Galois Theory for the Selmer Group of an Abelian Variety

43 Pages·2003·0.51 MB·English

by  Ralph Greenberg

Compositio Mathematica136: 255–297,2003. 255
#2003KluwerAcademic Publishers. Printed inthe Netherlands.
Galois Theory for the Selmer Group
?
of an Abelian Variety
RALPH GREENBERG
Departmentof Mathematics, University ofWashington, Seattle, WA 98195-4350,U.S.A.
e-mail: [email protected]
(Received: 5 January2001; accepted in finalform: 21December2001)
Abstract. This paper concerns the Galois theoretic behavior of the p-primary subgroup
Sel ðFÞ of the Selmer group for an Abelian variety A defined over a number field F in an
A p
extensionK=FsuchthattheGaloisgroupGðK=FÞisap-adicLiegroup.Herepisanyprime
suchthatAhaspotentiallygood,ordinaryreductionatallprimesofFlyingabovep.Theprin-
cipal results concern the kernel and the cokernel of the natural map sK=F0: SelAðF0Þp!
Sel ðKÞGðK=F0Þ whereF0 isanyfiniteextensionofFcontainedinK.Undervarioushypotheses
A p
ontheextensionK=F,itisprovedthatthekernelandcokernelarefinite.Morepreciseresults
about their structure are also obtained. The results are generalizations of theorems of
B. Mazur andM.Harris.
Mathematics SubjectClassifications (2000).11G05, 11G10, 11R23, 11R34.
Key words. Abelian variety,Galois theory, Selmergroup.
1. Introduction
Let A be an Abelian variety defined over a number field F. Let K denote the cyclo-
tomicZ -extensionofF,wherepisanyprime.ThustheGaloisgroupGðK=FÞisiso-
p
morphictoZ ,theadditivegroupofp-adicintegers.ForanyalgebraicextensionF0
p
of F, we let Sel ðF0Þ denote the p-primary subgroup of the Selmer group Sel ðF0Þ
A p A
for A over F0. The purpose of this article is to consider some generalizations of
the following classical theorem of Mazur.
THEOREM. Assume that A has good, ordinary reduction at all primes of F lying
over p. Let F0 be a finite extension of F contained in K. Then the natural map
Sel ðF0Þ !Sel ðKÞGðK=F0Þ has finite kernel and cokernel. The orders of the kernels
A p A p
and cokernels are bounded as F0 varies.
This is Mazur’s ‘Control Theorem’, which he proves for any Z -extension K=F
p
satisfying certain mild conditions. Actually the theorem is true for the full Selmer
groupsince one canshoweasilythat,foranyprimeq6¼p,theq-primary subgroups
?
SupportedpartiallybyaNationalScienceFoundationgrant.
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256 RALPHGREENBERG
of Selmer groups behave very well Galois theoretically. That is, the maps
Sel ðF0Þ ! Sel ðKÞGðK=F0Þ are isomorphisms.
A q A q
We will consider Galois extensions K=F such that GðK=FÞ is a p-adic Lie group.
For any field F0 such that F(cid:7)F0 (cid:7)K, we let s denote the natural restriction
K=F0
homomorphism
s : Sel ðF0Þ !Sel ðKÞGðK=F0Þ:
K=F0 A p A p
For our main theorems we will need to assume that K=F is ‘S-ramified’ for some
finite set S of primes of F. That is, every prime v of F not in S is unramified in
K=F. We will make this assumption throughout the article. Let p be a prime of F
lying over p. Let D and I denote the decomposition and inertia subgroups of
p p
GðK=FÞ for some prime of K lying over p. Both D and I are closed subgroups
p p
ofGðK=FÞandhencearealsop-adicLiegroups.Letd andi denotetheirLiealge-
p p
bras. They are subalgebras of the Lie algebra g of GðK=FÞ.
