{d}]
= [x + u, (u^{2} − d)(x^{2} − d)]
= [x + u, (u − w)(x − w)] .

The theorem is proved.

To consider the non-split case it is convenient to have a reformulation of the last theorem without conditions on the equation of E. Let E be a semisplit elliptic curve given by an affine equation

y^{2} = (x − a)g(x),

where a ∈ K and g(x) is a unitary irreducible polynomial over K. Denote the roots of g(x) by b and c. Let also E′ be a semisplit elliptic curve given by an equation

y^{2} = (x − w)(x^{2} − d),

where

w = a − \frac{b + c}{2} and d = \frac{(b − c)^{2}}{4} .

Clearly, the map

E ⟶ E′

(u, v) ↦ (u − \frac{b + c}{2}, v)

is an isomorphism of elliptic curves. It induces the commutative diagram

[figure: 0; _{2}Br K; _{2}Br E; κ; _{2}H^{1}(Γ, \overline{E}); 0; 0; _{2}Br K; _{2}Br E′; κ′; ≅; _{2}H^{1}(Γ, \overline{E}′); ≅; 0]

Let ε′ : _{2}H^{1}(Γ, \overline{E}′) → _{2}Br E′ be the section for the homomorphism κ′ : 2Br E′ → _{2}H^{1}(Γ, \overline{E}′) described in Proposition 4.1. Let ε : _{2}H^{1}(Γ, \overline{E}) → _{2}Br E be the section for the homomorphism κ : _{2}Br E → _{2}H^{1}(Γ, \overline{E}) defined by the following commutative square

\xymatrix{_{2}Br E & _{2}H^{1}(Γ, \overline{E}) \ar[l]^{ε} \\ _{2}Br E′ \ar[u]^{≅} & _{2}H^{1}(Γ, \overline{E}′) \ar[l]^{ε′} \ar[u]^{≅}}

Theorem 4.13 Let E be a semisplit elliptic curve defined by an equation

y^{2} = (x − a)g(x),

where a ∈ K, g(x) is a unitary irreducible quadratic polynomial over K and g(x) = (x − b)(x − c) over \overline{K}. Let ε : _{2}H^{1}(Γ, \overline{E}) → _{2}Br E be the section for the homomorphism κ : _{2}Br E → _{2}H^{1}(Γ, \overline{E}) defined above and let I = Im ε. Then

2Br E ≅ _{2}Br K ⊕ I

and every element in I is represented by either a quaternion algebra of the form

(r, x − a),

where r ∈ K^{∗}, or a biquaternion algebra of the form

(t, r^{2} − h^{2}t^{2}) ⊗ (t(x − h) + r, (r^{2} − t^{2}h^{2})g(x)),

where h = (b + c)/2 ∈ K, r, t ∈ K and t ≠ 0. Conversely, every algebra of the above types is unramified over E. It is trivial in I if and only if it is similar to a quaternion algebra

(x − h + u, (u + h − a)(x − a)) ,

where u is the abscissa of a point in E(K).

Proof. All statements follow from Theorem 4.12.

5 Non-split elliptic case

In this section we consider a non-split elliptic curve E given by an affine equation

y^{2} = f(x),

where f(x) is an irreducible unitary polynomial without multiple roots. Let a be a root of f(x). We define L = K(a) and Θ = Gal(\overline{K}/L). By construction, the curve E_{L} = E ×_{K} L is either split or semisplit over L. Let

ζ_{L} : H^{1}(Θ, M) ⟶ _{2}H^{1}(Θ, \overline{E})

be the homomorphism induced by the embedding M ⊂ \overline{E} and let

κ_{L} : _{2}Br E_{L} ⟶ _{2}H^{1}(Θ, \overline{E})

be the homomorphism defined either in Section 3 or 4. Let also

ϵ_{L} : H^{1}(Θ, M) ⟶ _{2}Br E_{L}

be the homomorphism defined either by formula (9) in the split case or by means of the homomorphism τ in the semisplit case (see Section 4). According to Propositions 3.5 and 4.1 there exists a section

ε_{L} : _{2}H^{1}(Θ, \overline{E}) ⟶ _{2}Br E_{L}

for the homomorphism κ_{L}, such that the composition ε_{L} ∘ ζ_{L} coincides with ϵ_{L}. We are now in the position to prove the existence of ϵ and ε with the same properties for the curve E/K in the non-split case.

Proposition 5.1 Let E be a non-split elliptic curve over K, char K ≠ 2. Let κ : _{2}Br E → _{2}H^{1}(Γ, \overline{E}) be the homomorphism defined in Section 2 and let ζ : H^{1}(Γ, M) → _{2}H^{1}(Γ, \overline{E}) be the homomorphism induced by the embedding M ⊂ \overline{E}. Let also ϵ be the composition

ϵ : H^{1}(Γ, M) ⟶^{res} H^{1}(Θ, M) ⟶^{ϵ_{L}} _{2}Br E_{L} ⟶^{cor} _{2}Br E

where ϵ_{L} is as above. Then there exists a homomorphism ε : _{2}H^{1}(Γ, \overline{E}) ⟶ 2Br E such that ε ∘ ζ = ϵ and κ ∘ ε = 1_{_{2}H^{1}(Γ,\overline{E})} is the identical map.

Proof. This is entirely analogous to the proof of Proposition 4.1. The only difference is that instead of τ we have to use the homomorphism H^{1}(Γ, M) →^{res} H^{1}(Θ, M).

Keeping the above notation we may reformulate Proposition 5.1 in terms of central simple algebras. We distinguish two cases.

Theorem 5.2 Suppose that the curve E_{L} is split. Let f(x) = (x−a)(x−b)(x− c), where a, b, c ∈ L = K(a). Let ε : _{2}H^{1}(Γ, \overline{E}) → _{2}Br E be the section for the homomorphism κ described in Proposition 5.1 and I = Im ε. Then

2Br E ≅ _{2}Br K ⊕ I

and any element in I has the form

cor_{L/K} [(r, x − b) ⊗ (s, x − c)]

where r, s ∈ L^{∗}. Conversely, any such a class of algebras is unramified over K(E) and it is trivial in I if and only if it coincides with a class

cor_{L/K} [(u − c, x − b) ⊗ (u − b, x − c)],

where u is the abscissa of a point in E(K).

