The Cassels-Tate pairing on polarized abelian varieties

41 Pages·2001·0.33 MB·English

by  Sarah Jaffe

AnnalsofMathematics,150(1999),1109{1149
The Cassels-Tate pairing
on polarized abelian varieties
By Bjorn Poonen and Michael Stoll*
Abstract
Let (A;‚) be a principally polarized abelian variety deflned over a global
fleld k, and let qq(A) be its Shafarevich-Tate group. Let qq(A) denote
nd
the quotient of qq(A) by its maximal divisible subgroup. Cassels and Tate
constructed a nondegenerate pairing
qq(A) £qq(A) ! Q=Z:
nd nd
If A is an elliptic curve, then by a result of Cassels the pairing is alternating.
But in general it is only antisymmetric.
Using some new but equivalent deflnitions of the pairing, we derive gen-
eral criteria deciding whether it is alternating and whether there exists some
alternating nondegenerate pairing on qq(A) . These criteria are expressed in
nd
terms of an element c 2 qq(A) that is canonically associated to the polar-
nd
ization ‚. In the case that A is the Jacobian of some curve, a down-to-earth
version of the result allows us to determine efiectively whether #qq(A) (if
flnite) is a square or twice a square. We then apply this to prove that a posi-
tive proportion (in some precise sense) of all hyperelliptic curves of even genus
g ‚ 2 over Q have a Jacobian with nonsquare #qq (if flnite). For example, it
appears that this density is about 13% for curves of genus 2. The proof makes
use of a general result relating global and local densities; this result can be
applied in other situations.
Contents
1. Introduction
2. Notation
3. Two deflnitions of the Cassels-Tate pairing
3.1. The homogeneous space deflnition
3.2. The Albanese-Picard deflnition
⁄MuchofthisresearchwasdonewhiletheflrstauthorwassupportedbyanNSFMathematical
Sciences Postdoctoral Research Fellowship at Princeton University. This work forms part of the
secondauthor’sHabilitationinDu˜sseldorf.
1991Mathematics Subject Classiflcation. 11G10,11G30,14H25,14H40.
1110 BJORN POONEN AND MICHAEL STOLL
4. The homogeneous space associated to a polarization
5. The obstruction to being alternating
6. Consequences of the pairing theorem
7. A formula for Albanese and Picard varieties
8. The criterion for oddness of Jacobians
9. The density of odd hyperelliptic Jacobians over Q
9.1. The archimedean density
9.2. The nonarchimedean densities
9.3. The passage from local to global
9.4. The global density
10. Examples of Shafarevich-Tate groups of Jacobians
10.1. Jacobians of Shimura curves
10.2. Explicit examples
11. An open question of Tate about Brauer groups
12. Appendix: Other deflnitions of the Cassels-Tate pairing
12.1. The Albanese-Albanese deflnition
12.2. The Weil pairing deflnition
12.3. Compatability
Acknowledgements
References
1. Introduction
The study of the Shafarevich-Tate group qq(A) of an abelian variety A
over a global fleld k 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 flrst papers
devoted to the study of qq, proved that in the case where E is an elliptic curve
over a number fleld, there exists a pairing
qq(E)£qq(E) ¡! Q=Z
that becomes nondegenerate after one divides qq(E) by its maximal divisible
subgroup. He proved also that this pairing is alternating; i.e., that hx;xi = 0
for all x. If, as is conjectured, qq(E) is always flnite, 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
qq(A)£qq(A_) ¡! Q=Z;
that is nondegenerate after division by maximal divisible subgroups. He also
proved that if qq(A) is mapped to qq(A_) via a polarization arising from a
k-rational divisor on A then the induced pairing on qq(A) is alternating. But
CASSELS-TATE PAIRING 1111
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 general,1 Flach [Fl] proved that the pairing is
antisymmetric, by which we mean hx;yi = ¡hy;xi for all x;y, which is slightly
weaker than the alternating condition.
