The Cassels-Tate pairing on polarized abelian varieties

32 Pages·2017·0.49 MB·English

THE CASSELS–TATE PAIRING ON POLARIZED ABELIAN VARIETIES
BJORN POONEN AND MICHAEL STOLL
Abstract. Let (A,λ) be a principally polarized abelian variety defined over a global field
k, and let X(A) be its Shafarevich–Tate group. Let X(A) denote the quotient of X(A)
nd
by its maximal divisible subgroup. Cassels and Tate constructed a non-degenerate pairing
X(A) ×X(A) →Q/Z.
nd nd
IfAisanellipticcurve,thenbyaresultofCassels’thepairingisalternating. Butingeneral
it is only antisymmetric.
Using some new but equivalent definitions of the pairing, we derive general criteria de-
ciding whether it is alternating and whether there exists some alternating non-degenerate
pairing on X(A) . These criteria are expressed in terms of an element c∈X(A) that is
nd nd
canonicallyassociatedtothepolarizationλ. InthecasethatAistheJacobianofsomecurve,
a down-to-earth version of the result allows us to determine effectively whether #X(A) (if
finite) is a square or twice a square. We then apply this to prove that a positive proportion
(insomeprecisesense)ofallhyperellipticcurvesofevengenusg ≥2overQhaveaJacobian
with non-square #X (if finite). For example, it appears that this density is about 13% for
curvesofgenus2. Theproofmakesuseofageneralresultrelatingglobalandlocaldensities;
this result can be applied in other situations.
Contents
1. Introduction 2
2. Notation 3
3. Two definitions of the Cassels–Tate pairing 6
3.1. The homogeneous space definition 6
3.2. The Albanese–Picard definition 6
4. The homogeneous space associated to a polarization 8
5. The obstruction to being alternating 10
6. Consequences of the pairing theorem 11
7. A formula for Albanese and Picard varieties 12
8. The criterion for oddness for Jacobians 13
9. The density of odd hyperelliptic Jacobians over Q 14
9.1. The archimedean density 15
9.2. The nonarchimedean densities 16
9.3. The passage from local to global 20
9.4. The global density 22
Date: November 6, 1998; minor correction August 23, 2014.
1991 Mathematics Subject Classification. 11G10, 11G30, 14H25, 14H40.
Keywordsandphrases. Abelianvarietiesoverglobalfields,Shafarevich–Tategroup,Cassels–Tatepairing,
Jacobian varieties, global densities.
Much of this research was done while the first author was supported by an NSF Mathematical Sciences
Postdoctoral Research Fellowship at Princeton University. This article has been published in Annals of
Math. 150 (1999), 1109–1149.
1
2 BJORN POONEN AND MICHAEL STOLL
10. Examples of Shafarevich–Tate groups of Jacobians 23
10.1. Jacobians of Shimura curves 23
10.2. Explicit examples 24
11. An open question of Tate about Brauer groups 26
12. Appendix: other definitions of the Cassels–Tate pairing 27
12.1. The Albanese–Albanese definition 27
12.2. The Weil pairing definition 27
12.3. Compatibility 28
Acknowledgements 31
References 31
1. Introduction
ThestudyoftheShafarevich–TategroupX(A)ofanabelianvarietyAoveraglobalfieldk
is fundamental to the understanding of the arithmetic of A. It plays a role analogous to that
of the class group in the theory of the multiplicative group over an order in k. Cassels [Ca],
in one of the first papers devoted to the study of X, proved that in the case where E is an
elliptic curve over a number field, there exists a pairing
X(E)×X(E) −→ Q/Z
that becomes nondegenerate after one divides X(E) by its maximal divisible subgroup. He
proved also that this pairing is alternating; i.e., that (cid:104)x,x(cid:105) = 0 for all x. If, as is conjectured,
X(E) is always finite, then this would force its order to be a perfect square. Tate [Ta2]
soon generalized Cassels’ results by proving that for abelian varieties A and their duals A∨
in general, there is a pairing
X(A)×X(A∨) −→ Q/Z,
that is nondegenerate after dividing by maximal divisible subgroups. He also proved that if
X(A) is mapped to X(A∨) via a polarization arising from a k-rational divisor on A then
the induced pairing on X(A) is alternating. But it is known that when dimA > 1, a k-
rational polarization need not come from a k-rational divisor on A. (See Section 4 for the
obstruction.) For principally polarized abelian varieties in general1, Flach [Fl] proved that
the pairing is antisymmetric, by which we mean (cid:104)x,y(cid:105) = −(cid:104)y,x(cid:105) for all x,y, which is slightly
weaker than the alternating condition.