Proof. Since ϵ is the composition

H^{1}(Γ, M) ⟶^{res} H^{1}(Θ, M) ⟶^{ϵ_{L}} _{2}Br E_{L} ⟶^{cor} _{2}Br E ,

it follows that ε is the composition

2H^{1}(Γ, \overline{E}) ⟶^{res} _{2}H^{1}(Θ, \overline{E}) ⟶^{ε_{L}} _{2}Br E_{L} ⟶^{cor} _{2}Br E

(an easy diagram chase). Hence

I = Im ε = cor(Im ε_{L}) .

According to Theorem 3.6 any element in Im ε_{L} is represented by an algebra of type (r, x − b) ⊗ (s, x − c) where r, s ∈ L^{∗}. Hence an element in I has the form cor_{L/K} [(r, x − b) ⊗ (s, x − c)] for some r, s ∈ L^{∗}. Let r, s ∈ L^{∗}. Consider the algebra (r, x − b) ⊗ (s, x − c) over L(E). It is unramified because its class lies in the image of the homomorphism ϵ_{L}. Therefore the class

α = cor_{L/K} [(r, x − b) ⊗ (s, x − c)] ∈ Br K(E)

is also unramified. Assume that α ∈ I. If

α = cor_{L/K} [(u − c, x − b) ⊗ (u − b, x − c)]

where u is the abscissa of a point in E(K), then α = 0 because

[(u − c, x − b) ⊗ (u − b, x − c)] = 0

in Im ε_{L}, by Theorem 3.6. Conversely, if α = 0 in I then α grows up (via ζ and ε) from the image of the connecting homomorphism δ. By the construction all homomorphisms δ, ζ, ε commute with restriction homomorphisms. It follows that α is equal to a class of algebras coming from E(K)/2, that is of type cor_{L/K} [(u−c, x−b)⊗(u−b, x−c)] where u is the abscissa of a point in E(K).

Theorem 5.3 Suppose that the curve E_{L} is semisplit. Let f(x) = (x − a)g(x), where a ∈ L, g(x) is an irreducible quadratic polynomial over L and g(x) = (x − b)(x − c) over \overline{K}. Let ε : _{2}H^{1}(Γ, \overline{E}) → _{2}Br E be the section for the homomorphism κ described in Proposition 5.1 and I = Im ε. Then

2Br E ≅ _{2}Br K ⊕ I

and every element in I is represented either by a class

cor_{L/K} [r, x − a],

where r ∈ L^{∗}, or a class of the form

cor_{L/K} [(t, r^{2} − h^{2}t^{2}) ⊗ (t(x − h) + r, (r^{2} − t^{2}h^{2})g(x))]

where h = (b + c)/2 ∈ L, r, t ∈ L and t ≠ 0. Conversely, every such a class is unramified over K(E). It is trivial in I if and only if it coincides with a class

cor_{L/K} [x − h + u, (u + h − a)(x − a)]

where u is the abscissa of a point in E(K).

Proof. The proof is similar to that of Theorem 5.2. The difference is just that we use Proposition 4.13 instead of Proposition 3.6. Indeed, we have I = cor(Im ε_{L}). According to Theorem 4.13 any element in Im ε_{L} is represented by either a quaternion algebra of the form A = (r, x − a), where r ∈ K^{∗}, or a biquaternion algebra of the form

B = (t, r^{2} − h^{2}t^{2}) ⊗ (t(x − h) + r, (r^{2} − t^{2}h^{2})g(x)),

where h = (b + c)/2 ∈ K, r, t ∈ K and t ≠ 0. Therefore an element in I is equal to either cor_{L/K} [A] or cor_{L/K} [B]. An algebra of the types A or B lies in Im ε_{L} and hence it is unramified. Therefore, classes cor_{L/K}[A] and cor_{L/K}[B] are also unramified. They are trivial in I if and only if they come from the image of the connecting homomorphism δ via the homomorphisms ζ and ε. Since δ, ζ and ε commute with the corresponding restriction homomorphisms, it follows (using the second assertion of Proposition 4.13) that the classes cor_{L/K}[A] and cor_{L/K}[B] are trivial in I if and only if they coincide with a class

cor_{L/K}[x − h + u, (u + h − a)(x − a)] ,

where u is the abscissa of a point in E(K).

The generators of _{2}Br E given in Theorems 5.2 and 5.3 are represented as classes cor_{L/K}[A], where A is a quaternion or biquaternion algebra over the cubic extension L(E)/K(E). We close this section by showing how one can rewrite these generators as tensor products of quaternion algebras defined over K(E). Let P/K be a cubic extension and let P = K(s) for some element s ∈ P.

Lemma 5.4 Every element a ∈ P can be written in the form

a = \frac{θ_{1} + θ_{2}s}{θ_{3} + θ_{4}s} ,

where θ_{1}, θ_{2}, θ_{3}, θ_{4} ∈ K.

Proof. Let V = {θ_{1} + θ_{2}s | θ_{1}, θ_{2} ∈ K} be a two-dimensional vector space over F. Since aV is also a two-dimensional vector space over K, the intersection V ∩ aV has dimension at least one. Let b ∈ V ∩ aV be a non-zero element. Then there exists θ_{1}, θ_{2}, θ_{3}, θ_{4} ∈ K such that

b = θ_{1} + θ_{2}s = (θ_{3} + θ_{4}s)a.

It follows that

a = \frac{θ_{1} + θ_{2}s}{θ_{3} + θ_{4}s} ,

as required.

Lemma 5.5 Let a, b ∈ K be such that a + b ≠ 0. Then

cor_{P/K}[a + s, b − s] = [a + b, (a + b) N_{P/K}((a + s)(b − s))] .

Proof. Let

t = \frac{a + s}{a + b} and l = \frac{b − s}{a + b} .

Then t + l = 1, whence [t, l] = [t, 1 − t] = 0 in Br P. Substituting t, l, we have

0 = [t, l] = [\frac{a + s}{a + b}, \frac{b − s}{a + b}] =

[a + s, b − s] + [a + b, b − s] + [a + s, a + b] + [a + b, a + b] .