DEFINITION. We say that K=F is admissible if, in addition to being a S-ramified
p-adic Lie extension for some S, we have d0 ¼i0 for all p lying over p.
p p
Here,foranyLiealgebral,weletl0denotethederivedLiesubalgebraofl.(Thatis,l0
istheQ -subspacespannedby½x;y(cid:9),x;y2l,whichisanidealofl.)Nowi0 isactu-
p p
ally an ideal of d . The equality d0 ¼i0 is equivalent to saying that the Lie algebra
p p p
d =i0 is Abelian. In attempting to prove the finiteness of the cokernel of s , this
p p K=F0
condition arises quite naturally as a hypothesis. Examples where it is satisfied are
rather abundant. Any Z -extension K=F is admissible. More generally, if the Lie
p
algebra g is Abelian (i.e., if GðK=FÞ contains a subgroup of finite index isomorphic
toZd forsomed),thenK=Fisadmissible.Theconditiond0 ¼i0 forpjpisobviously
p p p
satisfiedsinced isAbelianandsod0 ¼0.Anotherclassofexamplesarethosewhere
p p
the inertia subgroup I has finite index in GðK=FÞ for all pjp. Then d ¼i ¼g and
p p p
soagain d0 ¼i0 obviouslyholds. Animportant classofexamples,which we discuss
p p
below,arethosewhereGðK=FÞadmitsafaithful,finite-dimensionalp-adicrepresen-
tation which is of Hodge–Tate type at the primes p of F above p.
In our first theorem, we assume that the p-primary subgroup AðKÞ of AðKÞ is
p
finite. In this and other theorems, the hypothesis that A has potentially ordinary
reduction at the primes of F lying above p means that A achieves good, ordinary
reduction at those primes over some finite extension of F.
THEOREM 1. Assume that A has potentially ordinary reduction at all primes of F
lying over p. Assume that K=F is admissible and that AðKÞ is finite. Then, for every
p
finite extension F0 of F contained in K, the kernel and cokernel of s are finite.
K=F0
Note that the hypotheses in the above theorem are preserved when K=F is replaced
byK=F0 foranyfiniteextensionF0 ofFcontainedinK.Thus,itwouldbeenoughto
provejustthatkerðs Þandcokerðs Þarebothfinite.Thesameremarkappliesto
K=F K=F
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GALOISTHEORYFORTHESELMERGROUPOFANABELIANVARIETY 257
Theorems2and3below.Nevertheless,wewillusuallypresenttheproofsallowingF0
to vary in order to see how the order and structure of the kernel and cokernel
behave. Under the hypotheses of Theorem 1, the kernel of s is relatively easy
K=F0
to bound. In fact, it would suffice to just assume that K=F is a p-adic Lie extension
and that AðKÞ isfinite toconclude that kerðs Þisofbounded order asF0 varies.
p K=F0
ForthisoneneedsnohypothesisonthereductionofA.(SeeProposition3.1.)How-
ever, as will become clear later, one cannot expect cokerðs Þ to have bounded
K=F0
order as F0 varies unless one imposes rather stringent hypotheses.
ItisnotdifficulttoprovethatiftheLiealgebragofGðK=FÞissemisimple(i.e.,a
directproductofsimple,non-AbelianLiealgebras),thenAðKÞ isnecessarilyfinite.
p
(SeeProposition3.2forthisandsomeothersufficientconditionsforthefinitenessof
AðKÞ .)Itseemsthatforthep-adicLieextensionsK=Fwhichariseinvariousnatural
p
ways in number theory, the corresponding Lie algebra g is often reductive (i.e., a
directproductofasemisimpleLiealgebraandanabelianone).Inthiscase,itiscer-
tainlypossibleforAðKÞ tobeinfinite.Nevertheless,onecanuseTheorem1toprove
p
the following result.
THEOREM 2. Assume that A has potentially ordinary reduction at all primes of F
lyingoverp.AssumethatK=Fisadmissibleandthatgisreductive.Thenkerðs Þand
K=F0
cokerðs Þ are finite for every finite extension F0 of F contained in K.
K=F0
Supposethatr: G !GL ðQ Þisacontinuous,finite-dimensionalQ -representa-
F n p p
tionoftheabsoluteGaloisgroupG ¼GðQQ(cid:2)=FÞ.IfweletK¼QQ(cid:2)kerðrÞ,thenrinduces
F
anisomorphismofGðK=FÞtothecompactsubgroupimðrÞofGL ðQ Þ.Suchasub-
n p
group must be a p-adic Lie group of dimension d4n2. We suppose also that r is
unramified outside a finite set S of primes of F and so K=F is S-ramified. If r is a
completely reducible representation of G , then the Lie algebra of imðrÞ (which is
F
isomorphic to g) must be reductive. For every prime p of F lying over p, we can
restrictrtoadecompositionsubgroupobtainingtherepresentationrj ofthelocal
GaloisgroupG ¼GðQQ(cid:2) =F Þ.Wewillprovelater(Proposition4.7)thGaFptifrj isa
Hodge–Tate repFrpesentatipon,pthen the equality d0 ¼i0 does hold. As a conseqGFupence
p p
we obtain the following result, which perhaps is the most interesting theorem of
this article.
