Non-abelian descent and the arithmetic of Enriques surfaces

26 Pages·2005·0.28 MB·English

IMRN InternationalMathematicsResearchNotices
2005,No.52
Non-AbelianDescentand the Arithmetic
ofEnriques Surfaces
DavidHarariand Alexei Skorobogatov
D
ow
nloaded from
1TamhferenIunBtiattrlfraupouldreogurb-ecMnlteieamornnaihlnearopebpcsratornaucbchetisototnarttaeotdtiohanesafHloalplsoosiwenstp:sriiosnnictivppaloresiesatinbidelsewtoeovaedkreasncpurpmirbboexeirinmfipaeutlidroesnl.yApgrefouovnmiddeeats-- at Imimrn.oxfordjournals.org
rictermstheclassofsmoothprojectivevarietiesforwhichtheBrauer-Maninobstruction perial C
ippqsllueatesyhseewtdioohbnneylrywee´ohtthabelestehtBrGeruraacslutoiiemiorsn-iMcltaooarvnteehirxnieanomHgbsapsswltesriseutcheptxraiiiosnnntcoiifnpso-lrneaovbaatenrtldhiieaetwnioeeGnsalakywloaoitipnshpegar(rosnoexuaeimbp[1e.alT,ti7aiho,ni1nsg4?he]I)aon,smtrhleeeectfrtkeineocytpfereuoxnnladetmhaise-- ollege London Library on Septem
mraetinotnaallgsrouurfpa.cTehse(wcahsiechofaprreingceiopmaelthroicmaollgyesniemoupslyscpoancnesecotfeadb),ewliahnerveatrhieetiBersaaunedr-tMhaatnoinf ber 13, 2011
obstructionisexpectedtobetheonlyone,mightseemtosuggestthataslongasthegeo-
metric fundamental group is abelian, the Brauer-Manin obstruction should still be the
onlyone.
The Manin obstruction was linked to the classical abelian descent by Colliot-
The´le`neandSansuc[2].In[8],theauthorsintroducedthenon-abeliandescentasanew
toolforstudyingrationalpoints.Thepresentpaperenrichesthenon-abeliantheorywith
ageneralmethodforconstructingnon-abeliantorsors,andthenappliesittoanexample
whichanswerstheabovequestioninthenegative.
Received31May2005.Revisionreceived29September2005.
CommunicatedbyShinichiMochizuki.
3204 D.HarariandA.Skorobogatov
The Enriques surfaces seem to lie close to the frontier separating the varieties
whose arithmetic is controlled by the Brauer-Manin obstruction from those for which
it is not the case. These surfaces are cohomologically indistinguishable from rational
surfaces, but yet they possess a nontrivial geometric fundamental group Z/2. We con-
structanEnriquessurfaceoverthefieldofrationalnumberswhichisacounterexample
toweakapproximationthatcannotbeexplainedbytheBrauer-Maninobstruction.More
precisely,ifa,b,andcareintegerssatisfyingsomefairlymildconditions,thenthequo-
tientoftheKummersurfaceYgivenby
(cid:1) (cid:2)(cid:1) (cid:2)(cid:1) (cid:2)(cid:1) (cid:2)
y2 = x2−a x2−ab2 t2−a t2−ac2 (1.1)
D
bytheinvolutionwhichchangesthesignsofallthecoordinatesisanEnriquessurface ow
Xadweliitchpaoninatdsealitcispfioeisnatlwlthhiechglcoabnanloretcbiperaopcpitryoxcoimndatiteidonbsyparroavtiidoendablypothinete.lHemowenevtsero,ftthhise nloaded from
im
paBarnarutuimecrueglrOaircnroaeuElpnwerBxaiaryqmXuteo.ps(llNeos,ouowtkrefeaatthcceaathnthiiacsnsheooaxuoaZrsmaeerxpaiaslemk=iipsd5le,senu,btsgh=egees1Gse3tta,eoldcofib=rsaygt2rti.ohoAnuesapplahapncioltaoisndstotdrspii.vthiiyoanlolayflo[f1ne4aP]t:iuctrhXee.,)ntFhaoitsr- at Imrn.oxfordjournals.org
ural map BrX → BrY is not surjective; hence BrY can impose more conditions on the perial C