Taking cor_{P/F} and using Lemma 4.9 we obtain that

0 = cor_{P/K}[a + s, b − s] + [a + b, N_{P/K}(b − s)]+
[N_{P/K}(a + s), a + b] + [a + b, (a + b)^{3}] .

Therefore,

cor_{P/F} [a + s, b − s] = [a + b, N_{P/K}(b − s)] + [N_{P/K}(a + s), a + b] + [a + b, a + b] ,

as required.

Lemma 5.6 Let u_{1}, v_{1}, u_{2}, v_{2} ∈ K, v_{1} ≠ 0, v_{2} ≠ 0 and v_{1}u_{2} ≠ u_{1}v_{2}. Then

cor_{P/K}[u_{1} + v_{1}s, u_{2} + v_{2}s] = [v_{1}(v_{1}u_{2} − u_{1}v_{2}), N_{P/K}(u_{1} + v_{1}s)] +
[v_{2}(u_{1}v_{2} − v_{1}u_{2}), v_{1}(v_{1}u_{2} − u_{1}v_{2}) N_{P/K}(u_{2} + v_{2}s)] .

Proof. This is entirely analogous to the proof of Lemma 4.11 and so we omit the details to the reader.

Using Lemmas 5.4, 4.9, 5.5 and 5.6 one can easily produce explicit formulas to compute all algebras in Theorems 5.2 and 5.3. However we do not present them because of their bulk.

6 Elliptic curves over local fields

In the next few sections we demonstrate the efficiency of the above cohomological methods by considering an elliptic curve E defined over a local non-dyadic field K. To get an explicit description of _{2}Br E, by Theorems 3.6, 4.13, 5.2 and 5.3, we only need to explicitly describe all relations between the generators indicated in these theorems which is equivalent to the description of the image of the boundary map δ : E(K)/2 → H^{1}(Γ, M). For an elliptic curve over local fields there is a natural p-adic filtration on the group of K-points with finite quotients. Examining each quotient individually one can very quickly find generators for the group E(K)/2. This leads in turn to the required description of Im δ. All necessary facts for our further argument can be easily elicited from standard textbooks, for example from [Hu87] and [Sil85]. For the convenience of the reader we start with recalling them.

For the rest of the paper we use the following specific notation:

K – a local non-dyadic field, i.e. a finite extension of the p-adic field ℚ_{p}, p ≠ 2; v – the discrete valuation on K; 𝒪 = 𝒪_{K} – the ring of integers of K; 𝒪^{∗} = 𝒪_{K}^{∗} – the unit group of O; α = α_{K} ∈ 𝒪^{∗} – a non-square element; π = π_{K} – a uniformizer for O; k = 𝒪/π 𝒪 – the residue field of K.

Theorem 6.1 There is a natural isomorphism

H^{1}(Γ, \overline{E}) ≅ Hom_{cont}(E(K), ℚ/ℤ) .

Proof. See [Tate57] or [Mi86].

Corollary 6.2 | _{2}Br E| = 2 · \sqrt{|H^{1}(Γ, M)|} .

Proof. By Theorem 6.1, we have

| _{2}H^{1}(Γ, \overline{E})| = | _{2}Hom_{cont}(E(K), ℚ/ℤ)| =
|Hom_{cont}(E(K)/2, ℚ/ℤ)| = |E(K)/2| .

On the other hand, sequence (8) shows that

| _{2}H^{1}(Γ, \overline{E})| = |H^{1}(Γ, M)|/|E(K)/2| .

Therefore,

|E(K)/2|^{2} = |H^{1}(Γ, M)|

and the result follows.

Proposition 6.3 Let n be a natural number. Then

|E(K)/nE(K)| = |_{n}E(K)| · |𝒪/n𝒪|.

Proof. See, for example, [Mi86], p. 52.

Corollary 6.4 Let E be a non-split elliptic curve defined over a local nondyadic field K. Then _{2}Br E = _{2}Br K.

Proof. Clearly, we have

| _{2}Br E | = | _{2}Br K | · | _{2}H^{1}(Γ, \overline{E}) | = | _{2}Br K | · |E(K)/2 | .

Since E is non-split, it follows that every nontrivial element from M is not defined over K. Therefore, _{2}E(K) = 0 and, by Proposition 6.3, we obtain that E(K)/2 = 0. This implies that | _{2}Br E | = | _{2}Br K |, as required.

Let E be an elliptic curve over K and let

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}

be a Weierstrass equation for the curve E/K with all coefficients a_{i} ∈ 𝒪. Since its discriminant Δ is also an integer and since v is discrete we can look for an equation with v(Δ) as small as possible. A Weierstrass equation is called a minimal equation for E if v(Δ) is minimized subject to the condition a_{1}, a_{2}, a_{3}, a_{4}, a_{6} ∈ 𝒪. It is known (see [Sil85], Proposition 1.3, p. 172) that a minimal (Weierstrass) equation is unique up to a change of coordinates

x = u^{2}x′ + r , y = u^{3}y′ + u^{2}sx′ + t

with u ∈ 𝒪^{∗} and r, s, t ∈ 𝒪. Since, by our assumption, 2 ∈ 𝒪^{∗}, a coordinate change y → y′ = y + (a_{1}x + a_{3})/2 shows that we may always assume that a_{1} = a_{3} = 0, i.e. E is given by a minimal equation of the form

y^{2} = x^{3} + a_{2}x^{2} + a_{4}x + a_{6} .  (11)

Later we need to know when (11) is a minimal equation for E. Let

b_{2} = 4a_{2} , b_{4} = 2a_{4} , b_{6} = 4a_{6} , b_{8} = 4a_{2}a_{6} − a_{4}^{2} ,
c_{4} = b_{2}^{2} − 24b_{4} , c_{6} = b_{2}^{3} + 36b_{2}b_{4} − 216b_{6}

be the usual combinations of the a_{i}‘s and let

Δ = −b_{2}^{2}b_{8} − 8b_{4}^{3} − 27b_{6}^{2} + 9b_{2}b_{4}b_{6}

be the discriminant of equation (11) (see [Sil85], p. 46).