THEOREM 3. Assume that r is completely reducible and that rj is Hodge–Tate
GFp
for all primes p of F lying above p. Assume that the Abelian variety A has potentially
ordinaryreductionattheprimesofFlyingoverp.Then,foreveryfiniteextensionF0 of
F contained in K, the kernel and cokernel of s are finite.
K=F0
ThehypothesisinTheorem3areoftenknowntobetrueforp-adicrepresentations
rthatarisenaturallyinnumbertheory.Forexample,supposethatBisanarbitrary
Abelian variety defined over F and let VpðBÞ¼TpðBÞ(cid:12)Zp Qp, where TpðBÞ denotes
the Tate module for B. Then the representation r: GF !AutQpðVpðBÞÞ giving the
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258 RALPHGREENBERG
natural action of G on V ðBÞ is unramified outside the set S of primes of F lying
F p
above p or where B has bad reduction. Faltings has proven that r is completely
reducible. It is also known that rj is Hodge–Tate for the primes p of F above p.
GFp
(AresultofTatewhenBhasgoodreductionatp,extendedbyRaynaudtothegen-
eral case.) In this example, K¼FðB½p1(cid:9)Þ.
Asanotherexample,letF¼Qandletrf bethep-adicrepresentationofGQ asso-
ciated to a cusp form of level N. Here p is a prime of the field E generated by the
coefficients in the q-expansion for f. The representation r is of dimension 2 over
f
thecompletionE .Thenr isknowntobeirreducible.(Seethm2.3in[R].)Thatsuf-
p f
fices to imply that the Lie algebra g is reductive. Faltings has proven that rj is
Hodge–Tate. This means that the Q -representation r of dimension 2½Ef G:QQp (cid:9)
p p p
defined by r is Hodge–Tate. Since kerðrÞ¼kerðrÞ, Theorem 3 can be applied to
f f
K=Q, where K¼QQ(cid:2)kerðrfÞ.
Faltings has also proven that the Q -representations giving the action of G on
p Fp
thep-adic e´talecohomology ofanonsingular, projectivealgebraicvarietyXdefined
overF isHodge–Tate.IfXisdefinedoverthenumberfieldF,thenitisalsoexpec-
p
ted that the corresponding Q -representations of G are completely reducible.
p F
Inthisarticle,wewillsingleoutthecaseK¼FðA½p1(cid:9)Þ.Asmentionedabove,Theo-
rem 3 applies to this case (since we can take B¼A in the discussion following that
theorem). Thus, as a corollary, we have
THEOREM 4. Assume that A is an Abelian variety defined over F which has poten-
tially ordinary reduction at all primes of F over p. Let K¼FðA½p1(cid:9)Þ. Then, for every
finite extension F0 of F contained in K, kerðs Þ and cokerðs Þ are finite.
K=F0 K=F0
This theorem is equivalent to a result proved by M. Harris. (See the ‘effectivity
theorem’ of [H]. The statement there seems rather different, but can be shown to
be equivalent to Theorem 4.) Although this theorem is a consequence of Theorem
3, it seems worthwhile to treat it separately and as directly as possible. In the case
where dimðAÞ¼1, we will prove that kerðs Þ is actually of bounded order, where
K=Fn
F ¼FðA½pn(cid:9)Þ.AsimilarresultmaypossiblybetrueforAbelianvarietiesofarbitrary
n
dimension.
Under various sets of assumptions about A and K=F, one can show that
cokerðs ÞisnontrivialforallF0orthatthisgroupgrowsinsomeway.Suchresults
K=F0
would obviously give information about the structure of Sel ðKÞGðK=F0Þ and, hence,
A p
ofSel ðKÞ itself.Herearetwosampletheoremsofthatkind.Inboththeorems,we
A p
assumethatK=Fisap-adicLieextensionwhichisS-ramifiedforsomefinitesetSof
primesofF.WeassumethattheAbelianvarietyAhasgood,ordinaryreductionata
primepofFlyingoverp,butmakenoassumptionaboutthereductionofAatother
primes of F. Let f denote the residue field for p and AA~ denote the reduction of A
p p
modulo p. The group of points AA~ ðf Þ is of course finite. Let At denote the dual
p p
Abelian variety.