It seems to have been largely forgotten that the alternating property was
neverprovedingeneral: inafewplacesintheliterature, onecanflndtheclaim
thatthepairingisalwaysalternatingforJacobiansofcurvesovernumberflelds,
for example. In Section 10 we will give explicit examples to show that this is
not true, and that #qq(J) need not be a perfect square even if J is a Jacobian
of a curve over Q.2
One may ask what properties beyond antisymmetry the pairing has in
the general case of a principally polarized abelian variety (A;‚) over a global
fleld k. For simplicity, let us assume here that qq(A) is flnite, so that the
pairing is nondegenerate. Flach’s result implies that x 7! hx;xi is a homomor-
phism qq(A) ! Q=Z, so by nondegeneracy there exists c 2 qq(A) such that
hx;xi = hx;ci. Since Flach’s result implies 2hx;xi = 0, we also have 2c = 0
by nondegeneracy. It is then natural to ask, what is this element c 2 qq(A)[2]
that we have canonically associated to (A;‚)? An intrinsic deflnition of c is
given in Section 4, and it will be shown3 that c vanishes (i.e., the pairing is
alternating)ifandonlyifthepolarizationarisesfromak-rationaldivisoronA.
This shows that Tate’s and Flach’s results are each best possible in a certain
sense.
Ourpaperbeginswithasummaryofmostofthenotationandterminology
thatwillbeneeded, andwithtwodeflnitionsofthepairing. (Wegivetwomore
deflnitions and prove the compatibility of all four in an appendix.) Sections 4
and5givetheintrinsicdeflnitionofc,andshowthatithasthedesiredproperty.
(Actually, we work a little more generally: ‚ is not assumed to be principal,
and in fact it may be a difierence of polarizations.) Section 6 develops some
consequences of the existence of c; for instance if A is principally polarized and
qq(A) is flnite, then its order is a square or twice a square according as hc;ci
equals 0 or 1 in Q=Z. We call A even in the flrst case and odd in the second
2
case.
1ActuallyFlachconsidersthisquestioninamuchmoregeneralsetting.
2This is perhaps especially surprising in light of Urabe’s recent results [Ur], which imply, for
instance,fortheanalogoussituationofapropersmoothgeometricallyintegralsurfaceX overaflnite
fleld k ofcharacteristic p, thatiftheprime-to-ppartofBr(X)isflnite, theorderofthisprime-to-p
part is a square. (There exist \examples" of nonsquare Brauer groups in the literature, but Urabe
explainswhytheyareincorrect.) SeeSection11formorecommentsontheBrauergroup.
3The statement of this result needs to be modifled slightly if the flniteness of qq(A) is not
assumed.
1112 BJORN POONEN AND MICHAEL STOLL
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: hc;ci = N=2
2 Q=Z where N is the number of places v of k for which X has no k -rational
v
divisor of degree g¡1. Section 9 applies this criterion to hyperelliptic curves
of even genus g over Q, and shows that a positive proportion ‰ of these (in
g
a sense to be made precise) have odd Jacobian. It also gives an exact formula
for ‰ in terms of certain local densities, and determines the behavior of ‰ as
g g
g goes to inflnity. 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 ‰ of curves of genus 2 over Q with odd Jacobian.
2
Section10appliesthecriteriontoprovethatJacobiansofcertainShimura
curvesarealwayseven. Itgivesalsoafewotherexamples,includinganexplicit
genus 2 curve over Q for whose Jacobian we can prove unconditionally that
»
hc;ci = 1 and qq = Z=2Z, and another for which qq is flnite of square order,
2
but with h ; i not alternating on it.
Finally, Section 11 addresses the analogous questions for Brauer groups of
surfaces over flnite flelds, recasting an old question of Tate in new terms.
2. Notation
Many of the deflnitions in this section are standard. The reader is encour-
aged 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] =
S L
fm 2 M : nm = 0g. Let M = 1 M[n] = M(p), where for each
S tors n=1 p
prime p, M(p) = 1 M[pn] denotes the p-primary part of the torsion sub-
n=1
group of M. Let M be the maximal divisible subgroup of M; i.e., m is in
div
M if and only if for all n ‚ 1 there exists x 2 M such that nx = m. De-
div
note by M the quotient M=M . (The subscript nd stands for \nondivisible
nd div
part.") If
h ; i : M £M0 ¡! Q=Z
is a bilinear pairing between two abelian groups, then for any m 2 M, let
m? = fm0 2 M0 : hm;m0i = 0g, and for any subgroup V (cid:181) M, let V? =
T
v?. When M = M0, we say that h ; i is antisymmetric if ha;bi = ¡hb;ai
v2V
for all a;b 2 M, and alternating if ha;ai = 0 for all a 2 M. Note that a
bilinear pairing h ; i on M is antisymmetric if and only if m 7! hm;mi is a
homomorphism. If a pairing is alternating, then it is antisymmetric, but the
converse is guaranteed on M(p) only for odd p.