It seems to have been largely forgotten that the alternating property was never proved in
general: in a few places in the literature, one can find the claim that the pairing is always
alternating for Jacobians of curves over number fields, for example. In Section 10 we will
give explicit examples to show that this is not true, and that #X(J) need not be a perfect
square even if J is a Jacobian of a curve over Q.2
1Actually Flach considers this question in a much more general setting.
2ThisisperhapsespeciallysurprisinginlightofUrabe’srecentresults[Ur],whichimplyforinstanceforthe
analogous situation of a proper smooth geometrically integral surface X over a finite field k of characteristic
p, that if the prime-to-p part of Br(X) is finite, the order of this prime-to-p part is a square. (There exist
“examples” of non-square Brauer groups in the literature, but Urabe explains why they are incorrect.) See
Section 11 for more comments on the Brauer group.
THE CASSELS–TATE PAIRING 3
One may ask what properties beyond antisymmetry the pairing has in the general case of a
principally polarized abelian variety (A,λ) over a global field k. For simplicity, let us assume
here that X(A) is finite, so that the pairing is nondegenerate. Flach’s result implies that
x (cid:55)→ (cid:104)x,x(cid:105) is a homomorphism X(A) → Q/Z, so by nondegeneracy there exists c ∈ X(A)
such that (cid:104)x,x(cid:105) = (cid:104)x,c(cid:105). Since Flach’s result implies 2(cid:104)x,x(cid:105) = 0, we also have 2c = 0 by
nondegeneracy. It is then natural to ask, what is this element c ∈ X(A)[2] that we have
canonically associated to (A,λ)? An intrinsic definition of c is given in Section 4, and it
will be shown3 that c vanishes (i.e., the pairing is alternating) if and only if the polarization
arises from a k-rational divisor on A. This shows that Tate’s and Flach’s results are each
best possible in a certain sense.
Our paper begins with a summary of most of the notation and terminology that will be
needed, and with two definitions of the pairing. (We give two more definitions and prove the
compatibility of all four in an appendix.) Sections 4 and 5 give the intrinsic definition of c,
and show that it has the desired property. (Actually, we work a little more generally: λ is
not assumed to be principal, and in fact it may be a difference of polarizations.) Section 6
develops some consequences of the existence of c; for instance if A is principally polarized
and X(A) is finite, then its order is a square or twice a square according as (cid:104)c,c(cid:105) equals 0
or 1 in Q/Z. We call A even in the first case and odd in the second case.
2
The main goal of Sections 7 and 8 is to translate this into a more down-to-earth criterion
for the Jacobian of a genus g curve X over k: (cid:104)c,c(cid:105) = N/2 ∈ Q/Z where N is the number
of places v of k for which X has no k -rational divisor of degree g − 1. Section 9 applies
v
this criterion to hyperelliptic curves of even genus g over Q, and shows that a positive
proportion ρ of these (in a sense to be made precise) have odd Jacobian. It also gives an
g
exact formula for ρ in terms of certain local densities, and determines the behavior of ρ as
g g
g goes to infinity. The result relating the local and global densities is quite general and can
be applied to other similar questions. Numerical calculations based on the estimates and
formulas obtained give an approximate value of 13% for the density ρ of curves of genus 2
2
over Q with odd Jacobian.
Section 10 applies the criterion to prove that Jacobians of certain Shimura curves are
always even. It gives also a few other examples, including an explicit genus 2 curve over
Q for whose Jacobian we can prove unconditionally that (cid:104)c,c(cid:105) = 1 and X ∼= Z/2Z, and
2
another for which X is finite of square order, but with (cid:104) , (cid:105) not alternating on it.
Finally, Section 11 addresses the analogous questions for Brauer groups of surfaces over
finite fields, recasting a an old question of Tate in new terms.
2. Notation
Many of the definitions in this section are standard. The reader is encouraged to skim
this section and the next, and to proceed to Section 4.
If S is a set, then 2S denotes its power set.
Suppose that M is an abelian group. For each n ≥ 1, let M[n] = {m ∈ M : nm = 0}. Let
M = (cid:83)∞ M[n] = (cid:76) M(p), where for each prime p, M(p) = (cid:83)∞ M[pn] denotes the
tors n=1 p n=1
p-primary part of the torsion subgroup of M. Let M be the maximal divisible subgroup of
div
M. If M[p] is finite for every prime p, then M = (cid:84)∞ nM. Denote by M the quotient
div n=1 nd
3The statement of this result needs to be modified slightly if the finiteness of X(A) is not assumed.