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GALOISTHEORYFORTHESELMERGROUPOFANABELIANVARIETY 259
THEOREM 5. Assume that AA~ ðf Þ 6¼0. Assume also that p is infinitely ramified in
p p p
K=F and that AðKÞ ¼AtðKÞ ¼0. Then Sel ðKÞ is infinite.
p p A p
THEOREM 6. Assume that AA~ ðf Þ 6¼0. Assume also that there are infinitely many
p p p
primesofKlyingabovep,thattheresiduefieldk foranysuchprimeZisinfinite,and
Z
thatpisinfinitelyramifiedinK=F.ThenðSel ðKÞ Þ isisomorphictoaninfinite ðbut
A p div
countableÞ direct sum of copies of Q =Z .
p p
The hypothesis that AA~ ðf Þ 6¼0 plays an important role in [M]. Following
p p p
Mazur, one calls p an anomalous prime for A if A has good ordinary reduction
at p and AA~ ðf Þ 6¼0. For a given A=F, it seems likely that infinitely many such
p p p
primes should exist. The hypothesis that p is infinitely ramified in K=F simply
means that i 6¼0. Infinitely many primes lying above p exist if d 6¼g and the resi-
p p
due field for such primes is infinite if i 6¼d . It is also worth remarking that if
p p
GðK=FÞ is pro-p, then AðFÞ ¼AtðFÞ ¼0 easily implies that AðKÞ ¼AtðKÞ ¼ 0.
p p p p
In particular, if K=F is the cyclotomic Z -extension and if dimðAÞ¼1 (so that
p
A¼At), then just assuming that AðFÞ ¼0 and that some prime pjp is anomalous
p
for A would imply that Sel ðKÞ is infinite. (This is Proposition 8.5 in [M] when
A p
F¼Q. See also Proposition 5.3 in [G1].) If Sel ðKÞ is infinite, then either
A p
ðSel ðKÞ Þ 6¼0orSel ðKÞ½p(cid:9)isinfinite.Bothcasescanoccur.IfðSel ðKÞ Þ isinfi-
A p div A A p div
nite, then either AðKÞ(cid:12)Z ðQp=ZpÞ is infinite or ðSAðKÞpÞdiv is infinite. Again, both
cases canoccur.
To illustrate Theorem6,consideragainthecaseofan ellipticcurveA=F.Assume
that A does not have complex multiplication and that A has potentially ordinary
reduction at a prime p of F lying over p. Let K¼FðA½p1(cid:9)Þ. If one replaces F by
F0 ¼FðA½p(cid:9)Þ (or by F0 ¼FðA½4(cid:9)Þ if p¼2), then all the hypotheses in Theorem 6
are satisfied. A now has good, ordinary reduction at any prime p0 of F0 above p,
we have AA~p0ðfp0Þp 6¼0, and the Lie algebras g, dp0, and ip0 are distinct because they
have dimensions 4, 3, and 2, respectively. Hence ðSel ðKÞ Þ is an infinite direct
A p div
sum of copies of Q =Z . A proof of essentially the same result is given in [CH2].
p p
Thisarticleshouldberegardedasasequelto[CG].Inthatpaperonefindsarather
simple description of the local conditions occurring in the definition of the Selmer
group. This description makes it easy to study Galois theory for the Selmer group,
especially in the case where A has potentially ordinary reduction.
Ourproofswillbebasedonacertainexactsequencewhichwenowexplain.LetA
beanarbitraryAbelianvarietydefinedoverF.LetLbeanyalgebraicextensionofF.
ForanyprimevofF,letF denotethev-adiccompletionofF.IfZisanyprimeofL
v
lyingoverv,weletL denotetheunionoftheZ-adiccompletions ofallfinite exten-
Z
sionsofFcontainedinL(sothatL isanalgebraicextensionofF ).Wedenotebyk
Z v Z
the corresponding Kummer homomorphism
k : AðL Þ(cid:12)ðQ =Z Þ!H1ðL ;A½p1(cid:9)Þ:
Z Z p p Z
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260 RALPHGREENBERG
For each Z, we let H ðL Þ¼H1ðL ;A½p1(cid:9)Þ=imðk Þ. Then the p-Selmer group for A
A Z Z Z
over L is defined by
Sel ðLÞ ¼ker(cid:1)H1ðL;A½p1(cid:9)Þ!YH ðL Þ(cid:3);
A p A Z
Z
where the map is induced by restricting cocycles to decomposition groups. Here Z
runs over all primes of L, including the Archimedean primes (important only for
Q
p¼2). We will let P ðLÞ¼ H ðL Þ for brevity. Also we put G ðLÞ¼
A Z A Z A
im(cid:2)H1ðL;A½p1(cid:9)Þ!P ðLÞ(cid:3).
