The Brauer-Manin obstruction for subvarieties of abelian varieties over function fields

24 Pages·2010·0.83 MB·English

ANNALS OF
M
ATHEMATICS
The Brauer-Manin obstruction for
subvarieties of abelian varieties over
function fields
By Bjorn Poonen and Jose´ Felipe Voloch
SECOND SERIES, VOL. 171, NO. 1
January, 2010
anmaah

AnnalsofMathematics,171(2010),511–532
The Brauer-Manin obstruction for subvarieties
of abelian varieties over function fields
By BJORN POONEN and JOSÉ FELIPE VOLOCH
Abstract
Weprovethatforalargeclassofsubvarietiesofabelianvarietiesoverglobal
function fields, the Brauer-Manin condition on adelic points cuts out exactly the
rational points. This result is obtained from more general results concerning the
intersectionoftheadelicpointsofasubvarietywiththeadelicclosureofthegroup
ofrationalpointsoftheabelianvariety.
1. Introduction
Thenotationinthissectionremainsinforcethroughoutthepaper,exceptin
Section3.3,andinSection4whereweallowalsothepossibilitythatK isanumber
field.
Letk beafield. LetK beafinitelygeneratedextensionofk oftranscendence
degree1. Weassumethatk isrelativelyalgebraicallyclosedinK,sincethecontent
x
of our theorems will be unaffected by this assumption. Let K be an algebraic
closureofK. Wewillusethisnotationconsistentlyforanalgebraicclosure,and
we will choose algebraic closures compatibly whenever possible. Thus k is the
algebraic closure of k in Kx. Let Ks be the separable closure of K in Kx. Let
(cid:127) be the set of all nontrivial valuations of K that are trivial on k. Let (cid:127) be a
all
cofinitesubsetof(cid:127) . Ifk isfinite,we mayweakenthecofinitenesshypothesisto
all
assumeonlythat(cid:127)(cid:18)(cid:127) hasDirichletdensity1. Foreachv2(cid:127),letK bethe
all v
completion of K at v, and let F be the residue field. Equip K with the v-adic
v v
topology. Definetheringofade`lesAastherestricteddirectproductQ .K ;O /
v2(cid:127) v v
oftheK withrespecttotheirvaluationsubringsO . ThenAisatopologicalring,
v v
inwhichQ O isopenandhastheproducttopology.
v2(cid:127) v
IfAisanabelianvarietyoverK,thenA.K/embedsdiagonallyintoA.A/'
Q
A.K /. Define the adelic topology on A.K/ as the topology induced from
v v
511
512 BJORNPOONENandJOSÉFELIPEVOLOCH
A.A/. Foranyfixedv definethev-adictopologyonA.K/asthetopologyinduced
fromA.K /. LetA.K/betheclosureofA.K/inA.A/.
v
ForanyextensionoffieldsF0(cid:27)F andanyF-varietyX,letXF0 bethebase
extensionofX toF0. CallaK-varietyX constantifXŠY forsomek-varietyY,
K
andcallX isotrivialifXx ŠYx forsomevarietyY definedoverk.
K K
Fromnowon,X isaclosedK-subschemeofA. CallX coset-freeifXx does
K
notcontainatranslateofapositive-dimensionalabeliansubvarietyofAx.
K
When k is finite and (cid:127) D (cid:127) , the intersection X.A/\A.K/ (cid:26) A.A/ is
all
closely related to the Brauer-Manin obstruction to the Hasse principle for X=K;
seeSection4. Forcurvesovernumberfields,V.ScharaschkinandA.Skoroboga-
tovindependentlyraisedthequestionofwhethertheBrauer-Maninobstructionis
the only obstruction to the Hasse principle, and proved that this is so when the
Jacobian has finite Mordell-Weil group and finite Shafarevich-Tate group. The
connection with theadelic intersection is statedexplicitly in [Sch99], and isbased
on global duality statements originating in the work of Cassels: see Remark 4.4.
Seealso[Sko01],[Fly04],[Poo06],and[Sto07],whichcontainsmanyconjectures
andtheoremsrelatingdescentinformation,themethodofChabautyandColeman,
theBrauer-Maninobstruction,andGrothendieck’ssectionconjecture.