Proposition 6.5 Equation (11) with integer coefficients a_{2}, a_{4}, a_{6} is minimal if and only if either v(Δ) < 12 or v(c_{4}) < 4.

Proof. See [Sil85], page 186, Exercises 7.1.

We assume that our elliptic curve E is given by a minimal equation (11). Reducing its coefficients modulo π we obtain the curve (possibly singular) Ẽ over k:

y^{2} = x^{3} + ã_{2}x^{2} + ã_{4}x + ã_{6} .

The curve Ẽ is called the reduction of E modulo π. Next let P ∈ E(K). We can find homogeneous coordinates P = [x_{0}, y_{0}, z_{0}] with integers x_{0}, y_{0}, z_{0} such that at least one of them is in 𝒪^{∗}. Then the reduced point P̃ = [x̃_{0}, ỹ_{0}, z̃_{0}] is in Ẽ. This gives a reduction map

E(K) ⟶ Ẽ(k), P ⟶ P̃ .

Since the curve Ẽ can be singular, we denote its set of nonsingular points by Ẽ_{ns}(k) and we put

E_{0}(K) = {P ∈ E(K) | P̃ ∈ Ẽ_{ns}(k)}
E_{1}(K) = {P ∈ E(K) | P̃ = Õ} .

Proposition 6.6 The following natural sequence of abelian groups

0 → E_{1}(K) ⟶ E_{0}(K) ⟶ Ẽ_{ns}(k) → 0

is exact.

Proof. See [Sil85], Proposition 2.1, p. 174.

Proposition 6.7 The group E_{1}(K) is uniquely divisible by 2; in particular, we have E_{1}(K) = 2E_{1}(K).

Proof. See [Hu87], Corollary 1.3, p. 264.

Let E/K be an elliptic curve and let Ẽ/k be the reduced curve for a minimal Weierstrass equation. One says that

(a) E has good reduction over K if Ẽ is nonsingular;

(b) E has multiplicative reduction over K if Ẽ has a node; in this case the reduction is said to be split (respectively non-split) if the slopes of the tangent lines at the node are in k (respectively not in k);

(c) E has additive reduction over K if Ẽ has a cusp.

Proposition 6.8 Let E/K be an elliptic curve given by a minimal Weierstrass equation (11).

(a) E has good reduction if and only if v(Δ) = 0;

(b) E has multiplicative reduction if and only if v(Δ) > 0 and v(c_{4}) = 0;

(c) E has additive reduction if and only if v(Δ) > 0 and v(c_{4}) > 0.

Proof. See [Sil85], Proposition 5.1, p. 180.

7 Generators of E(K)/2 for a split elliptic curve over a local field

Let E be a split elliptic curve given by a minimal equation (11). Since M is a trivial Γ-module, it follows that all roots of the cubic polynomial f(x) = x^{3} + a_{2}x^{2} + a_{4}x + a_{6} are in K. Then these roots, clearly, belong to 𝒪, so that we may assume that E is given by a minimal equation of the form

y^{2} = (x − a)(x − b)(x − c)  (12)

with all a, b, c in 𝒪. In this coordinate system M consists of the points

O, P = (a, 0), Q = (b, 0), T = (c, 0) .

Recall also that, by Proposition 6.3, we have |E(K)/2| = |M| = 4.

7.1 Additive reduction

Lemma 7.1 The group E_{0}(K) is divisible by 2.

Proof. Since E has additive reduction, we have E_{0}(K)/E_{1}(K) ≅ k^{+}; in particular the finite group E_{0}(K)/E_{1}(K) is divisible by 2. Then the result follows from Proposition 6.7.

Proposition 7.2 The elements O, P, Q, T are representatives of E(K)/2.

Proof. In view of Lemma 7.1 we have E_{0}(K) ⊂ 2E(K) ⊂ E(K) and by [Sil85], Theorem 6.1, p. 183, the group E(K)/E_{0}(K) is finite of order at most 4. Since |E(K)/2| = 4, we get E_{0}(K) = 2E(K) and it remains to note that the points P, Q, T do not belong to E_{0}(K).

7.2 Multiplicative reduction

By our assumption, among the residues ã, b̃, c̃ there are exactly two coinciding elements; say ã = b̃. Changing coordinates, if necessary, we may assume that E is given by a minimal equation of the form

y^{2} = x(x + π^{m}β)(x + γ)

with β ∈ 𝒪^{∗}, m ≥ 1 and γ ∈ 𝒪^{∗}. Recall that in the case of non-split reduction γ coincides modulo squares with α; otherwise γ is a square in 𝒪^{∗}.

Lemma 7.3 There exists a point R_{1} = (u, v) ∈ E_{0}(K) such that

u = αt^{2} , u + π^{m}β = αq^{2} , u + γ = s^{2} , v = α t q s

with t, q, s in 𝒪^{∗}.

Proof. The proof is easy. Namely, we have to find a solution of the system

\{ \begin{matrix}αx^{2} + π^{m}β & = & αy^{2} \\ αx^{2} + γ & = & z^{2} \end{matrix}

According to standard facts from the theory of quadratic forms over finite and local fields the quadratic form α̃x^{2} − z^{2} represents −γ ∈ k^{∗}, whence, by the Hensel lemma, we can pick units t, s ∈ 𝒪^{∗} satisfying the second equation. Substitute t in the first equation. Since the residues of the elements αt^{2} + π^{m}β and α coincide modulo squares, again, applying the Hensel lemma we can find q ∈ 𝒪^{∗} satisfying the equation αt^{2} + π^{m}β = αy^{2}.

Remark 7.4 Since the abscissa u of R_{1} is not a square in K^{∗}, Proposition 3.3 shows that δ(R_{1}) ≠ (1, 1). Then it follows that R_{1} ∉ 2E(K).

Lemma 7.5 There exists a point R_{2} = (u, v) ∈ E(K) \ E_{0}(K) with u = πd, d ∈ 𝒪, and such that its image in the group E(K)/E_{0}(K) is not divisible by 2.

