SELF CUP PRODUCTS AND THE THETA CHARACTERISTIC TORSOR

BJORN POONEN AND ERIC RAINS

Abstract. We give a general formula relating self cup products in cohomology to connecting maps in nonabelian cohomology, and apply it to obtain a formula for the self cup product associated to the Weil pairing.

1. Introduction

We prove a general statement about cohomology, Theorem 2.5, that reinterprets the self cup product map

H^{1}(M) → H^{2}(M ⊗ M)
x ↦ x ∪ x

as the connecting map of cohomology for a certain sequence

1 → M ⊗ M → 𝒰 M → M → 1

involving a canonical nonabelian central extension 𝒰 M. The proof of Theorem 2.5 can be read with group cohomology in mind, but we prove it for an arbitrary site since later we need it for fppf cohomology. The sheaf 𝒰 M has other properties as well: for example, there is a map M → (𝒰 M)^{ab} that is universal for quadratic maps from M to a (variable) abelian sheaf.

As an application of Theorem 2.5, we answer a question of B. Gross, who asked when the self cup product associated to the Weil pairing on the 2-torsion of the Jacobian A := Jac X of a curve X is 0. The Weil pairing

e_{2} : A[2] × A[2] → 𝔾_{m}

induces a symmetric bilinear pairing

⟨ , ⟩: H^{1}(A[2]) × H^{1}(A[2]) → H^{2}(𝔾_{m}).

We prove the identity

⟨x, x⟩ = ⟨x, c_{T} ⟩,  (1)

where c_{T} is a particular canonical element of H^{1}(A[2]). Namely, c_{T} is the class of the torsor under A[2] parametrizing the theta characteristics on X.

We have been vague about the field of definition of our curve; in fact, it is not too much harder to work over an arbitrary base scheme. (See Theorem 3.9.) Moreover, we prove a version with Jacobians replaced by arbitrary abelian schemes A, in which c_{T} is replaced by an element c_{λ} ∈ H^{1}(Â[2]). (See Theorem 3.4.)

Date: September 23, 2011; references updated September 27, 2013.

2010 Mathematics Subject Classification. Primary 18G50; Secondary 11G10, 11G30, 14G25, 14K15.

Key words and phrases. Weil pairing, theta characteristic, self cup product.

B. P. was supported by National Science Foundation grants DMS-0841321 and DMS-1069236.

This article has been published in Math. Res. Letters 18 (2011), no. 06, 1305–1318.

With an eye towards applications of these theorems, we give many criteria for the vanishing of c_{λ} and c_{T}, some of which generalize earlier results of M. Atiyah and D. Mumford: see Proposition 3.6 and Remark 3.10. We also give an example over ℚ_{3} for which c_{T} ≠ 0, and an example over ℚ for which c_{T} is nonzero but locally trivial.

As further motivation, some of our results, namely Proposition 2.9, Proposition 3.6, and Theorem 3.9, are used in [PR12] to study the distribution of Selmer groups.

2. Some homological algebra

2.1. A tensor algebra construction. We will define a functor U from the category of ℤ-modules to the category of groups, the goal being Theorem 2.5. Let M be a ℤ-module. Let T M = ⨁_{i≥0} T^{i}M be the tensor algebra. Then T^{≥n}M = ⨁_{i≥n} T^{i}M is a 2-sided ideal of T M. Let T^{<n}M be the quotient ring T M/T^{≥n}M. Let UM be the kernel of (T^{<3}M)^{×} → (T^{<1}M)^{×} = ℤ^{×} = {±1}. The grading on T M gives rise to a filtration of UM, which yields the following central extension of groups

1 → M ⊗ M → UM →^{π} M → 1.  (2)

Elements of UM may be written as 1 + m + t where m ∈ M and t ∈ M ⊗ M, and should be multiplied as follows:

(1 + m + t)(1 + m′ + t′) = 1 + (m + m′) + ((m ⊗ m′) + t + t′).

The surjection UM → M admits a set-theoretic section s: M → UM sending m to 1 + m. If m, m′ ∈ M, then

s(m) s(m′) s(m + m′)^{−1} = m ⊗ m′  (3)

in M ⊗ M ⊆ UM.

A simple computation verifies the following universal property of UM:

Proposition 2.1. The map s: M → UM is universal for set maps σ: M → G to a group G such that (m, m′) ↦ σ(m)σ(m′)σ(m + m′)^{−1} is a bilinear function from M × M to an abelian subgroup of G.

A quadratic map q: M → G is a set map between abelian groups such that (m, m′) ↦ q(m + m′) − q(m) − q(m′) is bilinear. (Perhaps “pointed quadratic map” would be better terminology; for instance, the quadratic maps q: ℚ → ℚ are the polynomial functions of degree at most 2 sending 0 to 0.) Proposition 2.1 implies:

Corollary 2.2. The map M → (UM)^{ab} is universal for quadratic maps from M to an abelian group. The map M → (UM)^{ab} ⊗ 𝔽_{2} is universal for quadratic maps from M to an abelian group such that the image is killed by 2.

Remark 2.3.

(a) The commutator [1 + m + t, 1 + m′ + t] equals m ⊗ m′ − m′ ⊗ m, so we have an exact sequence of abelian groups

0 → S^{2}M → (UM)^{ab} → M → 0,

where SM = ⨁_{n≥0} S^{n}M is the symmetric algebra. In particular,

(UM)^{ab} ≃ ker ((S^{<3}M)^{×} → (S^{<1}M)^{×}) .