ebardlaeelluiicnntdpeWeorripenrttsheshteaiontawicotthntiheoiansctlpooarsfouttvorhiredesoeoEfdrntbohrvyieqensruoeeants-Koaif3nbr-vecaolotilivauoentnridoainnlegsp(cooeeif.nngat.t,.snatEhunanrniiqvBeurreXssa.slHutroofrwasceoevr,e),rw,cahaimnchobrieesgcsoetnna--- ollege London Library on Septem
saidgeenreedraalssiatunaotnio-anbwelhiaennatocrosmorpoovseitriothneoEfntorirqsuoressissuarftaocres.oTrhuinsdiserathpeargtricouulparwchaiscehoisf ber 13, 2011
an extension of relevant structure groups. This result reminiscent of Mumford’s con-
struction of theta-groups [10] is another main goal of this paper (see Theorem2.2 and
Proposition2.5).Ityieldsalargesupplyofnon-abeliantorsors;thenon-abeliandescent
method can then be deployed with potential applications to weak approximation and
the Hasse principle. The counterexample to weak approximation on the Enriques sur-
face which we construct in this paper can be explained by the descent obstruction as-
sociated to this non-abelian torsor, as defined in [8]. Another application is a link be-
tween the approach of [14] and that of [8]: we show that the “iterated Manin obstruc-
tion” of [14] is in fact equivalent to the descent obstruction given by the composition
of an abelian (e.g., universal) torsor with the corresponding e´tale Galois covering (see
Proposition2.10).
Non-AbelianDescentandtheArithmeticofEnriquesSurfaces 3205
2 Compositionoftorsors
2.1 Preliminaries
Let k be a field of characteristic 0 . Letk¯ be an algebraic closure of k, Γ = Gal(k¯/k). In
thispaper,byak-varietyweunderstandaseparatedk-schemeoffinitetype.IfXisak-
variety,wewriteXforthek¯-varietyobtainedfromXbytheextensionofthegroundfield
fromktok¯.Wedenotebyk¯[X]∗thegroupofinvertibleregularfunctionsonX.Acommu-
tativealgebraick-groupFofmultiplicativetypeisanextensionofafinitecommutative
k-group scheme by a k-torus. By(cid:3)F, we denote the module of characters of F; this is an
abeliangroupoffinitetypewithacontinuousactionofΓ.Ifanalgebraick-groupGacts
onak-varietyYpreservingthefibresofamorphismY →X,thenYisanX-torsorunderG
iflocallyinthee´taletopologyonXthevarietyYwiththeactionofGisisomorphictothe
directproductX× G.AllcohomologygroupsinthispaperareGaloisore´talecohomol- D
k ow
ko-ggyrgoruopu,pnso;twneecaelssosacroinlysiadbeerltiahne.GaloiscohomologysetH1(k,G),whereGisanalgebraic nloaded from
(tsipelei,cea.tgiI.vf,e[Y1ty6isp,(ea2,g.t2he2oe)mn])te:htreirceailslythinetfeoglrloawlvianrgieetxyacwtistheqk¯u[Yen]∗ce=ok¯f∗C,oalnlidoFt-iTshae´lke`-ngeroaunpdoSfanmsuulc- at Imimrn.oxfordjournals.org
WIfhZen→k0Yis−ia→slgaHetb1or(rkas,iocFra)lu−lyn→dcelHor1sF(e,Yd,t,hFt)ehn−e−χ→nχ(([H2Z.o]1)m)∈Γsh(cid:1)H(cid:3)oF,woPmsicΓthY((cid:3)Fa(cid:2),t−P−a∂→ictYoH)r2si(oskr,ciFas)lld−ee→dtetHrhme2(itYny,epFd)e.boyfiZts→(t2y.p1Ye). perial College London Library on S
uχ(p[Zto])ihsaosmnoorptohrissimon.T,fhoerveaxraimetpyleZ,iwshgeenomtheetrtiycpaelliyscinojnenceticvtee.dAifYa-tnodrsoonrlyunifdtehreakgerronueplooff eptember 13, 2011
multiplicativetypeisuniversalifitstypeisanisomorphism.