4 BJORN POONEN AND MICHAEL STOLL
M/M . (The subscript nd stands for “non-divisible part.”) If
div
(cid:104) , (cid:105) : M ×M(cid:48) −→ Q/Z
is a bilinear pairing between two abelian groups, then for any m ∈ M, let m⊥ = {m(cid:48) ∈ M(cid:48) :
(cid:104)m,m(cid:48)(cid:105) = 0}, and for any subgroup V ⊆ M, let V⊥ = (cid:84) v⊥. When M = M(cid:48), we say that
v∈V
(cid:104) , (cid:105) is antisymmetric if (cid:104)a,b(cid:105) = −(cid:104)b,a(cid:105) for all a,b ∈ M, and alternating if (cid:104)a,a(cid:105) = 0 for all
a ∈ M. Note that a bilinear pairing (cid:104) , (cid:105) on M is antisymmetric if and only if m (cid:55)→ (cid:104)m,m(cid:105)
is a homomorphism. If a pairing is alternating, then it is antisymmetric, but the converse is
guaranteed on M(p) only for odd p.
Ifk isafield,thenk andksep denotealgebraicandseparableclosures,andG = Gal(k/k) =
k
Gal(ksep/k) denotes its absolute Galois group. If k is a global field, then M denotes the set
k
of places of k. If moreover v ∈ M , then k denotes the completion, and G = Gal(ksep/k )
k v v v v
denotes the absolute Galois group of k .
v
Suppose that G is a profinite group acting continuously on an abelian group M. We
use Ci(G,M) (resp. Zi(G,M) and Hi(G,M)) to denote the group of continuous i-cochains
(resp.i-cocylesandi-cohomologyclasses)withvaluesintheG-moduleM. Ifk isunderstood,
we use Ci(M) as an abbreviation for Ci(G ,M), and similarly for Zi(M) and Hi(M). If
k
α ∈ Ci(G ,M), then α ∈ Ci(G ,M) denotes its local restriction.4 If v is a place of a global
k v v
field, we use inv to denote the usual monomorphism H2(G ,ksep∗) = Br(k ) → Q/Z (which
v v v v
is an isomorphism if v is nonarchimedean).
Varieties will be assumed to be geometrically integral, smooth, and projective, unless
otherwise specified. If X is a variety over k, let k(X) denote the function field of X. If K is
a field extension of k, then X denotes X× K, the same variety with the base extended to
K k
SpecK. Let Div(X) = H0(G ,Div(X )) denote the group of (k-rational) Weil divisors on
k ksep
X. If f ∈ k(X)∗ or f ∈ k(X)∗/k∗, let (f) ∈ Div(X) denote the associated principal divisor.
If D ∈ Div(X×Y), let tD ∈ Div(Y ×X) denote its transpose. Let Pic(X) denote the group
of invertible sheaves (= line bundles) on X, let Pic0(X ) denote the subgroup of Pic(X )
ksep ksep
of invertible sheaves algebraically equivalent to 0, and let Pic0(X) = Pic(X)∩Pic0(X ).
ksep
In older terminology, which we will find convenient to use on occasion, H0(Pic(X )) is
ksep
the group of k-rational divisor classes, and Pic(X) is the subgroup of divisor classes that
are actually represented by k-rational divisors. Define the N´eron-Severi group NS(X) =
Pic(X)/Pic0(X). If D ∈ Div(X), let L(D) ∈ Pic(X) be the associated invertible sheaf. Let
Div0(X) be the subgroup of Div(X) mapping into Pic0(X). If λ ∈ H0(NS(V )), let Picλ
ksep V/k
denotethecomponentofthePicardschemePic correspondingtoλ. Ifk hascharacteristic
V/k
p > 0, and dimV ≥ 2, then these schemes need not be reduced [Mu1, lecture 27], but in
any case the associated reduced scheme Picλ is a principal homogeneous space of the
V/k,red
Picard variety Pic0 , which is an abelian variety over k. Also, Picλ (ksep) equals the
V/k,red V/k,red
preimage of λ under Pic(V ) → NS(V ) with its G -action.
ksep ksep k
Let Z(X) (resp. Zi(X)) denote the group of 0-cycles on X (resp. the set of 0-cycles of
degree i on X). For any i ∈ Z, let Albi denote the degree i component of the Albanese
X/k
scheme. Let Y0(X) denote the kernel of the natural map Z0(X) → Alb0 (k). Then
X/k
4We will be slightly sloppy in writing this, because we often will intend the “M” in Ci(G ,M) to be a
k
propersubgroupofthe“M”inCi(G ,M);forinstancethesetwoM’smaybetheksep-pointsandksep-points
v v
of a group scheme A over k, in which case we abbreviate by Ci(G ,A) and Ci(G ,A) even though the two
k v
A’s represent different groups of points.