(b) Similarly, if 2M = 0, then (1 + m + t)^{2} = 1 + m ⊗ m, so we obtain an exact sequence of

𝔽_{2}-vector spaces

0 → ^{⋀2} M → (UM)^{ab} ⊗ 𝔽_{2} → M → 0,

and

(UM)^{ab} ⊗ 𝔽_{2} ≃ ker ((^{⋀<3} M)^{×} → (^{⋀<1} M)^{×}) .

2.2. Sheaves of groups. In the rest of Section 2, C is a site. Let 𝔊𝔭_{C} be the category of sheaves of groups on C, and let 𝔄𝔟_{C} be the category of sheaves of abelian groups on C. For M ∈ 𝔄𝔟_{C}, write H^{i}(M) for Ext^{i}(ℤ, M), where ℤ is the constant sheaf; in other words, H^{i}(−) is the i^{th} right derived functor of Hom(ℤ, −) on 𝔄𝔟_{C}. For M ∈ 𝔊𝔭_{C}, define H^{0}(M) as Hom(ℤ, M) and define H^{1}(M) in terms of torsors as in [Gir71, §III.2.4]. The definitions are compatible for M ∈ 𝔄𝔟_{C} and i = 0, 1 [Gir71, Remarque III.3.5.4].

Remark 2.4. The reader may prefer to imagine the case for which sheaves are G-sets for some group G, abelian sheaves are ℤG-modules, and H^{i}(M) is just group cohomology.

All the constructions and results of Section 2.1 have sheaf analogues. In particular, for M ∈ 𝔄𝔟_{C} we obtain 𝒰 M ∈ 𝔊𝔭_{C} fitting in exact sequences

1 → M ⊗ M → 𝒰 M → M → 1  (4)
0 → S^{2}M → (𝒰 M)^{ab} → M → 0,  (5)

and, if 2M = 0,

0 → ^{⋀2} M → (𝒰 M)^{ab} ⊗ 𝔽_{2} → M → 0.  (6)

2.3. Self cup products.

Theorem 2.5. For M ∈ 𝔄𝔟_{C}, the connecting map H^{1}(M) → H^{2}(M ⊗ M) induced by (4) (see [Gir71, §IV.3.4.1]) maps each x to x ∪ x.

Proof. Let x ∈ H^{1}(M) = Ext^{1}(ℤ, M). Let

0 → M → X →^{α} ℤ → 0  (7)

be the corresponding extension. Let X_{1} := α^{−1}(1), which is a sheaf of torsors under M. We will construct a commutative diagram

[figure: 1; M ⊗ M; 𝒰 M; π; M; 1; 1; M ⊗ M; G; δ; G′; 1; 0; M ⊗ M; X ⊗ M; ϵ; M; 0]  (8)

of sheaves of groups, with exact rows. The first row is (4). The last row, obtained by tensoring (7) with M, is exact since ℤ is flat. Let G be the sheaf of (u, t) ∈ 𝒰 M ⊕ (X ⊗ M) such that π(u) = ϵ(t) in M. The vertical homomorphisms emanating from G are the two projections. Let δ: G → 𝒰 X send (u, t) to u−t. Then ker δ = M ⊗M, embedded diagonally in G. Let G′ = δ(G). Explicitly, if e is a section of X_{1}, then G′ consists of sections of 𝒰 X of the form 1 + m − e ⊗ m + t with m ∈ M and t ∈ M ⊗ M. The vertical homomorphisms emanating from G′ are induced by the map G → M sending (u, t) to π(u) = ϵ(t).

A calculation shows that 1 + X_{1} + M ⊗ M is a right torsor X′ under G′, corresponding to some x′ ∈ H^{1}(G′). Moreover, 𝒰 X → X restricts to a torsor map X′ → X_{1} compatible with G′ → M, so H^{1}(G′) → H^{1}(M) sends x′ to x.

By [Gir71, §IV.3.4.1.1], the commutativity of (8) shows that the image of x under the connecting map H^{1}(M) → H^{2}(M ⊗ M) from the first row, equals the image of x′ under the connecting map H^{1}(G′) → H^{2}(M ⊗ M) from the second row, which equals the image of x under the connecting homomorphism H^{1}(M) → H^{2}(M ⊗ M) from the third row. This last homomorphism is y ↦ x∪y, so it maps x to x∪x (cf. [Yon58], which explains this definition of x ∪ y for extensions of modules over a ring). □

Example 2.6. If M = ℤ/2ℤ, then (4) is the sequence of constant sheaves

0 → ℤ/2ℤ → ℤ/4ℤ → ℤ/2ℤ → 0,

which induces the Bockstein morphism H^{1}(X, ℤ/2ℤ) → H^{2}(X, ℤ/2ℤ). So Theorem 2.5 recovers the known result that for any topological space X, the self-cup-product H^{1}(X, ℤ/2ℤ) → H^{2}(X, ℤ/2ℤ) is the Bockstein morphism. (See properties (5) and (7) of Steenrod squares in Section 4.L of [Hat10].)

Remark 2.7. In group cohomology, if we represent a class in H^{1}(M) by a cochain ζ, then one can check that the coboundary of s ∘ ζ equals the difference of ζ ∪ ζ and the image of ζ under the connecting map. A similar argument using Čech cochains gives an alternate proof of the general case of Theorem 2.5, as we now explain.