Inthispaperweanswer(mostcasesof)ageneralizationofthefunctionfield
analogue of a question raised for curves over number fields in [Sch99], concern-
ing whether the Brauer-Manin condition cuts out exactly the rational points; see
Theorem D. This question is still wide open in the number field case. Along the
way, we prove results about adelic intersections similar to the “adelic Mordell-
Lang conjecture” suggested in [Sto07, Question 3.12]. Again, these are open in
thenumberfieldcase. Inparticular,weprovethefollowingtheorems.
THEOREMA. IfcharkD0,thenX.K/DX.A/\A.K/.
THEOREMB. SupposethatcharkDp>0,thatAx hasnononzeroisotrivial
K
quotient,andthatA.Ks/Œp1(cid:141)isfinite. SupposethatX iscoset-free. ThenX.K/D
X.A/\A.K/.
Remark 1.1. The proposition in [Vol95] states that in the “general case” in
whichAisordinaryandtheKodaira-SpencerclassofA=K hasmaximalrank,we
haveA.Ks/Œp1(cid:141)D0.
CONJECTUREC. ForanyclosedK-subschemeX ofanyA,wehaveX.K/D
X.A/\A.K/,whereX.K/istheclosureofX.K/inX.A/.
Remark1.2. IfAx hasnononzeroisotrivialquotientandX iscoset-free,then
K
X.K/isfinite[Hru96,Th.1.1];thusX.K/DX.K/. HenceConjectureCpredicts
inparticularthatthehypothesisonA.Ks/Œp1(cid:141)inTheoremBisunnecessary.
BRAUER-MANINOBSTRUCTIONFORSUBVARIETIESOFABELIANVARIETIES 513
Remark1.3. Hereis anexample toshowthat thestatementX.K/DX.A/\
A.K/ can fail for a constant curve in its Jacobian. Let X be a curve of genus
(cid:21) 2 over a finite field k. Choose a divisor of degree 1 on X to embed X in its
Jacobian A. Let FWA!A be the k-Frobenius map. Let K be the function field
ofX. LetP 2X.K/bethepointcorrespondingtotheidentitymapX !X. Let
P 2X.F /bethereductionofP atv.
v v
For each v, the Teichmu¨ller map F !K identifies F with a subfield of
v v v
K . AnyQ2A.K /canbewrittenasQDQ CQ withQ 2A.F /andQ in
v v 0 1 0 v 1
thekernelofthereductionmapA.Kv/!A.Fv/;thenlimm!1Fm.Q1/D0,so
limn!1FnŠ.Q/Dlimn!1FnŠ.Q0/DQ0. Inparticular,takingQDP,wefind
that(cid:0)FnŠ.P/(cid:1) convergesinA.A/tothepoint.P /2X.A/DQ X.K /,where
n(cid:21)1 v v v
wehaveidentifiedP withitsimageundertheTeichmu¨llermapX.F /,!X.K /.
v v v
If.P /wereinX.K/,theninX.K /wewouldhaveP 2X.F /\X.K/DX.k/,
v v v v
whichcontradictsthedefinitionofP ifv isaplaceofdegreegreaterthan1overk.
v
Thus.P /isinX.A/\A.K/butnotinX.K/.
v
In the final section of this paper, we restrict to the case of a global function
field,andextendTheoremBtoprove(undermildhypotheses)thatforasubvari-
etyofanabelianvariety,theBrauer-Maninconditioncutsoutexactlytherational
points;seeSection4forthedefinitionsofX.A/Br andScel. Ourresultisasfollows.
THEOREM D. Suppose that K is a global function fieldof characteristic p,
thatAx hasnononzeroisotrivialquotient,andthatA.Ks/Œp1(cid:141)isfinite. Suppose
K
thatX iscoset-free. ThenX.K/DX.A/BrDX.A/\Scel.
Toourknowledge,TheoremDisthefirstresultgivingawideclassofvarieties
ofgeneraltypesuchthattheBrauer-Maninconditioncutsoutexactlytherational
points.
2. Characteristic0
Throughout this section, we assume chark D0. In this case, results follow
rathereasily.
PROPOSITION 2.1. For any v, the v-adic topology on A.K/ equals the dis-
cretetopology.