THE CASSELS–TATE PAIRING 5
Albi is a principal homogeneous space of the Albanese variety Alb0 and its k-
X/k,red X/k,red
points correspond (G -equivariantly) to elements of Zi(X ) modulo the action of Y0(X ).
k k k
If X is a curve (geometrically integral, smooth, projective, as usual) and i ∈ Z, let Pici(X)
denote the set of elements of Pic(X) of degree i, and let Pici be the degree i component
X/k
of the Picard scheme, which is a principal homogeneous space of the Jacobian Pic0 of
X/k
X. (Since dimX = 1, these are already reduced.) Points on Pici over k correspond to
X/k
divisor classes of degree i on X . It will be important to keep in mind that the injection
k
Pici(X) → Pici (k) is not always surjective. (In other words, k-rational divisor classes are
X/k
not always represented by k-rational divisors.)
If A is an abelian variety, then A∨ = Pic0 = Pic0 denotes the dual abelian variety.
A/k,red A/k
If X is a principal homogeneous space of A, then for each a ∈ A(k), we let t denote the
a
translation-by-a map on X, and similarly if x ∈ X(k), then t is the trivialization A → X
x
mapping 0 to x. If D ∈ DivA, let D = t D ∈ DivX. (Note: if a ∈ A(k), then t∗D = D .)
x x a −a
If L ∈ PicX, then φ denotes the homomorphism A → A∨ mapping a to t∗L⊗L−1. (There
L a
is a natural identification Pic0 = A∨.) We may also identify φ with the class of L in
X/k L
NS(X). If D ∈ DivX, then define φ = φ . A polarization on A (defined over k)
D L(D)
is a homomorphism A → A∨ (defined over k) which over ksep equals φ for some ample
L
L ∈ Pic(X ). (One can show that this gives the same concept as the usual definition,
ksep
in which k is used instead of ksep.) A principal polarization is a polarization that is an
isomorphism.
IfAisanabelianvarietyoveraglobalfieldk,thenletX(A) = X(k,A)betheShafarevich-
TategroupofAoverk, whoseelementsweidentifywithlocallytrivialprincipalhomogeneous
spaces of A (up to equivalence). By [Mi1], X(A)[n] is finite for all n ≥ 1.
Suppose that V and W are varieties over a field k, and that D is a divisor on V ×W. If
v ∈ V(k), let D(v) ∈ Div(W ) be the pullback of D under the map W → V ×W sending
k
w to (v,w), when this makes sense. For a ∈ Z(V ), define D(a) ∈ Div(W ) by extending
k k
linearly, when this makes sense. If a ∈ Y0(V ) and D(a) is defined, then D(a) = (f) for
k
some function f on W. (See the proof of Theorem 10 on p. 171 of [La1, VI, §4].) If in
addition a(cid:48) ∈ Z0(W ), and if f(a(cid:48)) is defined, we put D(a,a(cid:48)) = f(a(cid:48)) and say that D(a,a(cid:48)) is
k
defined. If a ∈ Y0(V ) and a(cid:48) ∈ Y0(W ), then we may interchange V and W to try to define
k k
tD(a(cid:48),a), and Lang’s reciprocity law (p. 171 of [La1] again) states that D(a,a(cid:48)) and tD(a(cid:48),a)
are defined and equal, provided that a×a(cid:48) and D have disjoint supports.
We let µ denote the standard Lebesgue measure on Rd, and let µ denote the Haar
∞ p
measure on Zd normalized to have total mass 1. For v = (v ,v ,...,v ) ∈ Zd, define
p 1 2 d
|v| := max |v |. If S ⊆ Zd, then the density of S is defined to be
i i
(cid:88)
ρ(S) := lim (2N)−d 1,
N→∞
v∈S,|v|≤N
if the limit exists. Define the upper density ρ(S) and lower density ρ(S) similarly, except
with the limit replaced by a limsup or liminf, respectively.