By [Gir71, Théorème 0.2.6], we may replace C by a site with one having an equivalent topos in order to assume that C has finite fiber products, and in particular, a final object S. Then the natural map Ȟ^{1}(M) → H^{1}(M) is an isomorphism [Gir71, Remarque III.3.6.5(5)], so any x ∈ H^{1}(M) is represented by a 1-cocycle m for some covering (S′_{i} → S)_{i∈I}. For simplicity, let us assume that the covering consists of one morphism S′ → S (the general case is similar). Let S′′ = S′ ×_{S} S′, and S′′′ = S′ ×_{S} S′ ×_{S} S′. Let π_{23}, π_{13}, π_{12} : S′′′ → S′′ be the projections. So m ∈ M(S′′) satisfies π_{13}^{∗}m = π_{12}^{∗}m + π_{23}^{∗}m. Applying the section M → 𝒰 M yields a 1cochain 1+m ∈ (𝒰 M)(S′′). Its 2-coboundary in (M ⊗M)(S′′′) ⊆ (𝒰 M)(S′′′) represents the image of x under the connecting map H^{1}(M) → H^{2}(M ⊗ M) [Gir71, Corollaire IV.3.5.4(ii)]. On the other hand, the definition of the 2-coboundary given in [Gir71, Corollaire IV.3.5.4] together with (3) shows that it is π_{12}^{∗}m ⊗ π_{23}^{∗}m ∈ (M ⊗ M)(S′′′), whose class in H^{2}(M ⊗ M) represents x ∪ x, by definition.

Let M, N ∈ 𝔄𝔟_{C}. Let β ∈ H^{0}(Hom(M ⊗ M, N)). (The bold face in Hom, Ext, etc., indicates that we mean the sheaf versions.) Using β, construct an exact sequence

0 → N → 𝒰_{β} → M → 0  (9)

as the pushout of (4) by β: M ⊗ M → N.

If β is symmetric, then 𝒰_{β} is abelian, and we let ϵ_{β} be the class of (9) in Ext^{1}(M, N). If β is symmetric and Ext^{1}(M, N) = 0, then applying Hom(−, N) to (5) yields

0 → Hom(M, N) → Hom((𝒰 M)^{ab}, N) → Hom(S^{2}M, N) → 0

and a connecting homomorphism sends β ∈ H^{0}(Hom(S^{2}M, N)) to an element c_{β} ∈ H^{1}(Hom(M, N)).

Corollary 2.8. Then the following maps H^{1}(M) → H^{2}(N) are the same, when defined:

(a) The composition

H^{1}(M) ⟶^{Δ} H^{1}(M) × H^{1}(M) ⟶^{∪} H^{2}(M ⊗ M) ⟶^{β} H^{2}(N).

(b) The connecting homomorphism H^{1}(M) → H^{2}(N) associated to (9).

(c) The pairing with ϵ_{β} under the Yoneda product

Ext^{1}(M, N) × H^{1}(M) → H^{2}(N)

(if β is symmetric).

(d) The pairing with c_{β} under the evaluation cup product

H^{1}(Hom(M, N)) × H^{1}(M) → H^{2}(N)

(if β is symmetric and Ext^{1}(M, N) = 0).

Proof. Theorem 2.5 and functoriality implies the equality of (a) and (b). Standard homological algebra gives equality of (b), (c), and (d). □

2.4. Commutator pairings.

Proposition 2.9. Let 1 → A → B →^{ρ} C → 1 be an exact sequence in 𝔊𝔭_{C}, with A central in B, and C abelian. Let q: H^{1}(C) → H^{2}(A) be the connecting map. Given c_{1}, c_{2} ∈ C, we can lift them locally to b_{1}, b_{2} and form their commutator [b_{1}, b_{2}] := b_{1}b_{2}b_{1}^{−1}b_{2}^{−1} ∈ A; this induces a homomorphism [ , ]: C ⊗C → A. For γ_{1}, γ_{2} ∈ H^{1}(C), we have that q(γ_{1} +γ_{2})−q(γ_{1})−q(γ_{2}) equals the image of −γ_{1} ∪ γ_{2} under the homomorphism H^{2}(C ⊗ C) → H^{2}(A) induced by [ , ].

Proof. That the commutator induces a homomorphism is a well-known simple computation. Pulling back 1 → A^{3} → B^{3} → C^{3} → 1 by the homomorphism C^{2} → C^{3} sending (c_{1}, c_{2}) to (c_{1}, c_{2}, c_{1} + c_{2}) and then pushing out by the homomorphism A^{3} → A sending (a_{1}, a_{2}, a_{3}) to a_{3} − a_{2} − a_{1} yields an exact sequence 1 → A → Q → C^{2} → 1. Here Q = B′/B′′ where B′ is the subgroup sheaf of (b_{1}, b_{2}, b_{3}) ∈ B^{3} satisfying ρ(b_{3}) = ρ(b_{1})+ρ(b_{2}), and B′′ is the subgroup sheaf of B^{3} generated by sections (a_{1}, a_{2}, a_{3}) ∈ A^{3} with a_{3} = a_{1} +a_{2}. The surjection Q → C^{2} admits a section σ: C^{2} → Q defined locally as follows: given (c_{1}, c_{2}) lifting to (b_{1}, b_{2}) ∈ B^{2}, send it to the image of (b_{1}, b_{2}, b_{1}b_{2}) in Q (this is independent of the choice of lifts, since we work modulo B′′). A calculation shows that

σ((c′_{1}, c′_{2})) σ((c_{1}, c_{2}) + (c′_{1}, c′_{2}))^{−1} σ((c_{1}, c_{2})) = [c′_{1}, c_{2}^{−1}] = −[c′_{1}, c_{2}]  (10)

in A, and the three factors on the left may be rotated since the right hand side is central in Q. Proposition 2.1 and (10) yield the middle vertical map in the commutative diagram