There is another useful exact sequence, also due to Colliot-The´le`ne and Sansuc
(see [2, (2.1.1)]) . LetT be a k-torus, and Z → Y a torsor under T, where both Y and Z
aregeometricallyintegral,andk¯[Y]∗ = k¯∗.ThefollowingsequenceofΓ-modulesisthen
exact:
1−→k¯∗ −→k¯[Z]∗ −→T(cid:3) −→PicY −→PicZ−→0. (2.2)
Moreover,uptosign,themapT(cid:3) →PicYcoincideswiththetypeofZ→Y.Itisclearfrom
(2.2)thatwhenthetypeisinjective,wehavek¯[Z]∗ =k¯∗.
3206 D.HarariandA.Skorobogatov
In the case when Z → Y is a torsor under a finite k-group F, and the condition
k¯[Z]∗ =k¯∗issatisfied,westillhaveanexactsequence(see[16,(2.5)])
0−→(cid:3)F−→PicY −→PicZ. (2.3)
Hereagain,(cid:3)F→PicYisthetypeofthetorsorZ→Y(see[16,page25]).
WewriteBrXforthecohomologicalBrauer-GrothendieckgroupH2(X,G ).Itis
m
known that if X is smooth of dimension at most 2, then this group coincides with the
groupofequivalenceclassesofAzumayaalgebrasonX.Let
(cid:4) (cid:5) (cid:4) (cid:5)
D
Br0X=Im Brk−→BrX , Br1X=Ker BrX−→BrX . (2.4) ownloaded from
im
HhTHoh11me((kkoH,,mPPoiicocchrXXps))ch;.hiisiflmdk¯-[oSXfe]∗rΓr-e=msopk¯d∗eu,cltterhsae,lntshetehqneueBknrecλrenXe(lc⊂fo.fB[1trh61,iXsCimosratohplleaisrinyBv2re.0r3sX.e9].i)mIgfaλivgee:soMaf λm→∗aHp1P(Bikcr,X1MXi)s→⊂a at Imrn.oxfordjournals.org
Foranumberfieldk,wewriteΩk forthesetofallplacesofk.LetAk bethering perial C
ofade`leXs(cid:1)oAfkk(cid:2).BFo=r(cid:2)a(cid:3)sPuvb(cid:4)gr∈ouXp(cid:1)ABk⊂(cid:2)(cid:6)(cid:6)(cid:6)Br(cid:5)X,dinefivnve(cid:1)α(cid:1)Pv(cid:2)(cid:2)=0, ∀α∈B(cid:6), (2.5) ollege London Library on S
v∈Ωk eptem
whereX(A )isthesetofadelicpointsofX,andinv :Brk →Q/Zisthelocalinvariantof ber 13, 2011
k v v
localclassfieldtheory.Byglobalreciprocity,wehaveX(k) ⊂ X(A )Br.WhenX isproper,
(cid:7) k
X(A )BrcontainstheclosureX(k)ofX(k)inX(A )= X(k )intheproducttopology.
k k v∈Ωk v
Finally,foratorsorf:Z→Xunderak-groupG,wewrite
(cid:2) (cid:7) (cid:8)(cid:6)
(cid:1) (cid:2) (cid:3) (cid:4) (cid:1) (cid:2)(cid:6)(cid:3)(cid:4) (cid:5)(cid:4) (cid:8) (cid:1) (cid:2)
X A f = P ∈X A (cid:6) Z ∈Im H1(k,G)−→ H1 k ,G . (2.6)
k v k Pv v
v∈Ωk
We have X(k) ⊂ X(A )f; moreover, X(k) ⊂ X(A )f when X is proper and G is linear (see
k k
[16,Proposition5.3.3]).