We will use the notations Ad and Pd for d–dimensional affine and projective space, re-
spectively.
6 BJORN POONEN AND MICHAEL STOLL
3. Two definitions of the Cassels–Tate pairing
In this section, we present the two definitions of the Cassels–Tate pairing that we will
use in the paper. The first is well-known [Mi5]. The second appears to be new, but it was
partly inspired by Remark 6.12 on page 100 of [Mi5]. In an appendix we will give two other
definitions, and show that all four are compatible.
3.1. The homogeneous space definition. Let A be an abelian variety over a global field
k. Suppose a ∈ X(A) and a(cid:48) ∈ X(A∨). Let X be the (locally trivial) homogeneous
space over k representing a. Then Pic0(X ) is canonically isomorphic as G -module to
ksep k
Pic0(A ) = A∨(ksep), so a(cid:48) corresponds to an element of H1(Pic0(X )), which we may
ksep ksep
map to an element b(cid:48) ∈ H2(ksep(X)∗/ksep∗) using the long exact sequence associated with
ksep(X)∗
0 −→ −→ Div0(X ) −→ Pic0(X ) −→ 0.
ksep∗ ksep ksep
Since H3(ksep∗) = 0, we may lift b(cid:48) to an element f(cid:48) ∈ H2(G ,ksep(X)∗). Then it turns out
k
that f(cid:48) ∈ H2(G ,ksep(X)∗) is the image of an element c ∈ H2(G ,ksep∗).5 We define
v v v v v v
(cid:88)
(cid:104)a,a(cid:48)(cid:105) = inv (c ) ∈ Q/Z.
v v
v∈M
k
See Remark 6.11 of [Mi5] for more details. The obvious advantage of this definition over the
others is its simplicity.
If λ : A → A∨ is a homomorphism, then we define a pairing
(cid:104) , (cid:105) : X(A)×X(A) −→ Q/Z
λ
by (cid:104)a,b(cid:105) = (cid:104)a,λb(cid:105).
λ
3.2. TheAlbanese–Picarddefinition. LetV beavariety(geometricallyintegral,smooth,
projective, as usual) over a global field k. Our goal is to define a pairing
(1) (cid:104) , (cid:105) : X(Alb0 )×X(Pic0 ) −→ Q/Z.
V V/k,red V/k,red
We will first need a partially-defined G -equivariant pairing
k
(2) [ , ] : Y0(V )×Div0(V ) −→ ksep∗.
ksep ksep
Temporarily we work instead with a variety V over a separably closed field K. Let
A = Alb0 and A(cid:48) = Pic0 . Let P denote a Poincar´e divisor on A×A(cid:48). Choose a
V/K,red V/K,red
basepointP ∈ V(K)todefineamapφ : V → A. LetP = (φ,1)∗P ∈ Div(V ×A(cid:48)). Suppose
0 0
y ∈ Y0(V) and D(cid:48) ∈ Div0(V). Choose z(cid:48) ∈ Z0(A(cid:48)) that sums to L(D(cid:48)) ∈ Pic0(V) = A(cid:48)(K).
Then D(cid:48) −tP (z(cid:48)) is the divisor of some function f(cid:48) on V. Define
0
[y,D(cid:48)] := f(cid:48)(y)+P (y,z(cid:48))
0
if the terms on the right make sense.
We now show that this pairing is independent of choices made. If we change z(cid:48), we can
changeitonlybyanelementy(cid:48) ∈ Y0(A(cid:48)),andthen[y,D(cid:48)]changesby−tP (y(cid:48),y)+P (y,y(cid:48)) =
0 0
0, by Lang reciprocity. If we change P by the divisor of a function F on A×A(cid:48), then [y,D(cid:48)]
changes by F(φ(y) × z(cid:48)) − F(φ(y) × z(cid:48)) = 0. If E ∈ Div(A(cid:48)), and if we change P by π∗E
2
(π being the projection A × A(cid:48) → A(cid:48)), then [y,D(cid:48)] is again unchanged. By the seesaw
2
5One can compute c by evaluating f(cid:48) at a point in X(k ), or more generally at an element of Z1(X )
v v v kv
(provided that one avoids the zeros and poles of f(cid:48)).
v
THE CASSELS–TATE PAIRING 7
principle [Mi4, Theorem 5.1], the translate of P by (a,0) ∈ (A×A(cid:48))(K) differs from P by
0 0
a divisor of the form (F)+π∗E; therefore the definition of [y,D(cid:48)] is independent of the choice
2
of P . It then follows that if V is a variety over any field k, then we obtain a G -equivariant
0 k
pairing (2).