Proof. Thequestionisisogeny-invariant,sowereducetothecasewhereAis
simple. LetA.F /denotethegroupofF -pointsontheNe´ronmodelofAoverO .
v v v
LetA1.K /bethekernelofthereductionmapA.K /!A.F /. TheLang-Ne´ron
v v v
theorem[LN59,Th.1]impliesthateitherAisconstantandA.K/=A.k/isfinitely
generated,orAisnonconstantandA.K/itselfisfinitelygenerated. Ineithercase,
the subgroup A1.K/WDA.K/\A1.K / is finitely generated. By the theory of
v
formalgroups(cf.[Ser92,p.118,Th.2]),A1.K /hasadescendingfiltrationby
v
514 BJORNPOONENandJOSÉFELIPEVOLOCH
opensubgroupsinwhichthequotientsofconsecutivetermsaretorsion-free(thisis
whereweusecharkD0),sotheinducedfiltrationonthefinitelygeneratedgroup
A1.K/ has only finitely many nonzero quotients. Thus A1.K/ is discrete. Since
A1.K /isopeninA.K /,thegroupA.K/isdiscreteinA.K /. (cid:3)
v v v
Remark2.2. TheliteraturecontainsresultsclosetoProposition2.1. Itismen-
tioned in the third subsection of the introduction to [Man63a] for elliptic curves
withnonconstantj-invariant,anditappearsforabelianvarietieswithK=k-trace
zeroin[BV93].
COROLLARY2.3. TheadelictopologyonA.K/equalsthediscretetopology.
Proof. The adelic topology is at least as strong as (i.e., has at least as many
opensetsas)thev-adictopologyforanyv. (cid:3)
Wecanimprovetheresultbyimposingconditionsinonlytheresiduefields
F insteadofthecompletionsK ,thatis,“flat”insteadof“deep”informationin
v v
thesenseof[Fly04]. Infact,wehave:
PROPOSITION 2.4. There exist v;v0 2(cid:127) of good reduction for A such that
A.K/!A.Fv/(cid:2)A.Fv0/isinjective.
Proof. LetB betheK=k-traceofA. Pickanyv2(cid:127)ofgoodreduction. The
kernel H of A.K/!A.F / meets B.k/ trivially. By Silverman’s specialization
v
theorem [Lan83, Ch. 12, Th. 2.3], there exists v0 2(cid:127) such that H injects under
reductionmodulov0. (cid:3)
ProofofTheoremA.ByCorollary2.3,X.A/\A.K/DX.A/\A.K/DX.K/.
(cid:3)
3. Characteristicp
Throughoutthissection,charkDp.
3.1. Field-theoreticlemmas.
LEMMA 3.1. For any v, if ˛ 2Kv is algebraic over K, then ˛ is separable
overK.
Proof. Replacing K by its relative separable closure in LWDK.˛/, we may
assume that Lis purelyinseparable overK. Thenthe valuation v onK admits a
uniqueextensionw toL,andtheinclusionofcompletionsK !L isanisomor-
v w
phism. By[Ser79,I.(cid:144)4,Prop.10](loc.cit. Hypothesis(F)holdsforlocalizations
offinitelygeneratedalgebrasoverafield),wehavean“nDPe f ”result,which
i i
inourcasesaysŒLWK(cid:141)DŒL WK (cid:141)D1. So˛2K. (cid:3)
w v
BRAUER-MANINOBSTRUCTIONFORSUBVARIETIESOFABELIANVARIETIES 515
If L is a finite extension of K, let A be the corresponding ring of ade`les,
L
definedasarestricteddirectproductoverplacesofL. Thereisanaturalinclusion
A,!A .
L
LEMMA3.2. LetLbeafiniteextensionofK. TheninALwehaveA\LDK.
Proof. Fix v 2 (cid:127). By [Bou98, VI.(cid:144)8.5, Cor. 3] and the fact that [Ser79,
Hypothesis(F)]holds,thenaturalmapK ˝ L!Q L isanisomorphism.
v K wjv w
HenceinQ L wehaveK \LDK. Theresultfollows. (cid:3)
wjv w v
3.2. Abelianvarieties.
LEMMA3.3. Foranyn2Z(cid:21)1,thequotientA.Kv/=nA.Kv/isHausdorff.
Proof. Equivalently, wemust show thatnA.K /is closed inA.K /. Suppose
v v
.P /isasequenceinnA.K /thatconvergestoP 2A.K /. WriteP DnQ with
i v v i i
Qi 2A.Kv/. Thenn.Qi (cid:0)QiC1/!0asi !1.