CASSELS-TATE PAIRING 1113
If k is a fleld, then k and ksep denote algebraic and separable closures,
and G = Gal(k=k) = Gal(ksep=k) denotes its absolute Galois group. If k is a
k
global fleld, then M denotes the set of places of k. If moreover v 2 M , then
k k
k denotes the completion, and G = Gal(ksep=k ) denotes the absolute Galois
v v v v
group of k .
v
Suppose that G is a proflnite 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-cocyles and i-cohomology classes) with
valuesintheG-moduleM. Ifkisunderstood,weuseCi(M)asanabbreviation
for Ci(G ;M), and similarly for Zi(M) and Hi(M). If fi 2 Ci(G ;M), then
k k
fi 2 Ci(G ;M)denotesitslocalrestriction.4 Ifv isaplaceofaglobalfleld,we
v v
use inv to denote the usual monomorphism H2(G ;ksep⁄) = Br(k ) ! Q=Z
v v v v
(which is an isomorphism if v is nonarchimedean).
Varieties will be assumed to be geometrically integral, smooth, and pro-
jective, unless otherwise specifled. If X is a variety over k, let k(X) de-
note the function fleld of X. If K is a fleld extension of k, then X de-
K
notes X £ K, the same variety with the base extended to SpecK. Let
k
Div(X) = H0(G ;Div(X )) denote the group of (k-rational) Weil divisors
k ksep
on X. If f 2 k(X)⁄ or f 2 k(X)⁄=k⁄, let (f) 2 Div(X) denote the associ-
ated principal divisor. If D 2 Div(X £Y), let tD 2 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 ) of invertible sheaves
ksep ksep
algebraically equivalent to 0, and let Pic0(X) = Pic(X)\Pic0(X ). In older
ksep
terminology, which we will flnd convenient to use on occasion, H0(Pic(X ))
ksep
is the group of k-rational divisor classes, and Pic(X) is the subgroup of divisor
classes that are actually represented by k-rational divisors. Deflne the N¶eron-
Severi group NS(X) = Pic(X)=Pic0(X). If D 2 Div(X), let L(D) 2 Pic(X)
be the associated invertible sheaf. Let Div0(X) be the subgroup of Div(X)
mapping into Pic0(X). If ‚ 2 H0(NS(V )), let Pic‚ denote the component
ksep V=k
of the Picard scheme Pic corresponding to ‚. If k has characteristic p > 0,
V=k
anddimV ‚ 2, thentheseschemesneednotbereduced[Mu1, Lect.27], butin
any case the associated reduced scheme Pic‚ is a principal homogeneous
V=k;red
space of the Picard variety Pic0 , which is an abelian variety over k. Also,
V=k;red
Pic‚ (ksep) equals the preimage of ‚ under Pic(V ) ! NS(V ) with its
V=k;red ksep ksep
G -action.
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 2 Z, let Albi denote the degree i
X=k
4Wewillbeslightlysloppyinwritingthis,becauseweoftenwillintendthe\M"inCi(Gk;M)
tobeapropersubgroupofthe\M"inCi(Gv;M);forinstancethesetwoM’smaybetheksep-points
andkvsep-pointsofagroupschemeAoverk,inwhichcaseweabbreviatebyCi(Gk;A)andCi(Gv;A)
eventhoughthetwoA’srepresentdifierentgroupsofpoints.
1114 BJORN POONEN AND MICHAEL STOLL
componentoftheAlbanesescheme. LetY0(X)denotethekernelofthenatural
map Z0(X) ! Alb0 (k). Then Albi is a principal homogeneous space of
X=k X=k
the Albanese variety Alb0 and its k-points correspond (G -equivariantly) to
X=k k
elements of Zi(X ) modulo the action of Y0(X ).
k k
If X is a curve (geometrically integral, smooth, projective, as usual) and
i 2 Z, let Pici(X) denote the set of elements of Pic(X) of degree i, and let
Pici be the degree i component of the Picard scheme, which is a principal
X=k
homogeneous space of the Jacobian Pic0 of X. (Since dimX = 1, these
X=k
are already reduced.) Points on Pici over k correspond to divisor classes
X=k
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
X=k
divisor classes are not always represented by k-rational divisors.)