Proof. The abscissa of every point from E(K)\E_{0}(K) is of the form πd with d ∈ 𝒪 because its residue is the node. Further, we have Δ = 16(π^{m}βγ(π^{m}β − γ))^{2} and π^{m}β − γ ∈ 𝒪^{∗}, so that v(Δ) is even. Then, by [Hu87], p. 266, the order of the finite group E(K)/E_{0}(K) is divisible by 2, whence such a point exists.

Remark 7.6 If the reduction is non-split, we can take R_{2} = (0, 0), because in this case the group E(K)/E_{0}(K) has order 2 (loc. cit.) and, of course, R_{2} = (0, 0) ∉ E_{0}(K).

Proposition 7.7 The points R_{1}, R_{2} from the above two lemmas are generators of E(K)/2E(K).

Proof. Since |E(K)/2| = 4, we have E(K)/2E(K) ≅ ℤ/2 × ℤ/2. By our construction and by Remark 7.4, the images of R_{1}, R_{2} in E(K)/2E(K) are not trivial and they do not coincide.

8 Generators of E(K)/2 for a semisplit elliptic curve over a local field

We may assume that E is given by a minimal equation of the form

y^{2} = (x − a)(x^{2} − d),  (13)

where a, d ∈ 𝒪 and the polynomial g(x) = x^{2} − d is irreducible over K. Let L = K(\sqrt{d}) be its splitting field and let Λ = Gal (\overline{K}/L). As it was mentioned in Section 4, the module M is isomorphic to the induced module M_{Γ}^{Λ}(ℤ/2). This gives the isomorphisms

H^{1}(Γ, M) ≅ L^{∗}/L^{∗}^{2} , H^{1}(Λ, M) ≅ L^{∗}/L^{∗}^{2} × L^{∗}/L^{∗}^{2} .

Recall also that under this identification the restriction map H^{1}(Γ, M) → H^{1}(Λ, M) is given by the formula l → (l^{σ}, l), where l ∈ L^{∗} and σ is the nontrivial automorphism L/K; in particular, res is injective (see Section 4). It then follows from the commutative square

\xymatrix{E(L)/2 \ar@{^{(}->}[r]^{δ_{L}} & L^{∗}/L^{∗}^{2} ⊕ L^{∗}/L^{∗}^{2} \\ E(K)/2 \ar[u]^{η} \ar@{^{(}->}[r]^{δ} & L^{∗}/L^{∗}^{2} \ar@{^{(}->}[u]^{res}}

that η : E(K)/2 → E(L)/2 is also injective. Applying Proposition 6.3 we have |E(K)/2| = | _{2}E(K)| = 2. Now we want to explicitly describe the image η(E(K)/2). The answer depends on the type of reduction.

8.1 Multiplicative reduction.

For an elliptic curve given by (13) one has Δ = 64d(a^{2} − d)^{2} and c_{4} = 16(a^{2} + 3d). Since, by Proposition 6.8, v(Δ) > 0 and v(c_{4}) = 0, we obtain that v(d) > 0 and a ∈ 𝒪^{∗}. Then, according to Proposition 6.5, (13) is a minimal equation for E_{L}. Hence E_{L} has multiplicative reduction (again by Proposition 6.8). Note that in view of v(d) > 0 and a ∈ 𝒪^{∗} we have a^{2} − d ∈ O^{∗}, whence v(Δ) = v(d). We say that we are in case:

(M1) if either v(d) is odd or 4 divides v(d) and E has non-split multiplicative reduction;

(M2) if v(d) is even and either E has split multiplicative reduction or 4 does not divide v(d).

Proposition 8.1 Let R_{1}, R_{2} be the points in E(L) introduced in 7.2. Then in case (M1) the nontrivial element of η(E(K)/2) coincides with R_{1} + 2E(L) and in case (M2) it coincides with R_{2} + 2E(L).

Proof. Consider case (M1). If v(d) is odd, then by, [Hu87], p. 266, the group E(K)/E_{0}(K) has an odd order. So we may choose a representative R of a unique nontrivial element in E(K)/2 among elements of E_{0}(K). Since E_{0}(K) ⊂ E_{0}(L) and η is injective, R coincides with R_{1} modulo 2E(L). Next suppose that 4 divides v(d) and E has non-split multiplicative reduction. Since v(d) is even, the extension L/K is unramified, so that [k_{L} : k] = 2, where k_{L} is the residue field of the local field L. It follows that E_{L} has split multiplicative reduction and, by [Hu87], p. 266, the group E(L)/E_{0}(L) is cyclic of order v(Δ_{L}) = v(Δ_{K}) = v(d); in particular, 4 divides |E(L)/E_{0}(L)|. Let R be a representative of the nontrivial element of E(K)/2. Since E has non-split multiplicative reduction, it follows that |E(K)/E_{0}(K)| = 2 (loc. cit.), hence R can be chosen among elements E(K)\E_{0}(K). To show that η(R) coincides with R_{1} modulo 2E(L) consider the 2-Sylow subgroup G in E(L)/E_{0}(L). It is clear that R + E_{0}(L) ∈ G and it has order 2. Then R + E_{0}(L) is divisible by 2 in G and so in E(L)/E_{0}(L). But, by our construction (see Lemma 7.5), the element R_{2} is not divisible by 2 in E(L)/E_{0}(L), so we obtain R +2E(L) ≠ R_{2} +2E(L) and similarly we have R +2E(L) ≠ R_{1} +R_{2} +2E(L). It follows that R + 2E(L) = R_{1} + 2E(L), as required. Consider case (M2). We have already mentioned that (13) is a minimal equation for E_{L}. It follows that E_{0}(K) ⊂ E_{0}(L) and that the natural embedding E(K) ⊂ E(L) induces the injection ψ : E(K)/E_{0}(K) → E(L)/E_{0}(L). Suppose that E has split multiplicative reduction and v(d) is even. Then L/K is unramified and again, by [Hu87], p. 266, the groups E(K)/E_{0}(K) and E(L)/E_{0}(L) are cyclic of the same order v(Δ) = v(Δ_{L}) = v(d) implying ψ is a bijection. Since v(d) is even, we can choose a representative R of the nontrivial element of E(K)/2 such that R + E_{0}(K) is not divisible by 2 in E(K)/E_{0}(K). Then it is not divisible by 2 in E(L)/E_{0}(L); hence R + 2E(L) = R_{2} + 2E(L). Suppose that E has non-split multiplicative reduction. Then according to [Hu87], p. 266, we have |E(K)/E_{0}(K)| = 2 and |E(L)/E_{0}(L)| = v(d). Since 4 does not divide v(d), the group ψ (E(K)/E_{0}(K)) is a 2-Sylow subgroup in E(L)/E_{0}(L). Hence again picking an element R with the same property as above we easily get R + 2E(L) = R_{2} + 2E(L).