LetO bethevaluationringofK ,andletAbetheNe´ronmodelofAover
v v
O . Applying[Gre66,Cor. 1]toAŒn(cid:141)showsthat foranysequence.R /inA.K /
v i v
withnR !0,thedistanceofR tothenearestpointofA.K /Œn(cid:141)tendsto0.
i i v
Thusbyinductiononi wemayadjusteachQ byapointinA.K /Œn(cid:141)sothat
i v
Qi (cid:0)QiC1 !0 as i !1. Since A.Kv/ is complete, .Qi/ converges to some
Q2A.K /,andnQDP. ThusnA.K /isclosed. (cid:3)
v v
Remark 3.4. In the case where k is finite, Lemma 3.3 is immediate since
A.K /iscompactanditsimageundermultiplication-by-nisclosed.
v
The following is a slight generalization of [Vol95, Lemma 2], with a more
elementaryproof.
PROPOSITION3.5. IfA.Ks/Œp1(cid:141)isfinite,thenforanyv,thev-adictopology
onA.K/isatleastasstrongasthetopologyinducedbyallsubgroupsoffinitep-
powerindex.
x x
Proof. Forconveniencechoosealgebraicclosures K;K ofK;K suchthat
v v
Ks (cid:18)Kx(cid:18)Kx . As in the proof of Proposition 2.1, there is an open subgroup U
v
ofA.K /suchthatB WDA.K/\U isfinitelygenerated. Itsufficestoshowthat
v
foreverye2Z(cid:21)0,thereexistsanopensubgroupV ofA.Kv/suchthatB\V (cid:18)
peA.K/.
ChoosemsuchthatpmA.Ks/Œp1(cid:141)D0. LetM DeCm. ThenB=pMB is
finite. ByLemma3.3,A.K /=pMA.K /isHausdorff,sotheimageofB=pMB
v v
in A.K /=pMA.K / is discrete. Hence there is an open subgroup V of A.K /
v v v
suchthatB\V Dker.B !A.K /=pMA.K //.
v v
Suppose b 2 B \V. Then b D pMc for some c 2 A.K /\A.Kx/. By
v
Lemma3.1,weobtainc 2A.Ks/. If(cid:27) 2Gal.Ks=K/,then(cid:27)c(cid:0)c 2A.Ks/ŒpM(cid:141)
516 BJORNPOONENandJOSÉFELIPEVOLOCH
is killed by pm. Thus pmc 2A.K/. So bDpepmc 2peA.K/. Hence B\V (cid:18)
peA.K/. (cid:3)
PROPOSITION 3.6. TheadelictopologyonA.K/isatleastasstrongasthe
topologyinducedbyallsubgroupsoffiniteindex.
Proof. As in the proof of Proposition 2.1, the Lang-Ne´ron theorem implies
thatA.A/hasanopensubgroupwhoseintersectionwithA.K/isfinitelygenerated.
Itsuffices tostudy thetopology inducedon thatfinitely generatedsubgroup, sowe
mayreducetothecaseinwhichk isfinitelygeneratedoverafinitefieldF . This
q
case is proved in [Mil72], which adapts and extends [Ser64] and [Ser71]. (The
paper[Mil72]usesnottheadelictopologyaswehavedefinedit,butthetopology
coming from the closed points of a finite-type Z-scheme with function field K.
Sincetheadelictopologyisstronger,[Mil72]containswhatwewant.) (cid:3)
(cid:16) (cid:17)
LEMMA 3.7. Suppose that A.K/ is finitely generated. Then A.K/ D
tors
A.K/ .
tors
Proof. Let
!
T WDker A.K/! Y A.Fv/ ;
A.F /
v2(cid:127) v tors
where A.F / is the group of F -points on the Ne´ron model of A. Since A.K/ is
v v
finitelygeneratedandthegroupsA.F /=A.F / aretorsion-free,thereisafinite
v v tors
subsetS (cid:26)(cid:127)suchthatT DA.K/\U fortheopensubgroup
!
U WDker A.A/! Y A.Fv/
A.F /
v2S v tors
of A.A/. The finitely generated group A.K/=T is contained in the torsion-free
groupQ A.Fv/ ,soA.K/=T isfree,andwehaveA.K/ŠT ˚F astopolog-
v2S A.Fv/tors
icalgroups,whereF isadiscretefreeabeliangroupoffiniterank.