Non-AbelianDescentandtheArithmeticofEnriquesSurfaces 3207
2.2 Ageneralresult
Wewillneedthefollowingauxiliarystatement.
Lemma2.1. LetY → Xbeatorsorunderanalgebraick-groupG.Assumethattheimage
ofanyk¯-morphismY → Gisak¯-point.ThenthecanonicalmapG(k¯) → Aut(Y/X)isan
isomorphism. (cid:1)
Proof. Letψ∈Aut(Y/X).ThecanonicalisomorphismY× Y =YׯGidentifiesthegraph
X k
ofψwiththegraphofamorphismg : Y → G.Nowbyassumption,wehaveg(y) = g for
0
someg ∈G(k¯)andanyy∈Y(k¯).Henceψ(y)=g y. (cid:2)
0 0
Colliot-The´le`nepointedouttousthattheconverseisfalse,forexample,forX =
Speck,Y =Spec(k⊕k)withtheactionofG=Z/2bypermutations.
Themainresultofthissectionisthefollowing. D
ow
Tahtoerosroermun2.d2e.rLHe,twFhaenrdeHXbiseaalsgmeboroathicakn-dgrgoeuopms,eptri:cZal→lyiYntaegtorraslokr-vuanrdieetryF.,AasnsdumYe→thXe imnloaded from
ftohlaltowthien(fg1o)clFlooonwrdieintaigcohndsihahg∈oralHdm(.k¯i)s,tchoemremeuxtiasttisvea:nisomorphismofk¯-varietiesϕh :Z→Zsuch at Imrn.oxfordjournals.org
perial C
pZY ϕhh ZYp (2.7) ollege London Library on S
eptem
quenceo(2f)aTlgheebirmaiacgke-gorfoaunpysk¯-morphismZ → Fisak¯-point.Thenthereisanexactse- ber 13, 2011
1−→F−→G−→H−→1 (2.8)
suchthattheactionofFonZextendstoanactionofGwhichinducestheactionofHon
thequotientY =Z/F.ThisactionmakesZ→XintoatorsorunderG. (cid:1)
Therefore,thetheoremgivesanaturalsufficientconditionforacompositionof
two torsors to be a torsor. In the proof, G is constructed in a certain canonical way,
namely, G(k¯) is the group of k¯-automorphisms of Z which are liftings of the automor-
phismsofYdefinedbytheelementsofH(k¯).
3208 D.HarariandA.Skorobogatov
Proofofthetheorem. DefineGasthesubsetofAut¯(Z)consistingoftheautomorphisms
k
φsuchthatthereexistsh∈H(k¯)makingthediagram
φ
Z Z
(2.9)
p p
h
Y Y
commutative.Toanyφ ∈ G,therecorrespondsexactlyoneh ∈ H(k¯)becausetheaction
ofHonY isfaithful.Since(h,y) (cid:5)→ hyisanactionofHonY,weseethatGisasubgroup
ofAut¯(Z).Forthesamereason,thenaturalmapπ:G→H(k¯)isahomomorphism.Obvi-
k
ously,F(k¯)isasubgroupofGcontainedinthekernelofπ.ByLemma2.1,condition(2)of
D
thetheoremimpliesthatAut(Z/Y)=F(k¯),henceweobtainanexactsequenceofgroups ownloaded from
aTIIchIt.ei1o.k¯1n-]v)o:a1frΓ−ie→otnieFts(hk¯Ze)ga−rn→oduGpX−Ac−π→uomt(HZe(cid:1)/fk¯Xr(cid:2)o).;mthviasraicettiioesndisedfienfiendeodvberykth,ethfeorlleofworiengthfeorremiuslaa(nca(f2t.u.[11r0a2)l, at Imperial Cimrn.oxfordjournals.org
(cid:1)γϕ(cid:2)(z)=γ(cid:9)ϕ(cid:1)γ−1z(cid:2)(cid:10), γ∈Γ, z∈Z(k¯), ϕ∈Aut(cid:1)Z/X(cid:2). (2.11) ollege London Library on S
eptem
Omnuetacthiveictkysoimf mtheedisaatmeleydthiaagtrtahmecwomithmuφtaatnivdityhorfetphlaecdeidagbryamγφabaonvdeiγmhp,lrieessptehceticvoemly-. ber 13, 2011
ThisshowsthatthesubgroupG⊂Aut¯(Z)isstableundertheactionoftheGaloisgroupΓ.