8.2 Additive reduction

Proposition 8.2 (1) If L/K is unramified, then E(K)/2 is generated by P = (a, 0).

(2) Let L/K be ramified. If a − \sqrt{d} is not a square in L^{∗}, then E(K)/2 is again generated by P = (a, 0). If a − \sqrt{d} = s^{2}, s ∈ L^{∗}, then E(K)/2 is generated by the point U = (u, w) ∈ E(K), where u = N_{L/K}(s) + a and w = N_{L/K}(s) Tr_{L/K}(s).

Proof. First let L/K be unramified. Then E_{L} has additive reduction and by Proposition 7.2, we have P ∉ 2E(L). It follows that P ∉ 2E(K), as required.

Next let L/K be ramified. Recall that, by Lemma 7.1, we have E_{0}(K) ⊂ 2E(K) and that E(K)/E_{0}(K) is a group of order at most 4 (see [Sil85], p. 183). If a − \sqrt{d} is not a square in L^{∗}, then, by Proposition 3.3, δ_{L}(P) ≠ (1, 1), hence P ∉ 2E(L) and the result follows. Let a−\sqrt{d} = s^{2}, s ∈ L^{∗}. Then it is easy to check that 2U = P. This implies that P ∈ 2E(K) \ E_{0}(K) and so |2E(K)/E_{0}(K)| ≥ 2. But |E(K)/2E(K)| = 2 and |E(K)/E_{0}(K)| ≤ 4. It follows that |2E(K)/E_{0}(K)| = 2, whence U ∉ 2E(K), as required.

For the description of _{2}Br E we will also need to know whether (δ_{L}∘η) (E(K)/2) belongs to the unramified part of the subset res (L^{∗}/L^{∗}^{2}) ⊂ L^{∗}/L^{∗}^{2} × L^{∗}/L^{∗}^{2}. In other words, we will need to know whether v_{L}(a + \sqrt{d}) and v_{L}(u + \sqrt{d}) are odd or even. Here u is the abscissa of the above point U. It turns out that the answer depends on the coefficients of the minimal equation (13) only. Let a = π^{m}a′, d = π^{2k+λ}d′ with a′, d′ ∈ 𝒪^{∗} and λ = 0, 1. Using Propositions 6.5 and 6.8 one can easily make sure that m > 0, 2k + λ > 0 and that m = 1 or 2k + λ ≤ 3. We will say that we are in case:

(A1) if one of the following conditions holds:

(a) λ = 0, i.e. L/K is unramified,

(b) λ = 1, m = 1, k = 0,

(c) λ = 1, m > 1;

(A2) if λ = 1, m = 1, k ≥ 1 and a − \sqrt{d} ∉ L^{∗}^{2}.

(A3) if λ = 1, m = 1, k ≥ 1 and a − \sqrt{d} ∈ L^{∗}^{2},

Lemma 8.3 (i) In case (A1) the group E(K)/2 is generated by P and v_{L}(a + \sqrt{d}) is odd. (ii) In case (A2) the group E(K)/2 is generated by P and v_{L}(a + \sqrt{d}) is even. (iii) In case (A3) the group E(K)/2 is generated by U and v_{L}(u + \sqrt{d}) is odd.

Proof. First examine case (A1).

(a) Here L/K is unramified and at least one of the numbers k and m equals 1. So, obviously, v_{L}(a + \sqrt{d}) = 1.

(b) Since L/K is ramified, we have v_{L}(a) = v_{L}(π) = 2 and v_{L}(\sqrt{d}) = 1. So v_{L}(a + \sqrt{d}) = 1.

(c) We have v_{L}(a) = 2m ≥ 4 and v_{L}(\sqrt{d}) = 2k +1. Since 2k +λ ≤ 3, we obtain that v_{L}(a + \sqrt{d}) = v_{L}(d) = 2k + 1 is odd. Case (A2). Since L/K is ramified, we have v_{L}(a) = v_{L}(π) = 2 and v_{L}(\sqrt{d}) = 2k + 1 ≥ 3. It follows that v_{L}(a + \sqrt{d}) = 2. Case (A3). Keeping the notation of Proposition 8.2 we have a − \sqrt{d} = s^{2} and u = N_{L/K}(s) + a. It easily follows that v_{L}(s) = 1. Further, letting σ be the nontrivial automorphism L/K we have

u + \sqrt{d} = N_{L/K}(s) + a + \sqrt{d} = ss^{σ} + s^{σ}s^{σ} = (s + s^{σ})s^{σ}.

Therefore, v_{L}(u + \sqrt{d}) = v_{L}(s + s^{σ}) + 1 and it remains to note that v_{L}(s + s^{σ}) is even because s + s^{σ} ∈ K.

9 Computing _{2}Br E over non-dyadic local fields: split case

Putting together the results of the previous sections one can easily obtain an explicit and very short description of the 2-torsion subgroup of Br E for split and semisplit elliptic curves (note that for non-split curves it was done in Corollary 6.4). Namely, let δ : E(K)/2 → H^{1}(Γ, M) be the boundary map. The description of generators of E(K)/2 and their images under the map δ given in Sections 7 and 8 enables us to explicitly construct a subgroup in H^{1}(Γ, M) that complements δ(E(K)/2). If we then restrict the section ϵ : H^{1}(Γ, M) → _{2}Br E constructed in Sections 3 and 4 at this subgroup, we immediately obtain a description of the second summand in the decomposition 2Br E = _{2}Br K ⊕ Im ϵ as, by Proposition 3.4, and Lemma 4.8, the equality ϵ (Im δ) = 0 holds. In this section we consider a split elliptic curve E given by a minimal equation of the form

y^{2} = x(x − b)(x − c) ,  (14)

with b, c in the integer ring 𝒪. Its 2-torsion consists of the points O, P = (0, 0), Q = (b, 0) and T = (c, 0). As in Section 3, we may identify

M = ⟨Q⟩ ⊕ ⟨T ⟩ ≅ ℤ/2 ⊕ ℤ/2

and

H^{1}(Γ, M) ≅ K^{∗}/K^{∗}^{2} ⊕ K^{∗}/K^{∗}^{2} .