If A is an abelian variety, then A_ = Pic0 = Pic0 denotes the dual
A=k;red A=k
abelian variety. If X is a principal homogeneous space of A, then for each
a 2 A(k), we let t denote the translation-by-a map on X, and similarly if
a
x 2 X(k), then t is the trivialization A ! X mapping 0 to x. If D 2 DivA,
x
let Dx = txD 2 DivX. (Note: if a 2 A(k), then t⁄aD = D¡a.) If L 2 PicX,
then `L denotes the homomorphism A ! A_ mapping a to t⁄aL›L¡1. (There
is a natural identiflcation Pic0 = A_.) We may also identify `L with the
X=k
class of L in NS(X). If D 2 DivX, then we deflne `D = `L(D). A polarization
on A (deflned over k) is a homomorphism A ! A_ (deflned over k) which over
ksep equals `L for some ample L 2 Pic(Xksep). (One can show that this gives
the same concept as the usual deflnition, in which k is used instead of ksep.)
A principal polarization is a polarization that is an isomorphism.
If A is an abelian variety over a global fleld k, then let qq(A) = qq(k;A)
be the Shafarevich-Tate group of A over k, whose elements we identify with
locally trivial principal homogeneous spaces of A (up to equivalence).
Suppose that V and W are varieties over a fleld k, and that D is a divisor
onV£W. Ifv 2 V(k),letD(v) 2 Div(W )bethepullbackofDunderthemap
k
W ! V £W sendingw to(v;w), whenthismakessense. Fora 2 Z(V ), deflne
k
D(a) 2 Div(W ) by extending linearly, when this makes sense. If a 2 Y0(V )
k k
and D(a) is deflned, then D(a) = (f) for some function f on W. (See the
proof of Theorem 10 on p. 171 of [La1, VI, x4].) If in addition a0 2 Z0(W ),
k
0 0 0 0
and if f(a) is deflned, we put D(a;a) = f(a) and say that D(a;a) is deflned.
If a 2 Y0(V ) and a0 2 Y0(W ), then we may interchange V and W to try to
k k
deflne tD(a0;a), and Lang’s reciprocity law (p. 171 of [La1] again) states that
D(a;a0) and tD(a0;a) are deflned and equal, provided that a£a0 and D have
disjoint supports.
We let „1 denote the standard Lebesgue measure on Rd, and let „p
denote the Haar measure on Zd normalized to have total mass 1. For v =
p
(v ;v ;:::;v ) 2 Zd, deflne jvj := max jv j. If S (cid:181) Zd, then the density of S is
1 2 d i i
CASSELS-TATE PAIRING 1115
deflned to be X
‰(S) := lim (2N)¡d 1;
N!1
v2S;jvj•N
if the limit exists. Deflne the upper density ‰(S) and lower density ‰(S) simi-
larly, except with the limit replaced by a limsup or liminf, respectively.
WewillusethenotationAd andPd ford-dimensionala–neandprojective
space, respectively.
3. Two deflnitions of the Cassels-Tate pairing
In this section, we present the two deflnitions of the Cassels-Tate pairing
used here. The flrst is well-known [Mi4]. The second appears to be new, but
it was partly inspired by Remark 6.12 on page 100 of [Mi4]. In an appendix
we will give two other deflnitions, and show that all four are compatible.
3.1. The homogeneous space deflnition. Let A be an abelian variety over
a global fleld k. Suppose a 2 qq(A) and a0 2 qq(A_). Let X be the (lo-
cally trivial) homogeneous space over k representing a. Then Pic0(X ) is
ksep
canonically isomorphic as G -module to Pic0(A ) = A_(ksep), so that a0
k ksep
corresponds to an element of H1(Pic0(X )), which we may map to an ele-
ksep
ment b0 2 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 b0 to an element f0 2 H2(G ;ksep(X)⁄).
k
Then it turns out that f0 2 H2(G ;ksep(X)⁄) is the image of an element
v v v
c 2 H2(G ;ksep⁄).5 We deflne
v v v
X
ha;a0i = inv (c ) 2 Q=Z:
v v
v2Mk
See Remark 6.11 of [Mi4] for more details. The obvious advantage of this
deflnition over the others is its simplicity.