We claim that the topology of T is that induced by the subgroups nT for
n(cid:21)1. Forn(cid:21)1,the subgroupnT isopeninT byProposition3.6. Ift 2T,then
somepositivemultipleoft isinthekernelofA.K /!A.F /,andthenp-power
v v
multiplesofthismultipletendto0. ApplyingthistoafinitesetofgeneratorsofT,
weseethatanyopenneighborhoodof0inT containsnT forsomen2Z .
>0
ItfollowsthatTx ŠT ˝Zy. Now
(cid:16) (cid:17)
A.K/ D.Tx˚F/ DTx Š.T ˝Zy/ ŠT : (cid:3)
tors tors tors tors
tors
Remark3.8. Whenk isfinite,aneasierproofofLemma3.7ispossible: Com-
bined with the fact that A.A/ is profinite, Proposition 3.6 implies that A.K/ Š
A.K/˝Zy;thetorsionsubgroupofthelatterequalsA.K/ .
tors
BRAUER-MANINOBSTRUCTIONFORSUBVARIETIESOFABELIANVARIETIES 517
Thefollowingpropositionisafunctionfieldanalogueof[Sto07,Prop.3.6].
Our proof must be somewhat different, however, since [Sto07] made use of strong
“image of Galois” theorems whose function field analogues have recently been
disproved[Zar07].
PROPOSITION3.9. SupposethatA.Ks/Œp1(cid:141)isfinite. LetZ beafiniteK-sub-
schemeofA. ThenZ.A/\A.K/DZ.K/.
Proof. InthisfirstparagraphweshowthatreplacingK byafiniteextension
Ldoesnotdestroy thehypothesisthatA.Ks/Œp1(cid:141)isfinite. Thisis obviousifL
is separable over K, so assume that L is purely inseparable. Choose n 2 Z(cid:21)0
with Lpn (cid:18) K. Then .Ls/pn (cid:18) Ks, so pnA.Ls/Œp1(cid:141) (cid:18) A.Ks/Œp1(cid:141). Thus
pnA.Ls/Œp1(cid:141)isfinite. Butmultiplication-by-pn hasfinitefibers,soA.Ls/Œp1(cid:141)
itselfisfinite.
Nextweclaimthatifweprovetheconclusionafterbaseextensiontoafinite
extensionL,thenthedesiredconclusionoverK holds. Namely,supposethatwe
proveZ.A /\A.L/DZ.L/. Then
L
Z.A/\A.K/(cid:18)Z.A /\A.L/DZ.L/;
L
so
Z.A/\A.K/(cid:18)Z.A/\Z.L/DZ.K/;
wherethelastequalityusesLemma3.2.
ThuswemayreplaceK byafiniteextensiontoassumethatZ consistsofa
finite set of K-points of A. (The same idea was used in [Sto07].) A point P 2
A.K/ is represented by a sequence .Pn/n(cid:21)1 in A.K/ such that for every v, the
limit limn!1Pn exists in A.Kv/. If in addition P 2Z.A/, then there is a point
Qv2Z.K/whoseimageinZ.Kv/equalslimn!1Pn2A.Kv/. ThePn(cid:0)Qv are
eventually containedinthe kernelofA.K/!A.F /, whichisfinitely generated,
v
so there are finitely generated subfields k (cid:18)k, K (cid:18)K with K =k a function
0 0 0 0
fieldsuchthatalltheP andthepointsofZ.K/areinA.K /. ByProposition3.5,
n 0
thesequence.Pn(cid:0)Qv/n(cid:21)1 iseventuallydivisibleinA.K0/byanarbitrarilyhigh
power of p. For any other v0 2 (cid:127), the same is true of .Pn (cid:0)Qv0/n(cid:21)1. Then
Qv0 (cid:0)Qv 2 A.K0/ is divisible by every power of p. Since A.K0/ is finitely
generated, Qv0 (cid:0)Qv is a torsion point in A.K0/. This holds for every v0 2 (cid:127),
and A.K / is finite. Thus RWDP (cid:0)Q 2A.K / is a torsion point in A.K /.