Remark. If V is a curve, then an element of Y0(V ) is the divisor of a function, and the
ksep
pairing (2) is simply evaluation of the function at the element of Div0(V ).
ksep
We now return to the definition of (1). It will be built from the two exact sequences
0 −→ Y0(V ) −→ Z0(V ) −→ A(ksep) −→ 0
ksep ksep
(3) ksep(V)∗
0 −→ −→ Div0(V ) −→ A(cid:48)(ksep) −→ 0
ksep∗ ksep
and the two partially-defined pairings
[ , ] : Y0(V )×Div0(V ) −→ ksep∗
ksep ksep
ksep(V)∗
(4) Z0(V )× −→ ksep∗,
ksep ksep∗
the latter defined by lifting the second argument to a function on V , and “evaluating” on
ksep
the first argument. We may consider ksep(V)∗ to be a subgroup of Div0(V ), and then the
ksep∗ ksep
two pairings agree on Y0(V )×ksep(V)∗, so there will be no ambiguity if we let ∪ denote the
ksep ksep∗
cup-product pairing on cochains associated to these. We have also the analogous sequences
and pairings for each completion of k.
Suppose a ∈ X(A) and a(cid:48) ∈ X(A(cid:48)). Choose α ∈ Z1(A(ksep)) and α(cid:48) ∈ Z1(A(cid:48)(ksep))
representing a and a(cid:48), and lift these to a ∈ C1(Z0(V )) and a(cid:48) ∈ C1(Div0(V )) so that for
ksep ksep
all σ,τ ∈ G , all G -conjugates of a have support disjoint from the support of a(cid:48). Define
k k σ τ
η := da∪a(cid:48) −a∪da(cid:48) ∈ C3(ksep∗).
We have dη = 0, but H3(ksep∗) = 0, so η = d(cid:15) for some (cid:15) ∈ C2(ksep∗).
Since a is locally trivial, we may for each place v ∈ M choose β ∈ A(ksep) such that
k v v
α = dβ . Choose b ∈ Z0(V ) mapping to β and so that for all σ ∈ G , all G -conjugates
v v v kvsep v k v
of b have support disjoint from the support of a(cid:48) . Then6
v σ
γ := (a −db )∪a(cid:48) −b ∪da(cid:48) −(cid:15) ∈ C2(G ,ksep∗)
v v v v v v v v v
is a 2-cocycle representing some
c ∈ H2(G ,ksep∗) = Br(k ) −in→vv Q/Z.
v v v v
Define
(cid:88)
(cid:104)a,a(cid:48)(cid:105) = inv (c ).
D v v
v∈M
k
One checks using d(x∪y) = dx∪y +(−1)degxx∪dy [AW, p. 106] that this is well-defined
and independent of choices. In proving independence of the choice of β one uses the local
v
triviality of a(cid:48), which has not been used so far.
This definition appears to be the most useful one for explicit calculations when A is a
Jacobian of a curve of genus greater than 1. This is because the present definition involves
6Since it is only the difference of the terms a and db that is in Y0(V ), it would make no sense to
v v ksep
replace (a −db )∪a(cid:48) by a ∪a(cid:48) −db ∪a(cid:48).
v v v v v v v
8 BJORN POONEN AND MICHAEL STOLL
only divisors on the curve, instead of m2-torsion or homogeneous spaces of the Jacobian,
which are more difficult to deal with computationally.
Remark. The reader may have recognized the setup of two exact sequences with two pairings
as being the same as that required for the definition of the augmented cup-product (see [Mi5,
p.10]). Hereweexplaintheconnection. SupposethatwehaveexactsequencesofG -modules
k
0 −→ M −→ M −→ M −→ 0
1 2 3
0 −→ N −→ N −→ N −→ 0
1 2 3
0 −→ P −→ P −→ P −→ 0
1 2 3
and a G -equivariant bilinear pairing M ×N → P that maps M ×N into P . The pairing
k 2 2 2 1 1 1
inducesapairingM ×N → P . Ifa ∈ ker[Hi(M ) → Hi(M )]anda(cid:48) ∈ ker[Hj(N ) → Hj(N )],
3 1 3 1 2 1 2
then a comes from some b ∈ Hi−1(M ), which can be paired with a(cid:48) to give an element of
3
Hi+j−1(P ). If one changes b by an element c ∈ Hi−1(M ), then the result is unchanged,
3 2
since the change can be obtained by pairing c with the (zero) image of a(cid:48) in Hj(N ) under
2
the cup-product pairing associated with M ×N → P . Thus we have a well-defined pairing7
2 2 2
(cid:2) (cid:3) (cid:2) (cid:3)
(5) ker Hi(M ) → Hi(M ) ×ker Hj(N ) → Hj(N ) −→ Hi+j−1(P ).