[figure: 1; C^{2} ⊗ C^{2}; 𝒰 (C^{2}); C^{2}; 1; 1; A; Q; C^{2}; 1]  (11)

with exact rows, and the left vertical map sends (c_{1}, c_{2})⊗(c′_{1}, c′_{2}) to −[c′_{1}, c_{2}]. The connecting map for the first row sends (γ_{1}, γ_{2}) ∈ H^{1}(C^{2}) to (γ_{1}, γ_{2}) ∪ (γ_{1}, γ_{2}) ∈ H^{2}(C^{2} ⊗ C^{2}), by Theorem 2.5. The connecting map for the second row is a composition H^{1}(C^{2}) → H^{1}(C^{3}) → H^{2}(A^{3}) → H^{2}(A), so it maps (γ_{1}, γ_{2}) to q(γ_{1} + γ_{2}) − q(γ_{1}) − q(γ_{2}). Finally, the left vertical map sends (γ_{1}, γ_{2}) ∪ (γ_{1}, γ_{2}) ∈ H^{2}(C^{2} ⊗ C^{2}) to the image of −γ_{1} ∪ γ_{2} under the commutator pairing H^{2}(C ⊗ C) → H^{2}(A). So compatibility of the connecting maps yields the result. □

Remark 2.10. Yu. Zarhin [Zar74] proved Proposition 2.9 in the special case of group cohomology, by an explicit calculation with cocycles. Using the approach of Remark 2.7, that argument can be adapted to give a second proof of Proposition 2.9 in the general case.

3. Abelian schemes

3.1. The relative Picard functor. Let A → S be an abelian scheme. Let Pic_{A/S} be its relative Picard functor on the big fppf site of S. Trivialization along the identity section shows that Pic_{A/S}(T) ≃ Pic(A ×_{S} T)/ Pic T for each S-scheme T (see Proposition 4 on page 204 of [BLR90]). We generally identify line sheaves with their classes in Pic. For an S-scheme T and a ∈ A(T), let

τ_{a}: A_{T} → A_{T}
x ↦ a + x

be the translation-by-a morphism. Given a line sheaf ℒ on A, the theorem of the square implies that

ϕ_{ℒ} : A → Pic_{A/S}  (12)
a ↦ τ_{a}^{∗}ℒ ⊗ ℒ^{−1}

is a homomorphism. If we vary the base and vary ℒ, we obtain a homomorphism of fppfsheaves

Pic_{A/S} → Hom(A, Pic_{A/S})
ℒ ↦ ϕ_{ℒ} .

Its kernel is denoted Pic_{A/S}^{0}. Using the fact that Pic_{A/S} is an algebraic space, and the fact that Pic_{A/S}^{0} is an open subfunctor of Pic_{A/S} (which follows from [SGA 6, Exposé XIII, Théorème 4.7]), one can show that Pic_{A/S}^{0} is another abelian scheme Â over S [FC90, p. 3]. The image of ϕ_{ℒ} is contained in Â, so we may view ϕ_{ℒ} as a homomorphism A → Â. Moreover, ϕ_{ℒ} equals its dual homomorphism ϕ̂_{ℒ} . In fact, we have the following exact sequence of fppf-sheaves,

0 → Â → Pic_{A/S} → Hom_{self-dual}(A, Â) → 0,  (13)

in which the surjectivity of the last map ℒ ↦ ϕ_{ℒ} is [DP94, Proposition 1.2].

Remark 3.1. Let k be a field, let k_{s} be a separable closure contained in an algebraic closure \overline{k}, and let G_{k} = Gal(k_{s}/k). For an abelian variety A over k, the group Hom_{self-dual}(A, Â) of global sections of Hom_{self-dual}(A, Â) may be identified with the G_{k}-invariant subgroup of the Néron-Severi group NS A_{k_{s}}. (For the case k = \overline{k} see [Mum70], in particular Corollary 2 on page 178 and Theorem 2 on page 188 and the remark following it. The general case follows because any homomorphism defined over \overline{k} is in fact defined over k_{s}, since it maps the Zariski-dense set of prime-to-(char k) torsion points in A(k_{s}) to points in Â(k_{s}).)

For any homomorphism of abelian schemes λ: A → B, let A[λ] := ker λ.

3.2. Symmetric line sheaves. Multiplication by an integer n on A induces a pullback homomorphism [n]^{∗}: Pic_{A/S} → Pic_{A/S}. Let Pic_{A/S}^{Sym} be the kernel of [−1]^{∗} − [1]^{∗} on Pic_{A/S}. More concretely, because A → S has a section, we have Pic_{A/S}^{Sym}(T) = Pic^{Sym}(A ×_{S} T)/ Pic T for each S-scheme T, where Pic^{Sym}(A×_{S} T) is the group of isomorphism classes of symmetric line sheaves on A ×_{S} T. Since [−1]^{∗} acts as −1 on Â and as +1 on Hom_{self-dual}(A, Â), and since multiplication-by-2 on Â is surjective, the snake lemma applied to [−1]^{∗} − [1]^{∗} acting on (13) yields an exact sequence

0 → Â[2] → Pic_{A/S}^{Sym} → Hom_{self-dual}(A, Â) → 0.  (14)

3.3. The Weil pairing. We recall some facts and definitions that can be found in [Pol03, §10.4], for example. (In that book, S is Spec k for an algebraically closed field k, but the arguments there apply also to T -valued points of abelian schemes over an arbitrary base scheme S, functorially in the S-scheme T, so by Yoneda’s lemma they yield the desired results over S.) Given an abelian scheme A over S, there is a Weil pairing

e_{2} : A[2] × Â[2] → 𝔾_{m}.  (15)

For any homomorphism λ: A → Â, define e_{2}^{λ}: A[2] × A[2] → 𝔾_{m} by e_{2}^{λ}(x, y) = e_{2}(x, λy). If ℒ ∈ Pic_{A/S}(S), let e_{2}^{ℒ} = e_{2}^{ϕ_{ℒ}} ; this is an alternating bilinear pairing, and hence it is also symmetric.