According to Proposition 3.3 the connecting homomorphism

δ : E(K)/2 ↪ K^{∗}/K^{∗}^{2} ⊕ K^{∗}/K^{∗}^{2}

is given by the formula

δ(u, v) = \{ \begin{matrix}(u − c, u − b) & if u ≠ b and u ≠ c, \\ (b − c, b(b − c)) & if u = b, \\ (c(c − b), c − b) & if u = c, \\ (1, 1) & if u = ∞, \end{matrix}  (15)

where (u, v) ∈ E(K). Let

C_{α} = [α, x−c] , C_{π} = [π, x−c] , B_{α} = [α, x−b] and B_{π} = [π, x−b] (16)

be the classes of quaternion algebras over K(E). We distinguish the following three cases.

9.1 Good reduction

We start with the following

Lemma 9.1 δ(E(K)/2) is generated by the pairs (α, 1) and (1, α).

Proof. Let K^{nr}/K be a maximal unramified extension. It suffices to show that the images of our pairs under the natural map ζ : H^{1}(Γ, M) → _{2}H^{1}(Γ, \overline{E}) are trivial. To do so, first recall that, by [LT58] and [L56], we have

H^{1}(Gal (K^{nr}/K), E(K^{nr})) = H^{1}(Gal (\overline{k}/k), Ẽ) = 0 .

This implies that res : H^{1}(Γ, \overline{E}) → H^{1}(K^{nr}, \overline{E}) is injective. On the other hand, obviously we have (res ∘ζ)(α, 1) = (res ∘ζ)(1, α) = 1, so the result follows.

Proposition 9.2 We have

2Br E = _{2}Br K ⊕ {0, B_{π}, C_{π}, B_{π} + C_{π}} .

Proof. It suffices to note that the subgroup generated by the pairs (π, 1) and (1, π) complements the subgroup δ(E(K)/2) and that ϵ takes these pairs to the classes B_{π} and C_{π}.

9.2 Additive reduction

We may assume that v(b) ≥ 1, v(c) ≥ 1 and that at least one of these numbers is 1. Let b = π^{m}d and c = πe, where d and e are units and m ≥ 1. Proposition 7.2 shows that E(K)/2 is generated by the points P, Q, T. Applying (15) we get

Lemma 9.3 δ(E(K)/2) is generated by the pairs

δ(P) = (−πe, −π^{m}d) and δ(T) = (πe(πe − π^{m}d), πe − π^{m}d) .

Proposition 9.4 We have

2Br E = _{2}Br K ⊕ {0, B_{α}, C_{α}, B_{α} + C_{α}} .

Proof. It easily follows from Lemma 9.3 that the subgroup generated by the pairs (α, 1) and (1, α) complements δ(E(K)/2) in K^{∗}/K^{∗}^{2} ⊕ K^{∗}/K^{∗}^{2} and it remains to note that ϵ takes these pairs to the classes B_{α} and C_{α}.

9.3 Non-split multiplicative reduction

We may assume that E is given by a minimal equation of the form

y^{2} = x(x + π^{m}β)(x + α) ,

with m ≥ 1 and β ∈ 𝒪. Note that in the notation of formulas (15) and (16) we have that

b = −π^{m}β  and  c = −α .

Lemma 9.5 δ(E(K)/2) is generated by the pairs (1, α) and (α, π^{m}β).

Proof. Let R_{1}, R_{2} be two points introduced in 7.2. It then follows from Lemma 7.3, Remark 7.6 and formula (15) that δ(R_{1}) = (1, α) and δ(R_{2}) = (α, π^{m}β), as required.

Proposition 9.6 We have

2Br E = _{2}Br K ⊕ {0, B_{π}, C_{π}, B_{π} + C_{π}} .

Proof. The subgroup generated by the pairs (π, 1) and (1, π) complements δ(E(K)/2), so the result follows.

9.4 Split multiplicative reduction

We may assume that E is given by a minimal equation of the form

y^{2} = x(x + π^{m}β)(x + 1).

Lemma 9.7 δ(E(K)/2) is generated by the pairs (1, α) and (1, π).

Proof. As above, we have δ(R_{1}) = (1, α). Further, it follows from the construction that the abscissa of the point R_{2} = (u, v) is of the form u = πd. So applying formula (15), we obtain that δ(R_{2}) = (1, πu + π^{m}β). But |δ(E(K)/2)| = 4, whence v(πu + π^{m}β) is odd and the result follows.

Proposition 9.8 We have

2Br E = _{2}Br K ⊕ {0, B_{α}, B_{π}, B_{απ}} .

Proof. This follows from the fact that the subgroup generated by the pairs (α, 1) and (π, 1) complements δ(E(K)/2).

10 Computing _{2}Br E over non-dyadic local fields: semisplit case

We keep the notation introduced in Section 8. Assume that E is given by a minimal equation of the form (13). Then E(K)/2 and H^{1}(Γ, M) are groups of order 2 and 4 respectively, so that δ(E(K)/2) can be complemented inside H^{1}(Γ, M) by a single element. We will find such an element among elements cor (H^{1}(Λ, M)). Recall that δ_{L} denotes the homomorphism E(L)/2 ↪ H^{1}(Λ, M).

Lemma 10.1 Let θ ∈ H^{1}(Λ, M) satisfies the condition (res ∘ cor) (θ) ∉ (δ_{L} ∘ res) (E(K)/2). Then cor (θ) complements δ (E(K)/2).

Proof. By our assumption,

res (cor (θ)) ∉ (δ_{L} ∘ res) (E(K)/2) = (res ∘δ) (E(K)/2) ,

so that cor (θ) does not lie in δ(E(K)/2).