1 2 1 2 3
If we replace each M and N by complexes with terms in degrees 0 and 1, replace each P
i i i
by a complex with a single term, in degree 1, and replace cohomology by hypercohomology,
then we obtain a pairing analogous to (5) defined using the augmented cup-product. One
obtains the definition of the Cassels–Tate pairing above by noting that:
(1) If A is the ad`ele ring of k,
k
X(A) = ker(cid:2)H1(G ,A(ksep)) → H1(G ,A(ksep ⊗ A ))(cid:3)
k k k k
by Shapiro’s lemma;
(2) The analogous statement holds for X(A(cid:48)); and
(3) If P = ksep∗ and P = (ksep⊗ A )∗, then the cokernel P has H2(G ,P ) = Q/Z by
1 2 k k 3 k 3
class field theory.
4. The homogeneous space associated to a polarization
Let A be an abelian variety over a field k. To each element of H0(NS(A )) we can
ksep
associate a homogeneous space of A∨ that measures the obstruction to it arising from a
k-rational divisor on A. From
0 −→ A∨(ksep) −→ Pic(A ) −→ NS(A ) −→ 0
ksep ksep
we obtain the long exact sequence
(6) 0 −→ A∨(k) −→ Pic(A) −→ H0(NS(A )) −→ H1(A∨(ksep)) −→ H1(Pic(A )).
ksep ksep
(We have H0(Pic(A )) = PicA because A(k) (cid:54)= ∅.) For λ ∈ H0(NS(A )), define c to be
ksep ksep λ
the image of λ in H1(A∨(ksep)). By the proof of Theorem 2 in Section 20 of [Mu3], we have
2λ = φ where L = (1,λ)∗P ∈ PicA is the pullback of the Poincar´e bundle P on A×A∨ by
L
(1,λ) : A → A×A∨. Hence 2c = 0.8
λ
7The “diminished cup-product”?!
8One could also obtain this by using the fact that multiplication by −1 on A induces (−1)2 = +1 on
NS(A ) and −1 on H1(A∨(ksep)).
ksep
THE CASSELS–TATE PAIRING 9
Lemma 1. If k is a local field, then c = 0 for all λ ∈ H0(NS(A )).
λ ksep
Proof. Recall that “Tate9 local duality” [Ta1] gives a pairing
H0(A(ksep))×H1(A∨(ksep)) −→ H2(ksep∗) (cid:44)→ Q/Z
that is perfect, at least after dividing by the connected component on the left in the
archimedean case. It can be defined as follows: if P ∈ H0(A(ksep)) = A(k) and z ∈
H1(A∨(ksep)), then use the long exact sequence associated to
ksep(A)∗
0 −→ −→ Div0(A ) −→ A∨(ksep) −→ 0
ksep∗ ksep
(cid:16) (cid:17)
to map z to H2 ksep(A)∗ , and “evaluate” the result on a degree zero k-rational 0-cycle on
ksep∗
A representing P to obtain an element of H2(ksep∗).
Suppose λ ∈ H0(NS(A )). By (6), c is in the kernel of H1(A∨) → H1(Pic(A )). The
ksep λ ksep
long exact sequences associated with
ksep(A)∗
0 −−−→ −−−→ Div0(A ) −−−→ A∨(ksep) −−−→ 0
ksep∗ ksep
(cid:13)  
(cid:13)  
(cid:13) (cid:121) (cid:121)
ksep(A)∗
0 −−−→ −−−→ Div(A ) −−−→ Pic(A ) −−−→ 0
ksep∗ ksep ksep
(cid:16) (cid:17)
then show that c maps to zero in H2 ksep(A)∗ , so every P ∈ A(k) pairs with c to give 0
λ ksep∗ λ
in Q/Z. Hence, by Tate local duality, c = 0. (cid:3)
λ
Corollary 2. If k is a global field and λ ∈ H0(NS(A )), then c ∈ X(A∨)[2].
ksep λ
Proposition 3. If X is a principal homogeneous space of A representing c ∈ H1(A), and
if λ = φ for some L ∈ PicX, then c is the image of c under the map H1(A) → H1(A∨)
L λ
induced by λ.