3.4. Quadratic refinements of the Weil pairing.

Proposition 3.2. There is a (not necessarily bilinear) pairing of fppf sheaves

𝔮: A[2] × Pic_{A/S}^{Sym} → μ_{2} ⊂ 𝔾_{m}

such that:

(a) The pairing is additive in the second argument: 𝔮(x, ℒ ⊗ ℒ′) = 𝔮(x, ℒ)𝔮(x, ℒ′).

(b) The restriction of 𝔮 to a pairing

𝔮: A[2] × Â[2] → 𝔾_{m}

is the Weil pairing e_{2}. In particular, this restriction is bilinear.

(c) In general, 𝔮(x + y, ℒ) = 𝔮(x, ℒ)𝔮(y, ℒ)e_{2}^{ℒ} (x, y).

Proof. This is essentially proved in [Pol03, §13.1]. The book [Pol03] assumes that S = Spec k for an algebraically closed field k, but again (as in Section 3.3) the arguments there apply to scheme-valued points in the general case, so Yoneda’s lemma completes the proof. □

We can summarize Proposition 3.2 in the commutative diagram

\xymatrix{0 \ar[r] & Â[2] \ar[r] \ar[d]_{e_{2}}^{≀} & Pic_{A/S}^{Sym} \ar[r] \ar[d]_{𝔮} & Hom_{self-dual}(A, Â) \ar[r] \ar[d]_{e_{2}^{•}} & 0 \\ 0 \ar[r] & Hom(A[2], 𝔾_{m}) \ar[r] & Hom((𝒰 A[2])^{ab} ⊗ 𝔽_{2}, 𝔾_{m}) \ar[r] & Hom(^{⋀2} A[2], 𝔾_{m}) \ar[r] & 0,}

(16)

which we now explain. The top row is (14). The bottom row is obtained by applying Hom(−, 𝔾_{m}) to (6) for M = A[2], and using Ext^{1}(A[2], 𝔾_{m}) = 0. The vertical maps are the map y ↦ e_{2}(−, y), the map sending ℒ to the homomorphism (𝒰 A[2])^{ab} ⊗ 𝔽_{2} → 𝔾_{m} corresponding to 𝔮(−, ℒ) (see Corollary 2.2), and the map λ ↦ e_{2}^{λ}, respectively. Commutativity of the two squares are given by (b) and (c) in Proposition 3.2, respectively.

The top row of (16) gives a homomorphism

Hom_{self-dual}(A, Â) → H^{1}(Â[2])
λ ↦ c_{λ}.  (17)

By definition, c_{λ} may be interpreted geometrically as the class of the torsor under Â[2] that parametrizes symmetric line sheaves ℒ with ϕ_{ℒ} = λ (cf. [Pol03, §13.5]). Thus c_{λ} is the obstruction to finding ℒ ∈ Pic^{Sym} A with ϕ_{ℒ} = λ.

Remark 3.3. The map H^{1}(Â[2]) → H^{1}(Â) sends c_{λ} to the element called c_{λ} in [PS99, §4]. If k is a global field and S = Spec k, then this and [PS99, Corollary 2] imply that our c_{λ} lies in the 2-Selmer group of Â.

Theorem 3.4. For any λ ∈ Hom_{self-dual}(A, Â) and any x ∈ H^{1}(A[2]), we have

x ∪_{e_{2}^{λ}} x = x ∪_{e_{2}} c_{λ}  (18)

in H^{2}(𝔾_{m}), where the cup products are induced by the pairings underneath.

Proof. The rightmost vertical map in (16) maps λ to e_{2}^{λ}. These are mapped by the horizontal connecting homomorphisms to c_{λ} ∈ H^{1}(Â[2]) and c_{e_{2}^{λ}} ∈ H^{1}(Hom(A[2], 𝔾_{m})), which are identified by the leftmost vertical map e_{2}. Apply Corollary 2.8 with M = A[2], N = 𝔾_{m}, and β = e_{2}^{λ}, using Ext^{1}(A[2], 𝔾_{m}) = 0: map (a) gives the left hand side of (18) and map (d) gives the right hand side of (18) (written backwards) because of the identification of c_{λ} with c_{e^{λ}} via e_{2}. □

3.5. Criteria for triviality of the obstruction. The following lemma serves only to prove Proposition 3.6(a) below.

Lemma 3.5. Let k be a field, and let G be a finite cyclic group.

(a) Let A be a finite-dimensional kG-module. Let A^{∗} := Hom_{k}(A, k) be the dual representation, and let A^{G} be the subspace of G-invariant vectors. Then dim A^{G} = dim(A^{∗})^{G}.

(b) If 0 → A → B → C → 0 is an exact sequence of finite-dimensional kG-modules, and the surjection B^{∗} → A^{∗} admits a section as G-sets, then the connecting homomorphism C^{G} → H^{1}(G, A) is 0.

Proof.

(a) Let g be a generator of G. If M is a matrix representing the action of g on A, the action of g^{−1} on A^{∗} is given by the transpose M^{t}. Then dim A^{G} = dim ker(M − 1) = dim ker(M^{t} − 1) = dim(A^{∗})^{G}, where the middle equality uses the fact that a matrix has the same rank as its transpose.