Let α_{L} and π_{L} be a non-square unit and a uniformizer of the integer ring 𝒪_{L} of L = K(\sqrt{d}) respectively.

10.1 Good Reduction

Proposition 10.2 _{2}Br E = _{2}Br K ⊕ {0, [π, x − a]}.

Proof. Clearly, (δ_{L} ∘ res) (E(K)/2) belongs to the unramified part of H^{1}(Λ, M) ≅ L^{∗}/L^{∗}^{2} ⊕ L^{∗}/L^{∗}^{2}. Since we have good reduction, d is a unit, whence π_{L} = π. We put θ = (1, π). The equation (res ∘ cor)(θ) = (π, π) shows that θ satisfies the condition of Lemma 10.1. It then follows from Theorem 4.12 that _{2}Br E is generated by _{2}Br K and

(cor ∘ ϵ_{L}) [1, π] = cor [π, x + \sqrt{d}] = [π, x^{2} − d] = [π, x − a].

10.2 Additive reduction

Proposition 10.3 (1) In cases (A1) and (A3) we have

2Br E = _{2}Br K ⊕ \{0, cor [α_{L}, x − \sqrt{d}]\}.

(2) In case (A2) we have

2Br E = _{2}Br K ⊕ \{0, cor [π_{L}, x − \sqrt{d}]\}.

Proof. It suffices to note that, by Lemma 8.3, in the first (resp. second) case the pair θ = (1, α_{L}) (resp. θ = (1, π_{L})) satisfies the condition of Lemma 10.1.

10.3 Multiplicative Reduction

Proposition 10.4 In case (M1) we have

2Br E = _{2}Br K ⊕ \{0, cor [π_{L}, x − \sqrt{d}]\}.

and in case (M2) we have

2Br E = _{2}Br K ⊕ \{0, cor [α_{L}, x − \sqrt{d}]\} .

Proof. Denote a representative of a unique nontrivial element in E(K)/2E(K) by R. Consider first case (M1). Let L^{nr} be a maximal unramified extension of L. According to Proposition 8.1 we have η(R) = R_{1} + 2E(L). Since, by construction, R_{1} ∈ E_{0}(L) and E_{0}(L^{nr})/2E_{0}(L^{nr}) = 0 (see [Sil85], p. 187), it follows that δ_{L}(η(R)) belongs to the unramified part of the group H^{1}(Λ, M) ≅ L^{∗}/L^{∗}^{2} ⊕ L^{∗}/L^{∗}^{2}. Therefore one can take θ = (1, π_{L}) and the result follows. In case (M2) we have η(R) = R_{2} + 2E(L). Since v(d) is even, the extension L/K is unramified and E_{L} has split multiplicative reduction. We know that the abscissa u of R_{2} is of the form u = πu′, so that δ_{L}(R_{2}) = (πu′ + \sqrt{d}, πu′ − \sqrt{d}). It is easy to make sure that v(πu′ + \sqrt{d}) is odd. Then θ = (1, α_{L}) satisfies the condition of Lemma 10.1 and the result follows.

References

[AM72] Artin M., Mumford D.: Some elementary examples of unirational varieties which are not rational. Proc. London Math. Soc. 3, 75 – 95 (1972)

[CEP71] Cassels J., Ellison W., Pfister A.: On sums of squares and on elliptic curves over function fields. J. of Number Theory 3, 125 – 149 (1971)

[CG00] Chernousov V., Guletskiǐ V.: 2-torsion of the Brauer group of an elliptic curve: generators and relations. Preprint 00 – 37, Universität Bielefeld (2000)

[Co88] Colliot-Thélène, J.-L. (with the collaboration of Sansuc, J.-J.): 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

[CSS98] Colliot-Thélène, J.-L., Skorobogatov A., Swinnerton-Dyer P.: Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points. Invent. math. 134, 579 – 650 (1998)

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

[G99] Guletskiǐ, V.: Algebras of Exponent 2 over an Elliptic Curve. Preprint 99 – 112, Universität Bielefeld (1999)

[GMY97] Guletskiǐ, V., Margolin, G., Yanchevskiǐ V.: Presentation of twotorsion part in Brauer groups of curves by quaternion algebras (in Russian). Dokl. NANB. 41, no. 6, 4–8 (1997)

[Hu87] Husemöller, D.: Elliptic Curves, Berlin Heidelberg New York: Springer 1987

[LT58] Lang, S., Tate, J.: Principal homogeneous spaces over abelian varieties. Amer. J. of Math. 80, 659 – 684 (1958)

[L56] Lang, S.: Algebraic groups over finite fields. Amer. J. of Math. 78, 555 – 563 (1956)

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

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

[Mi86] Milne, J.S.: Arithmetics Duality Theorems. Progress in Math. Vol. 1.: Academic Press 1986

[P82] Pierce, R.S.: Associative algebras. Graduate Texts in Mathematics 88: Springer-Verlag 1982

[Pu98] Pumplün, S.: Quaternion algebras over elliptic curves. Comm. Algebra 26, no. 12, 4357–4373 (1998)

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

[Serre64] Serre, J.-P.: Cohomologie Galoisienne, Berlin Heidelberg New York: Springer-Verlag 1964

[Serre79] Serre, J.-P.: Local Fields, New York Heidelberg Berlin: Springer-Verlag 1979

[Sil85] Silverman, J.: The Arithmetic of Elliptic Curves, Berlin Heidelberg New York: Springer-Verlag 1985

[S99] Skorobogatov A.: Beyond the Manin obstruction. Invent. math. 135, 399–424 (1999)

[Tate57] Tate, J.: W C-groups over 𝔭-adic fields. Séminaire Bourbaki, Décembre 1957, no. 1556

[YM96] Yanchevskiǐ, V., Margolin G.: The Brauer groups and the torsion of local elliptic curves. St. Petersburg Math. J. 7, no. 3, 473–505 (1996)

V. Chernousov Fakultät für Mathematik Universität Bielefeld Postfach 100131 33615 Bielefeld Germany chernous@mathematik.unibielefeld.de V. Guletskiǐ Institute of Mathematics Surganova str. 11 220072 Minsk, Belarus guletskii@im.bas-net.by