A
Now suppose that K=F is a Galois extension and F0 is an intermediate field. We
then obtain the following commutative diagram with exact rows.
0 !Sel ðF0Þ !H1ðF0;A½p1(cid:9)Þ !G ðF0Þ !0
A p A
s h g
K=F0 K=F0 K=F0
0(cid:15)!Sel ðKÞGðK=F0Þ(cid:15)!H1ðK;A½p1(cid:9)ÞGðK=F0Þ(cid:15)!G ðKÞGðK=F0Þ
A p A
The snake lemma then gives the exact sequence
0(cid:15)!kerðs Þ(cid:15)!kerðh Þ(cid:15)!kerðg Þ(cid:15)!cokerðs Þ(cid:15)!cokerðh Þ
K=F0 K=F0 K=F0 K=F0 K=F0
ð1Þ
As mentioned above, this exact sequence will be the basis of our proofs. In the lit-
eratureithasoftenbeenusedinasimilarway,especiallyinthecaseofZ -extensions.
p
(See [CM], [P] for example.) We also use certain basic results about compact p-adic
Lie groups, recalled in Section 2. In the subsequent two sections we will study
kerðh Þ and cokerðh Þ, and kerðg Þ. This will of course give information
K=F0 K=F0 K=F0
aboutthekernelandcokernelofs ,whichisthesubjectofSection5.Ineachsec-
K=F0
tionwefirstconsiderthetwoimportantspecialcaseswhereGðK=FÞffiZ andwhere
p
K¼FðA½p1(cid:9)Þ.
2. Cohomology of Compact p-Adic Lie Groups
Wewillcollect hereseveraluseful results.Let Gbe acompactp-adicLie group. Let
d denote the dimension of G. In the following lemma, we regard Z=pZ as a trivial
G-module.
LEMMA 2.1. ðiÞ Let V be a closed subgroup of G. Then H1ðV;Z=pZÞ is finite. Its
order is bounded. There exists an open subgroup U of G such that jH1ðV;Z=pZÞj4pd
for all closed subgroups V of U.
ðiiÞ Let V be a closed subgroup of G. Then H2ðV;Z=pZÞ is finite. Its order is
bounded.
Proof. We will use the notation and results of [DSMS]. Let P be a Sylow pro-p
subgroupofG.ThenPisanopensubgroupandtherestrictionmapHnðG;Z=pZÞ!
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GALOISTHEORYFORTHESELMERGROUPOFANABELIANVARIETY 261
HnðP;Z=pZÞ is injective, for any n51. It suffices to prove Lemma 2.1 for open
subgroupsV ofP.Pisapro-pp-adicanalyticgroup,andsoPhasfiniterankinthe
sense of [DSMS]. This means that dðVÞ¼dimZ=pZðH1ðV;Z=pZÞÞ, which is the car-
dinality of a minimal topological generating set for V, is finite and bounded as V
varies over closed subgroups of P. Also P contains an open subgroup U, which is
uniformly powerful (thm. 9.34 of [DSMS]). Thus, if V is any closed subgroup of U,
thendðVÞ4dðUÞ¼dimðUÞ¼d(thms9.36,9.38,Proposition4.4of[DSMS]).These
results prove (i).
Asfor(ii),tðVÞ¼dimðH2ðV;Z=pZÞÞisthenumberofrelationsforaminimaltopo-
logicalgeneratingsetforV,whichisfinite(thm4.25).Forauniformlypowerfulsub-
group U, we have tðUÞ¼dðd(cid:15)1Þ=2 (thm 4.26). Using this, one can give an explicit
upper bound for tðVÞ valid for all closed subgroups of G. (See exercise 9, p. 83 of
[DSMS].) &
Remark. We will apply this lemma to the subgroups V ¼GðK=F0Þ of the p-adic
LiegroupG¼GðK=FÞ.Thesesubgroupsareopen(andhenceclosed)if½F0: F(cid:9)<1.
Infact,GðK=FÞhasabaseofopensubgroupsV suchthatH1ðV;Z=pZÞhasorderpd
and H2ðV;Z=pZÞ has order pdðd(cid:15)1Þ=2, where d¼dimðGÞ. For arbitrary closed sub-
groups V, the bound on the order of HiðV;Z=pZÞ depends only on G.