(b) The section gives the 0 at the right in

0 → (C^{∗})^{G} → (B^{∗})^{G} → (A^{∗})^{G} → 0.

Taking dimensions and applying (a) yields dim B^{G} = dim A^{G} + dim C^{G}. This together with the exactness of

0 → A^{G} → B^{G} → C^{G} → H^{1}(G, A)

implies that the connecting homomorphism C^{G} → H^{1}(G, A) is 0. □

Proposition 3.6. Let λ: A → Â be a self-dual homomorphism of abelian varieties over a field k. Suppose that at least one of the following hypotheses holds:

(a) char k ≠ 2 and the image G of G_{k} → Aut A[2](k_{s}) is cyclic.

(b) k is a perfect field of characteristic 2.

(c) k is ℝ or ℂ.

(d) k is a nonarchimedean local field of residue characteristic not 2, and A has good reduction (i.e., extends to an abelian scheme over the valuation ring of k).

(e) k is a finite field.

(f) λ(A[2]) is an étale group scheme of rank at most 4. Then c_{λ} = 0.

Proof.

(a) Apply Lemma 3.5(b) to the bottom row of (16), viewed as a sequence of 𝔽_{2}G-modules; it applies since the dual sequence is (6) for M := A[2], and the section s of Section 2.1 yields a G-set section A[2] → (𝒰 A[2])^{ab} ⊗ 𝔽_{2}. Thus the top horizontal map in

[figure: H^{0}(G, Hom(^{⋀2} A[2], 𝔾_{m})); H^{1}(G, Â[2]); H^{0}(G_{k}, Hom(^{⋀2} A[2], 𝔾_{m})); δ; H^{1}(G_{k}, Â[2])]

is 0. Thus δ = 0. Now (16) shows that c_{λ} = δ(e_{2}^{λ}) = 0.

(b) Let M := A[2], and let M^{∨} := Hom(M, 𝔾_{m}) = Â[2] be its Cartier dual. The bottom row of (16) yields an exact sequence

H^{0}(Hom((𝒰 M)^{ab} ⊗ 𝔽_{2}, 𝔾_{m})) → H^{0}(Hom(^{⋀2} M, 𝔾_{m})) →^{δ} H^{1}(M^{∨}).  (19)

It suffices to prove that δ = 0, or that the first map is surjective. Equivalently, by the universal property of (𝒰 M)^{ab} ⊗ 𝔽_{2}, we need each alternating pairing b: M × M → μ_{2} to be q(x + y) − q(x) − q(y) for some quadratic map q: M → μ_{2}.

In fact, we will prove this for every finite commutative group scheme M over k with 2M = 0. Since k is perfect, there is a canonical decomposition M = M_{el} ⊕M_{le} ⊕M_{ll} into étale-local, local-étale, and local-local subgroup schemes. Then M^{∨} = (M_{le})^{∨} ⊕ (M_{el})^{∨} ⊕ (M_{ll})^{∨}. The homomorphism M → M^{∨} induced by the alternating pairing must map M_{el} to (M_{le})^{∨}, and M_{le} to (M_{el})^{∨}, and M_{ll} to (M_{ll})^{∨}. In particular, b = b_{e} + b_{ll} where b_{e} and b_{ll} are alternating pairings on M_{el} ⊕ M_{le} and M_{ll}, respectively. The pairing b_{e} necessarily has the form

(m_{el}, m_{le}), (m′_{el}, m′_{le}) ↦ B(m_{el}, m′_{le})B(m′_{el}, m_{le})

for some bilinear pairing B: M_{el} × M_{le} → μ_{2}. Then b_{e} comes from the quadratic map (m_{el}, m_{le}) ↦ B(m_{el}, m_{le}).

It remains to consider the case M = M_{ll}. Then M^{∨}(k_{s}) = 0. By [Mil06, Proposition III.6.1 and the paragraph preceding it], H^{1}(M^{∨}) = H^{1}(G_{k}, M^{∨}(k_{s})) = H^{1}(G_{k}, 0) = 0, so δ = 0.

(c) Follows from (a).

(d) The assumptions imply k(A[2]) is unramified over k (see [ST68, Theorem 1], for example), so (a) applies.

(e) Follows from (a) and (b).

(f) By definition of e_{2}^{λ}, the right kernel of e_{2}^{λ} contains the kernel K of A[2] ↠^{λ} λ(A[2]). Since e_{2}^{λ} is alternating, the left kernel of e_{2}^{λ} contains K too. Thus e_{2}^{λ} induces a nondegenerate alternating pairing

e′: λ(A[2]) × λ(A[2]) → 𝔾_{m}.

In particular, the étale group scheme λ(A[2]) has square order, which by assumption is 1 or 4. Let

q′: λ(A[2]) → 𝔾_{m}

be the morphism taking 0 to 1 and all other k_{s}-points of λ(A[2]) to −1. Then q′ is a quadratic form satisfying the identity q′(x + y) − q′(x) − q′(y) = e′(x, y). Now q := q′ ∘ λ is a quadratic form on A[2] refining e_{2}^{λ}, so c_{λ} = 0. □

3.6. Formula for the obstruction in the case of a line sheaf on a torsor. Let P be a torsor under A. For a ∈ A(S), let τ_{a}: P → P be the translation. Also, for x ∈ P (S), let τ_{x}: A → P be the torsor action. The maps τ_{x}^{∗} for local choices of sections x induce a well-defined isomorphism Pic_{P/S}^{0} ≃ Pic_{A/S}^{0} since any τ_{a}^{∗} is the identity on Pic_{A/S}^{0}. Let ℒ ∈ Pic_{P/S}(S). Generalizing (12), we define

ϕ_{ℒ} : A → Pic_{P/S}^{0} ≃ Pic_{A/S}^{0}
a ↦ τ_{a}^{∗}ℒ ⊗ ℒ^{−1}.