If ‚ : A ! A_ is a homomorphism, then we deflne a pairing
h ; i : qq(A)£qq(A) ¡! Q=Z
‚
by ha;bi = ha;‚bi.
‚
3.2. The Albanese-Picard deflnition. Let V be a variety (geometrically
integral, smooth, projective, as usual) over a global fleld k. Our goal is to
5Onecancomputecv byevaluatingfv0 atapointinX(kv),ormoregenerallyatanelementof
Z1(Xkv)(providedthatoneavoidsthezerosandpolesoffv0).
1116 BJORN POONEN AND MICHAEL STOLL
deflne a pairing
(1) h ; i : qq(Alb0 )£qq(Pic0 ) ¡! Q=Z:
V V=k V=k;red
We will flrst need a partially-deflned G -equivariant pairing
k
(2) [ ; ] : Y0(V )£Div0(V ) ¡! ksep⁄:
ksep ksep
TemporarilyweworkinsteadwithavarietyV overaseparablyclosedfleld
K. Let A = Alb0 and A0 = Pic0 . Let P denote a Poincar¶e divisor
V=K V=K;red
on A£A0. Choose a basepoint P 2 V(K) to deflne a map ` : V ! A. Let
0
P = (`;1)⁄P 2 Div(V £A0). Suppose y 2 Y0(V) and D0 2 Div0(V). Choose
0
z0 2 Z0(A0) that sums to L(D0) 2 Pic0(V) = A0(K). Then D0¡tP (z0) is the
0
0
divisor of some function f on V. Deflne
0 0 0
[y;D ] := f (y)+P (y;z )
0
if the terms on the right make sense.
We now show that this pairing is independent of choices made. If we
change z0, we can change it only by an element y0 2 Y0(A0), and then [y;D0]
changes by ¡tP (y0;y) + P (y;y0) = 0, by Lang reciprocity. If we change
0 0
P by the divisor of a function F on A £ A0, then [y;D0] changes by
F(`(y)£z0)¡F(`(y)£z0) = 0. If E 2 Div(A0), and if we change P by …⁄E
2
(… being the projection A£A0 ! A0), then [y;D0] is again unchanged. By
2
the seesaw principle [Mi3, Th. 5.1], the translate of P by (a;0) 2 (A£A0)(K)
0
⁄
difiers from P by a divisor of the form (F)+… E; therefore the deflnition of
0 2
0
[y;D ] is independent of the choice of P . It then follows that if V is a variety
0
over any fleld k, then we obtain a G -equivariant pairing (2).
k
Remark. If V is a curve, then an element of Y0(V ) is the divisor of a
ksep
function,andthepairing(2)issimplyevaluationofthefunctionattheelement
of Div0(V ).
ksep
We now return to the deflnition of (1). It will be built from the two exact
sequences
(3) 0 ¡! Y0(V ) ¡! Z0(V ) ¡! A(ksep) ¡! 0;
ksep ksep
ksep(V)⁄
0 ¡! ¡! Div0(V ) ¡! A0(ksep) ¡! 0;
ksep⁄ ksep
and the two partially-deflned pairings
(4) [ ; ] : Y0(V )£Div0(V ) ¡! ksep⁄;
ksep ksep
ksep(V)⁄
Z0(V )£ ¡! ksep⁄;
ksep ksep⁄
the latter deflned by lifting the second argument to a function on V , and
ksep
ksep(V)⁄
\evaluating"ontheflrstargument. Wemayconsider tobeasubgroup
ksep⁄
CASSELS-TATE PAIRING 1117
of Div0(V ), and then the two pairings agree on Y0(V )£ksep(V)⁄, so there
ksep ksep ksep⁄
will be no ambiguity if we let [ denote the cup-product pairing on cochains
associated to these. We have also the analogous sequences and pairings for
each completion of k.