0 tors v 0 0
Lemma3.7appliedtoK yieldsR2A.K / . HenceP DRCQ 2A.K/,and
0 0 tors v
soP 2Z.A/\A.K/DZ.K/. (cid:3)
LEMMA3.10. Fixv2(cid:127). Let(cid:128)v betheclosureofA.K/inA.Kv/. Thenfor
everye2Z(cid:21)0,themapA.K/=peA.K/!(cid:128)v=pe(cid:128)v issurjective.
Proof. Let O be the valuation ring of K , and let m be its maximal ideal.
v v v
LetAoverOv betheNe´ronmodel. Forr 2Z(cid:21)1,letGr bethekernelofA.Kv/D
518 BJORNPOONENandJOSÉFELIPEVOLOCH
A.Ov/ ! A.Ov=mrv/. It follows from [Ser92, p. 118, Th. 2] that Gr=GrC1 is
isomorphic to .O =m /dimA, which is killed by p, so that each G is an abelian
v v r
pro-p-group, and hence a topological Z -module. There are only finitely many
p
pointsoforderp inA.K /,andT G Df0g,sosomeG containsnonontriv-
v r(cid:21)1 r r
ial p-torsion points, and hence is torsion-free. In particular, A.K / has an open
v
ı
subgroupA .K /thatisatorsion-freetopologicalZ -module,andwemaychoose
v p
Aı.K /sothatAı.K/WDA.K/\Aı.K /isfinitelygenerated.
v v
Thegroup(cid:128)ıWD(cid:128) \Aı.K /istheclosureofAı.K/,sothereisanisomor-
v v v
phismoftopologicalgroups(cid:128)vıŠZp˚m forsomem2Z(cid:21)0. Inparticular,forany
e2Z(cid:21)0,thegrouppe(cid:128)vı isopenin(cid:128)vı,whichisopenin(cid:128)v. Sothelargergroup
pe(cid:128) also is open in (cid:128) . But the image of A.K/ in the discrete group (cid:128) =pe(cid:128)
v v v v
isdense,sothemapA.K/=peA.K/!(cid:128) =pe(cid:128) issurjective. (cid:3)
v v
3.3. AuniformMordell-Langconjecture. We thank Zoe´ Chatzidakis, Fran-
c¸oiseDelon,andThomasScanlonformanyoftheideasusedinthissection. See
[Del98] for the definitions of separable, p-basis, p-free, p-components, etc. By
iteratedp-componentswemeanp-componentsofp-componentsof...ofp-com-
ponents(allwithrespecttoagivenp-basis).
The goal of this section is to deduce a uniform version (Proposition 3.16)
of the function field Mordell-Lang conjecture from a version in [Hru96]. Under
somehypotheses,theuniformversionassertsthefinitenessoftheintersectionof
a subvariety X of an abelian variety A with any coset of a subgroup peA.F/ of
A.F/,whereF isallowedtorangeoverp-basis-preservingextensionsofaninitial
groundfieldK.
Remark 3.11. The p-basis condition on F, or something like it, is neces-
saryforthetruthofProposition3.16;withnocondition,F mightbealgebraically
closed, and then peA.F/DA.F/, so the desired finiteness would fail assuming
dimX >0. Thep-basisconditionisusedintheproofofProposition3.16toimply
separabilityof F overK, whichguarantees thata nonisotrivialityhypothesis onA
overK ispreservedbybaseextensiontoF;seeLemma3.13anditsproof.
LEMMA3.12. LetBbeap-basisforafieldK ofcharacteristicp. LetLbe
anextensionofK suchthatBisalsoap-basisforL. Supposethatc isanelement
ofLthatisnotalgebraicoverK. Thenthereexistsaseparablyclosedextension
F ofLsuchthatBisap-basisofF andtheAut.F=K/-orbitofc isinfinite.
Proof. FixatranscendencebasisT forL=K. Let(cid:127)beanalgebraicallyclosed
extensionofK suchthatthetranscendencebasisof(cid:127)=K isidentifiedwiththeset
Z(cid:2)T. Identify L with a subfield of (cid:127) in such a way that each t 2 T maps to
the transcendence basis element for (cid:127)=K labelled by .0;t/ 2 Z(cid:2)T. The map
of sets Z(cid:2)T !Z(cid:2)T mapping .i;t/ to .i C1;t/ extends to an automorphism