Now let V be a finite-dimensional Q -vector space on which G acts continuously.
p
LetTbeaG-invariantZ -latticeinV.LetM¼V=T.ThenM½p(cid:9)ffiðZ=pZÞdimðVÞ.Let
p
i¼1 or 2. By a simple devissage argument, it follows from Lemma 2.1 that
HiðV;M½p(cid:9)Þ is finite and of bounded order as V varies over all closed subgroups
of G. But HiðV;MÞ½p(cid:9) is a homomorphic image of HiðV;M½p(cid:9)Þ and, therefore, also
has bounded order. It follows that the p-primary group HiðV;MÞ is cofinitely gene-
ratedasaZ -module.ItalsofollowsthattheZ -corankofHiðV;MÞisboundedasV
p p
varies over all closed subgroups of G. Here is a more precise result for open sub-
groups. &
LEMMA2.2. LetgbetheLiealgebraofG.Leti¼1or2.ForeveryopensubgroupV
of G, we have
corankZ ðHiðV;MÞÞ4 dimQ ðHiðg;VÞÞ:
p p
ThereexistsanopensubgroupU ofGsuchthatequalityholdsforallopensubgroupsV
of U.
Proof. We have corankZpðHiðV;MÞÞ¼dimQpðHiðV;VÞÞ. If V1, V2 are any two
opensubgroupsofGwithV (cid:7)V ,thentherestrictionmapHiðV ;VÞ!HiðV ;VÞ
2 1 1 2
isinjective.ThereexistsanopensubgroupU ofGsuchthatHiðV;VÞ¼Hiðg;VÞfor
all open subgroups V of U. Lemma 2.2 follows from these remarks. &
Remark. As a consequence, if Hiðg;VÞ¼0, then HiðV;MÞ is finite for all open
subgroups V of G. In particular, this applies if g is a semisimple Lie algebra.
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262 RALPHGREENBERG
3. The Kernel and Cokernel of h
K=F0
By the inflation-restriction exact sequence we have
kerðh ÞffiH1ðK=F0;AðKÞ Þ:
K=F0 p
Asagroup,AðKÞ ffiðQ =Z Þt(cid:17)(afinitegroup),where04t42g,g¼dimðAÞ.Note
p p p
that kerðs Þ¼kerðh Þ\Sel ðF0Þ . This isoften smaller than kerðh Þ, but we
K=F0 K=F0 A p K=F0
willpostponethediscussionofthisissue.Theinflation-restrictionsequencealsogives
cokerðh ÞffikerðH2ðK=F0;AðKÞ Þ!H2ðF0;A½p1(cid:9)ÞÞ:
K=F0 p
But H2ðF0;A½p1(cid:9)ÞffiL H2ðF0 ;A½p1(cid:9)Þ, where v0 varies over the real primes of F0.
v0 v0
(This follows from Corollary 6.24 in [Mi].) It follows that H2ðF0;A½p1(cid:9)Þ¼0 if p is
oddandisafinite elementary2-group ifp¼2.Wewillsimplyusetheupperbound
jcokerðh Þj4jH2ðK=F0;AðKÞ j
K=F0 p
which is an equality if p is odd or if F0 is totally complex.
I. K=F is a Z -extension
p
Assume that s is a topological generator of GðK=FÞffiZ . The finite extensions F0
0 p
ofFcontainedinKformatowerF¼F (cid:19)F (cid:19)(cid:20)(cid:20)(cid:20)(cid:19)F (cid:19)(cid:20)(cid:20)(cid:20),whereF =Fiscyclic
0 1 n n
of degree pn and GðK=F Þ is generated topologically by spn. We have
n 0
H1ðK=F ;AðKÞ ÞffiAðKÞ =ðspn (cid:15)1ÞAðKÞ :
n p p 0 p
We consider spn (cid:15)1 as an endomorphism of the Abelian group AðKÞ . Its kernel is
0 p
the finite group AðF Þ . This implies that the restriction of spn (cid:15)1 to the maximal
n p 0
divisible subgroup ðAðKÞ Þ is surjective. Hence it follows that
p div
ðAðKÞ Þ (cid:7)ðspn (cid:15)1ÞAðKÞ (cid:7)AðKÞ
p div 0 p p
for all n, and therefore H1ðK=F ;AðKÞ Þ is finite. Its order is bounded above by the
n p
index ½AðKÞ : ðAðKÞ Þ (cid:9) with equality for n(cid:21)0. Thus jkerðh Þj is finite and
p p div K=F0
bounded as F0 varies.