Suppose a 2 qq(A) and a0 2 qq(A0). Choose fi 2 Z1(A(ksep)) and fi0 2
Z1(A0(ksep)) representing a and a0, and lift these to a 2 C1(Z0(V )) and
ksep
a0 2 C1(Div0(V )) so that for all (cid:190);¿ 2 G , all G -conjugates of a have
ksep k k (cid:190)
0
support disjoint from the support of a . Deflne
¿
· := da[a0¡a[da0 2 C3(ksep⁄):
We have d· = 0, but H3(ksep⁄) = 0, so · = d† for some † 2 C2(ksep⁄).
Sinceaislocallytrivial,wemayforeachplacev 2 M choosefl 2 A(ksep)
k v v
(cid:190)su2chGth,atalfil vG=-cdoflnvj.ugCahteososoefbbv 2haZve0(sVukpvseppo)rmt adpispjoiningttforoflmv athnedssuoptphoarttfoofraa0l.l
k v v (cid:190)
Then6
(cid:176) := (a ¡db )[a0 ¡b [da0 ¡† 2 C2(G ;ksep⁄)
v v v v v v v v v
is a 2-cocycle representing some
c 2 H2(G ;ksep⁄) = Br(k ) ¡in!vv Q=Z:
v v v v
Deflne X
ha;a0i = inv (c ):
V v v
v2Mk
One checks using d(x[y) = dx[y+(¡1)degxx[dy [AW, p. 106] that this is
well-deflnedandindependentofchoices. Inprovingindependenceofthechoice
0
of fl one uses the local triviality of a, which has not been used so far.
v
This deflnition 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 deflnition involves only divisors on the curve, instead of m2-torsion
or homogeneous spaces of the Jacobian, which are more di–cult to deal with
computationally.
Remark. Thereadermayhaverecognizedthesetupoftwoexactsequences
with two pairings as being the same as that required for the deflnition of the
augmented cup-product (see [Mi4, p. 10]). Here we explain the connection.
Suppose that we have exact sequences of G -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
6Sinceitisonlythedifierenceofthetermsav anddbv thatisinY0(Vksep),itwouldmakeno
sensetoreplace(av¡dbv)[a0v byav[a0v¡dbv[a0v.
1118 BJORN POONEN AND MICHAEL STOLL
andaG -equivariantbilinearpairingM £N ! P thatmapsM £N intoP .
k 2 2 2 £ 1 1 1⁄
The pairing induces a pairing M £N ! P . If a 2 ker Hi(M ) ! Hi(M )
£ 3⁄ 1 3 1 2
and a0 2 ker Hj(N ) ! Hj(N ) , then a comes from some b 2 Hi¡1(M ),
1 2 3
which can be paired with a0 to give an element of Hi+j¡1(P ). If one changes
3
b by an element c 2 Hi¡1(M ), then the result is unchanged, since the change
2
can be obtained by pairing c with the (zero) image of a0 in Hj(N ) under
2
the cup-product pairing associated with M £ N ! P . Thus we have a
2 2 2
well-deflned pairing7
h i h i
(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,
i i
replace each P by a complex with a single term, in degree 1, and replace
i
cohomology by hypercohomology, then we obtain a pairing analogous to (5)
deflned using the augmented cup-product. One obtains the deflnition of the
Cassels-Tate pairing above by noting that:
1. If A is the ad(cid:181)ele ring of k,
k
h i
qq(A) = ker H1(G ;A(ksep)) ! H1(G ;A(ksep› A ))
k k k k
by Shapiro’s lemma;
2. The analogous statement holds for qq(A0); and
3. IfP = ksep⁄ andP = (ksep› A )⁄,thenthecokernelP hasH2(G ;P )
1 2 k k 3 k 3
= Q=Z by class fleld theory.
4. The homogeneous space associated to a polarization
Let A be an abelian variety over a fleld k. To each element of
H0(NS(A )) we can associate a homogeneous space of A_ that measures
ksep
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 ))
ksep
¡! H1(A_(ksep)) ¡! H1(Pic(A )):
ksep
(We have H0(Pic(A )) = PicA because A(k) 6= ;.) For ‚ 2 H0(NS(A )),
ksep ksep
deflne c to be the image of ‚ in H1(A_(ksep)). By the proof of Theorem 2
‚
7The\diminishedcup-product"?!