Proof. Pick P ∈ X(ksep), and pick D ∈ Div(X) representing L. Then D ∈ Div(A ) and
−P ksep
λ = φ , as we see by using t to identify Pic0(A ) with Pic0(X ). By definition c is
D−P P ksep ksep λ
represented by ξ ∈ Z1(A∨) where
ξ := the class of σ(D )−D
σ −P −P
= the class of D −D
−σP −P
= the class of (D ) −D ,
−P −(σP−P) −P
which by definition represents the image of c under φ = λ. (cid:3)
D−P
Corollary 4. If (J,λ) is the canonically polarized Jacobian of a curve X, then the element
c is represented by the principal homogeneous space Picg−1 ∈ H1(J).
λ X/k
Proof. The polarization comes from the theta divisor Θ, which is canonically a k-rational
divisor on the homogeneous space Picg−1. (cid:3)
X/k
Combining Corollary 4 with Lemma 1 shows that if X is a curve of genus g over a local
field k , then X has a k -rational divisor class of degree g − 1, a fact originally due to
v v
Lichtenbaum [Li].
9The archimedean case is related to older results of Witt [Wi]. See p. 221 of [Sc].
10 BJORN POONEN AND MICHAEL STOLL
Question. Are all polarizations on an abelian variety A of the form φ for some L ∈ PicX,
L
for some principal homogeneous space X of A?
The answer to the question is yes for the canonical polarization on a Jacobian (as men-
tioned above) or a Prym. (For Pryms, the result is contained in Section 6 of [Mu2], which
describes a divisor Ξ on a principal homogeneous space P+ such that Ξ gives rise to the
˜
polarization.) One can deduce from Lemma 1 that if π : C → C is an unramified double
˜
cover of curves of genus 2g−1 and g, respectively, with C, C, and π all defined over a local
field k of characteristic not 2, then there is a k-rational divisor class D of degree 2g −2 on
˜
C such that π D is the canonical class on C.
∗
5. The obstruction to being alternating
Inthissection, weshowthatif(A,λ)isaprincipallypolarizedabelianvarietyoveraglobal
field k, then c (or rather its class in X(A∨) ) measures the obstruction to (cid:104) , (cid:105) being
λ nd λ
alternating. More precisely and more generally, we have the following “pairing theorem”:
Theorem5. SupposethatAisanabelianvarietyoveraglobalfieldk andλ ∈ H0(NS(A )).
ksep
Then for all a ∈ X(A),
(cid:104)a,λa+c (cid:105) = (cid:104)a,λa−c (cid:105) = 0.
λ λ
Proof. We will use the homogeneous space definition of (cid:104) , (cid:105). Write λ = φ for some
D
D ∈ Div(A ). Let X be the homogeneous space of A corresponding to a.
ksep
Now fix P ∈ X(ksep). For any σ ∈ G , λ = σλ = φ . Thus λa is represented by
k (σD)
ξ ∈ Z1(A∨), where
ξ := the class of (σD) −(σD) ∈ Pic0(A ).
σ −(σP−P) ksep
Under t , ξ corresponds to
(σP) σ
ξ(cid:48) := the class of (σD) −(σD) ∈ Pic0(X ).
σ P (σP) ksep
By definition, c is represented by γ ∈ Z1(A∨), where
λ
γ := the class of (σD)−D ∈ Pic0(A ).
σ ksep
Under t , γ corresponds to
P σ
γ(cid:48) := the class of (σD) −D ∈ Pic0(X ).
σ P P ksep
(Recall that the identification Pic0(A ) ∼= Pic0(X ) is independent of the trivializa-
ksep ksep
tion chosen.) Thus λa − c is represented by an element of Z1(A∨) corresponding to
λ
α(cid:48) ∈ Z1(Pic0(X )), where
ksep
α(cid:48) := ξ(cid:48) −γ(cid:48) = the class of D −σ(D ) ∈ Pic0(X ).
σ σ σ P P ksep
But α(cid:48) visibly lifts to an element of Z1(Div0(X )) (that even becomes a coboundary when
ksep
injected into Z1(Div(X ))), so the element b(cid:48) in the definition of Section 3.1 is zero. Hence
ksep
(cid:104)a,λa−c (cid:105) = 0. The other equality follows since 2c = 0. (cid:3)
λ λ