Since GalðK=F0Þ is isomorphic to Z and so has p-cohomological dimension 1, it
p
follows that H2ðK=F0;AðKÞ Þ¼0. Consequently, we have that cokerðh Þ¼0 for
p K=F0
all F0.
II. K¼FðA½p1(cid:9)Þ
A theorem of Serre (corollaire of Theoreme 2, [Se]), implies the finiteness of
HnðGðK=F0Þ;A½p1(cid:9)Þ for all n50 and all finite extension F0 of F contained in K.
Inparticular,kerðh Þisfinite.Butitsorderturnsouttobeunbounded.Morepre-
K=F0
cisely, we have the following result: Let F ¼FðA½pn(cid:9)Þ for n51. Then
n
kerðh ÞffiðZ=pnZÞ2gðm(cid:15)1Þ ð2Þ
K=Fn
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GALOISTHEORYFORTHESELMERGROUPOFANABELIANVARIETY 263
forn(cid:21)0.Herem¼m denotesthedimensionofthep-adicLiegroupGðK=FÞ.One
A
can show easily that m>1 and hence indeed jkerðh Þjisunbounded asF0 varies.
K=F0
To justify (2), consider the subgroup Z of GðK=FÞ defined as follows. Let
s2GðK=FÞ. Then, s2Z ,s acts on T ðAÞ as multiplication by a scalar
p
dðsÞ21þ2pZ . According to a result of Bogomolov [B], d defines an isomorphism
p
of Z to an open subgroup of 1þ2pZ , i.e., Z ffiZ . Let M¼KZ. For n51, let
p p
M ¼F M¼MðA½pn(cid:9)Þ. We assume hereon that n is sufficiently large. Then
n n
1þpnZ (cid:7)dðZÞ.WehaveGðK=M Þ¼Z ,whereZ ¼d(cid:15)1ð1þpnZ Þ.Wealso have
p n n n p
AðM Þ ¼AðF Þ ¼A½pn(cid:9).Now sinceZ istopologically cyclicandAðM Þ isfinite,
n p n p n n p
onesees easily thatH1ðZ ;A½p1(cid:9)Þ¼0and,hence,theinflation-restriction sequence
n
gives an isomorphism
H1ðM =F ;A½pn(cid:9)Þ!(cid:24) H1ðK=F ;A½p1(cid:9)Þ:
n n n
Therefore, kerðh Þ¼HomðGðM =F Þ;A½pn(cid:9)Þ. We have an isomorphism
K=Fn n n
GðM =F ÞffiGðM=M\F Þ¼H , say. This is an open subgroup of the p-adic Lie
n n n n
group GðM=FÞ, which has dimension m(cid:15)1. Using Lemma 2.1, one sees that H
n
can be generated topologically by m(cid:15)1 elements. Also, ½F : F (cid:9)¼pm and
nþ1 n
½M : M (cid:9)¼p from which it follows that ½H : H (cid:9)¼pm(cid:15)1. Now GðF =F Þ is
nþ1 n n nþ1 2n n
Abelian and of exponent pn. It follows that H =H is Abelian, of exponent pn,
n 2n
and of order pnðm(cid:15)1Þ. The above remarks imply that H =H ffiðZ=pnZÞm(cid:15)1 and that
n 2n
H ¼ðH ;H ÞHpn, which justifies (2).
2n n n n
Serre’s theorem referred to earlier states that H2ðK=F0;A½p1(cid:9)Þ, and hence
cokerðh Þ,arefinite.Alternatively,onecanprovethefinitenessasfollows.Define
K=F0
M as above. For any F0, let M0 ¼F0M. Then K=M0 ffiZ and so we have
p
H2ðK=M0;A½p1(cid:9)Þ¼0. Since H1ðK=M0;A½p1(cid:9)Þ¼0 also, we obtain an isomorphism
H2ðM0=F0;AðM0Þ Þ!(cid:24) H2ðK=F0;A½p1(cid:9)Þ:
p
Now AðM0Þ is finite. If V is the Sylow pro-p subgroup of the p-adic Lie group
p
GðM0=F0Þ, then H2ðV;Z=pZÞ is finite. The finiteness of H2ðV;AðM0Þ Þ and hence
p
of cokerðh Þ follows by devissage. Lemma 2.1 (ii) asserts that we have
K=F0
dimZ=pZðH2ðV;Z=pZÞÞ<C, for some C. Then an upper bound for jcokerðhK=F0Þj
would be jAðM0Þ jC. But note that jAðM0Þ j is unbounded as F0 varies.
p p
III. Arbitrary K=F
Here is one general result valid for any Abelian variety defined over F.