We may view ϕ_{ℒ} as an element of Hom_{self-dual}(A, Â). If x ∈ P (S), then ϕ_{τ_{x}^{∗}ℒ} = ϕ_{ℒ} .

Proposition 3.7. Let P be a torsor under A, equipped with an order-2 automorphism ι: P → P compatible with [−1]: A → A. The fixed locus P^{ι} of ι is a torsor under A[2]; let c ∈ H^{1}(A[2]) be its class. Let ℒ ∈ Pic_{P/S}(S) be such that ι^{∗}ℒ ≃ ℒ, and let λ = ϕ_{ℒ} : A → Â. Then c_{λ} = λ(c) in H^{1}(Â[2]).

Proof. If x is a section of P^{ι}, then [−1]^{∗}τ_{x}^{∗}ℒ ≃ τ_{x}^{∗}ι^{∗}ℒ ≃ τ_{x}^{∗}ℒ, so we obtain a map

γ: P^{ι} → Pic_{A/S}^{Sym}
x → τ_{x}^{∗}ℒ .

For sections a ∈ A[2] and x ∈ P^{ι}, we have

γ(a + x) = τ_{a+x}^{∗}ℒ = τ_{a}^{∗}(τ_{x}^{∗}ℒ) = ϕ_{τ_{x}^{∗}ℒ} (a) ⊗ τ_{x}^{∗}ℒ = λ(a) ⊗ γ(x)

in Pic_{A/S}^{Sym}. In other words, γ is a torsor map (with respect to λ: A[2] → Â[2]) from the torsor P^{ι} (under A[2]) to the torsor (under Â[2]) of line sheaves in Pic_{A/S}^{Sym} with Néron-Severi class λ. Taking classes of these torsors yields λ(c) = c_{λ}. □

3.7. Application to Jacobians. Let X → S be a family of genus-g curves, by which we mean a smooth proper morphism whose geometric fibers are integral curves of genus g. (If g ≠ 1, then the relative canonical sheaf or its inverse makes X → S projective: see Remark 2 on page 252 of [BLR90].) By the statement and proof of Proposition 4 on page 260 of [BLR90],

(1) There is a decomposition of functors Pic_{X/S} ≃ ∐_{n∈ℤ} Pic_{X/S}^{n}.

(2) The subfunctor Pic_{X/S}^{0} is (represented by) a projective abelian scheme A over S.

(3) The subfunctor Pic_{X/S}^{g−1} is (represented by) a smooth projective scheme P over S, a torsor under A. (If g = 1, then P = A.)

(4) The scheme-theoretic image of the “summing” map X^{g−1} → P is an effective relative Cartier divisor on P (take this to be empty if g = 0). Let Θ be the associated line sheaf on P.

(5) The homomorphism λ := ϕ_{Θ} : A → Â is an isomorphism.

(6) Define ι: P → P by ℱ ↦ ω_{X/S} ⊗ℱ^{−1}; then ι^{∗}Θ ≃ Θ. (To prove this, one can reduce to the case where S is a moduli scheme of curves with level structure, and then to the case where S is the spectrum of a field, in which case it is a consequence of the Riemann-Roch theorem.)

Definition 3.8. The theta characteristic torsor T is the closed subscheme of P = Pic_{X/S}^{g−1} parametrizing classes whose square is the canonical class ω_{X/S} ∈ Pic_{X/S}^{2g−2}(S).

Equivalently, T = P^{ι}. Let c_{T} ∈ H^{1}(A[2]) be the class of this torsor.

Theorem 3.9. Let X → S be a family of genus-g curves, and let A, λ, c_{T} be as above. Then c_{λ} = λ(c_{T}) in H^{1}(Â[2]), and for any x ∈ H^{1}(A[2]) we have

x ∪_{e_{2}^{λ}} x = x ∪_{e_{2}^{λ}} c_{T}  (20)

in H^{2}(𝔾_{m}).

Proof. Proposition 3.7 with P = Pic_{X/S}^{g−1} and ℒ = Θ yields c_{λ} = λ(c_{T}). So (18) in Theorem 3.4 becomes (20). □

Remark 3.10. If S = Spec k for a field k of characteristic not 2, and the action of G_{k} on A[2](k_{s}) factors through a cyclic quotient, then Proposition 3.6(a) gives c_{λ} = 0, so c_{T} = 0, recovering the result of M. Atiyah [Ati71, §5] that under these hypotheses X has a rational theta characteristic.

Similarly, if S = Spec k for a perfect field k of characteristic 2, then Proposition 3.6(b) gives c_{λ} = 0, so c_{T} = 0. In fact, the proof produces a canonical k-point of T . This generalizes an observation of Mumford [Mum71, p. 191] that a curve over an algebraically closed field of characteristic 2 has a canonical theta characteristic.

Additional criteria for the existence of a rational theta characteristic are given in [Sha13].

3.8. Hyperelliptic Jacobians.

Proposition 3.11. If E is an elliptic curve, then x ∪_{e_{2}^{λ}} x = 0 for all x ∈ H^{1}(E[2]). The same

holds for the Jacobian of any hyperelliptic curve X if it has a rational Weierstrass point or its genus is odd. In particular, this applies to y^{2} = f(x) with f separable of degree n ≢ 2 (mod 4) over a field of characteristic not 2.