PROPOSITION3.1. LetK=FbeaGaloisextensionsuchthatGðK=FÞisap-adicLie
group.AssumethatAðKÞ isfinite.Thenkerðh Þandcokerðh Þarefiniteandhave
p K=F0 K=F0
bounded order as F0 varies over all extensions of F contained in K.
Proof. AðKÞ is a finite GðK=FÞ-module. It is enough to bound the order of
p
HiðV;AðKÞ Þforallopenpro-psubgroupsV ofGðK=FÞ,wherei¼1or2.ButAðKÞ
p p
hasaV-compositionserieswithcorrespondingquotientsisomorphictoZ=pZ.Thus,
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264 RALPHGREENBERG
by devissage, one can bound the order of HiðV;AðKÞ Þ by jAðKÞ jdiðVÞ, where
p p
diðVÞ¼dimZ=pZðHiðV;Z=pZÞÞ. By Lemma 2.1, diðVÞ is bounded. &
There are various hypotheses which imply that AðKÞ is finite, some of which are
p
includedinthefollowingresult.Anumberofotherresultscanbefoundinarticlesof
Zarhin. (See [Z1], [Z2], and some of the references there.) We assume only that A is
an abelian variety defined over F and that GðK=FÞ is a p-adic Lie group.
PROPOSITION 3.2. AðKÞ is finite if any of the following conditions are satisfied.
p
ðiÞ There exists a nonarchimedean prime Z of K not lying over p such that the corre-
sponding residue field k is finite.
Z
ðiiÞ TheLiealgebragissolvable,Ahaspotentiallyordinaryreductionatallprimesof
Flyingabovep,andtheresiduefieldk isfiniteforallprimesZofKlyingabovep.
Z
ðiiiÞ The Lie algebra g is semisimple.
ProofofðiÞ. SupposethatvistheprimeofFlyingbelowZandthatvjl,l6¼p.The
statedconditionisactuallyequivalenttoassertingthatK isafiniteextensionofF .
Z v
For GðK =F Þ is a p-adic Lie group of dimension 42. If it has positive dimension,
Z v
then K must contain the cyclotomic Z -extension of F (which is the only Z -
Z p v p
extensionofF andisunramified). Then theresiduefieldk wouldbeinfinite.Since
v Z
K isafiniteextensionofF ,itisnowobviousthatAðK Þ isfinite,andhencesois
Z v Z tors
AðKÞ and, in particular, AðKÞ .
tors p
Proof of ðiiÞ. Replacing F and K by finite extensions, we can assume that A has
good,ordinaryreductionatallprimesvofFlyingabovep.Theotherconditionsstill
hold. Assume that AðKÞ is infinite. Let W ¼H0ðK;V ðAÞÞ. That is, W ¼
p p p p
TpðAðKÞpÞ(cid:12)Zp Qp, where TpðAðKÞpÞ denotes the Tate module of AðKÞp. Then
dimðW Þ51andwecanconsidertherepresentationr: GðK=FÞ!AutðW Þinduced
p p
from the action of GðK=FÞ on AðKÞ . Since AA~ ðk Þ is finite, it follows that
p v Z
W (cid:19)kerðV ðAÞ!V ðAA~ ÞÞ. Let At denote the dual Abelian variety for A. Then At
p p p v
alsohasgoodreductioninvandtheactionofGFv onVpðAAftvtvÞisunramified.Sincewe
are assuming that A has ordinary reduction at v, the Weil pairing
V ðAÞ(cid:17)V ðAtÞ!Q ð1Þ induces an isomorphism
p p p
kerðVpðAÞ!VpðAA~vÞÞffiHomðVpðfAAtvtvÞ;Qpð1ÞÞ
as representation spaces for G . Let w: G !Z(cid:17) denote the cyclotomic character
Fv F p
and let s¼r(cid:12)w(cid:15)1, which gives the action of G on HomðQ ð1Þ;W Þ. Then s is a
F p p
finite-dimensional representation of G and its restriction sj gives the action of
F GFv
GFv on some nonzero subspace of HomðVpðAAftvtvÞ;QpÞ and, hence, is unramified and
has infinite image. It follows that L¼FF(cid:2)kerðsÞ is an infinite p-adic Lie extension of
F which is unramified at all primes of F lying over p. A conjecture of Fontaine
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