Proof. For an elliptic curve E, the trivial line sheaf 𝒪_{E} is a theta characteristic defined over k. Now suppose that X is a hyperelliptic curve of genus g, so it is a degree-2 cover of a genus-0 curve Y. The class of a point in Y (k_{s}) pulls back to a k-point of Pic_{X/k}^{2}, and if g is odd, multiplying by (g − 1)/2 gives a k-point of T . On the other hand, if X has a rational Weierstrass point P, then 𝒪((g −1)P) is a theta characteristic defined over k. So T is trivial in all these cases. Now apply Theorem 3.9. □

Example 3.12. Suppose that X is a genus 2 curve, the smooth projective model of y^{2} = f(x) where f is a degree-6 separable polynomial over a field k of characteristic not 2. Let Δ be the set of zeros of f in k_{s}. As explained in [Mum71, p. 191], the group A[2] and its torsor T can be understood explicitly in terms of Δ. Namely, for m ∈ ℤ/2ℤ, let W_{m} be the quotient of the sum-m part of the permutation module 𝔽_{2}^{Δ} ≃ 𝔽_{2}^{2g+2} by the diagonal addition action of 𝔽_{2}. Then the G_{k}-module A[2] may be identified with W_{0}, and its torsor T may be identified with W_{1}. Using this, one can show:

(a) For f(x) = (x^{2} + 1)(x^{2} − 3)(x^{2} + 3) over ℚ_{3}, we have c_{T} ≠ 0.

(b) For f(x) = x^{6} + x + 6 over ℚ, we have

0 ≠ c_{T} ∈ �^{1}(ℚ, A[2]) := ker (H^{1}(ℚ, A[2]) → ∏_{p≤∞} H^{1}(ℚ_{p}, A[2])) .

(Proof: The discriminant of f is −ℓ, where ℓ is the prime 362793931. For p ∉ {2, ℓ}, the element c_{T} maps to 0 in H^{1}(ℚ_{p}, A[2]) by Proposition 3.6(c,d), and f(x) has a zero in each of ℚ_{2} and ℚ_{ℓ}, so the same is true at those places. On the other hand, the Galois group of f over ℚ is S_{6}, so c_{T} ≠ 0.)

Acknowledgements

We thank Benedict Gross for the suggestion to look at the self cup product on H^{1}(k, A[2]) induced by the Weil pairing e_{2} on a Jacobian. We also thank Brian Conrad and the referee for comments.

References

[Ati71] Michael F. Atiyah, Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup. (4) 4 (1971), 47–62. MR0286136 (44 #3350) ↑3.10

[BLR90] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR1045822 (91i:14034) ↑3.1, 3.7

[DP94] Pierre Deligne and Georgios Pappas, Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant, Compositio Math. 90 (1994), no. 1, 59–79 (French). MR1266495 (95a:11041) ↑3.1

[FC90] Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendix by David Mumford. MR1083353 (92d:14036) ↑3.1

[Gir71] Jean Giraud, Cohomologie non abélienne, Springer-Verlag, Berlin, 1971 (French). Die Grundlehren der mathematischen Wissenschaften, Band 179. MR0344253 (49 #8992) ↑2.2, 2.5, 2.3, 2.7

[Hat10] Allen Hatcher, Algebraic topology, June 30, 2010. Downloaded from

        http://www.math.cornell.edu/~hatcher/AT/ATpage.html . ↑2.6

[Mil06] J. S. Milne, Arithmetic duality theorems, Second edition, BookSurge, LLC, 2006. MR881804 (88e:14028) ↑b [Mum70] David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathemat-

ics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay, 1970. MR0282985 (44 #219) ↑3.1

[Mum71] , Theta characteristics of an algebraic curve, Ann. Sci. École Norm. Sup. (4) 4 (1971),

181–192. MR0292836 (45 #1918) ↑3.10, 3.12

[Pol03] Alexander Polishchuk, Abelian varieties, theta functions and the Fourier transform, Cambridge Tracts in Mathematics, vol. 153, Cambridge University Press, Cambridge, 2003. MR1987784 (2004m:14094) ↑3.3, 3.4, 3.4

[PR12] Bjorn Poonen and Eric Rains, Random maximal isotropic subspaces and Selmer groups, J. Amer. Math. Soc. 25 (2012), no. 1, 245–269, DOI 10.1090/S0894-0347-2011-00710-8. MR2833483 ↑1

[PS99] Bjorn Poonen and Michael Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109–1149. MR1740984 (2000m:11048) ↑3.3

[ST68] Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR0236190 (38 #4488) ↑d

[SGA 6] Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966– 1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR0354655 (50 #7133) ↑3.1

[Sha13] Shahed Sharif, A descent map for curves with totally degenerate semi-stable reduction, J. Théor. Nombres Bordeaux 25 (2013), no. 1, 211–244 (English, with English and French summaries). MR3063838 ↑3.10

[Yon58] Nobuo Yoneda, Note on products in Ext, Proc. Amer. Math. Soc. 9 (1958), 873–875. MR0175957 (31 #233) ↑2.3

[Zar74] Ju. G. Zarhin, Noncommutative cohomology and Mumford groups, Mat. Zametki 15 (1974), 415– 419 (Russian). MR0354612 (50 #7090) ↑2.10

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA

E-mail address: poonen@math.mit.edu
URL: http://math.mit.edu/~poonen/

Department of Mathematics, California Institute of Technology, Pasadena, CA 91125

E-mail address: rains@caltech.edu