k
Lemma2.3. Let z ∈ Z(k¯). The map θ : G → p−1(H(k¯) · p(z )) defined by g (cid:5)→ gz is a
0 0 0
bijection. (cid:1)
Proof. θisinjective.Supposethatg z = g z .Thisimpliesinparticularthatg andg
1 0 2 0 1 2
aremappedtothesameh∈H(k¯),thatis,g g−1isinAut(Z/Y).SinceAut(Z/Y)=F(k¯),we
1 2
haveg g−1 ∈F(k¯).Butg g−1fixesz ,thusg =g .
1 2 1 2 0 1 2
θ is surjective. Let z ∈ p−1(H(k¯)·p(z )). Then there exists h ∈ H(k¯) such that
1 0
hp(z )=p(z ).Letϕ beak¯-automorphismofZsuchthatp◦ϕ =h◦p.Thenp(ϕ (z ))=
0 1 h h h 0
p(z ).Itremainstomodifyϕ byanelementofF(k¯)toobtainφ∈Gsuchthatφ(z )=z .
1 h 0 1
(cid:2)
Non-AbelianDescentandtheArithmeticofEnriquesSurfaces 3209
Endoftheproofofthetheorem.Wehaveanobviouscommutativediagram
(cid:1) (cid:1) (cid:2) (cid:1) (cid:2)(cid:2)
G θ p−1 H k¯ ·p z
0
π p (2.12)
(cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2)
H k¯ H k¯ ·p z
0
ThemapH(k¯)→H(k¯)·p(z )isbijectivebecausetheactionofHonYisfree.Thistogether
0
with the bijectivity of θ shows that π : G → H(k¯) is surjective. We now have a Galois
equivariantextensionofgroups
(cid:1) (cid:2) (cid:1) (cid:2)
1−→F k¯ −→G−→H k¯ −→1. (2.13)
D
ow
aLneteGxtbeensthioenk¯o-vfaHribeytyFp.S−i1n(Hce(k¯th)·epg(rzo0)u)p.TvhaeribetijyecGtiiosnqθuamsaipkreosjGecitnivteoaanndaltgheebarcatiicok¯n-ogfroΓuopn, nloaded from
im
Riaatnneidmsexk¯cato[reZknn]t∗2sii.n=4ou.nok¯Wou∗f,seH,tshwebeeeygfcFrrao,onaumnpddLeGefit(mhnk¯aem)tGcaGoa2ians.1ccttithsdheoeasnqtuwwZohiwtteihientnAhttuhrotee(fqZcGuo/inXbrne)yedetcqhptueirsdoippacpecoertmditoipewnso.i.ntIehtniastoncflaeGtaurirsathlaaattcotGriuoi(cid:2)sns, at Imperial Crn.oxfordjournals.org
eoafsΓy. tovTehreiffyo.llowingpropositiongivessufficientconditionsforTheorem2.2whichare ollege London Library on S
Proposition2.5. Letp:Z→Ybeatorsorunderak-groupFofmultiplicativetype,andlet eptem
cYa→llyXinbteeagrtaolrsko-vraurniedteyr,aanndaltgheabtrYaiicsks-ugcrhouthpaHt.k¯A[Ys]s∗u=mek¯∗th(ae.tgX.,ipsraopsmeroaonthdagneodmgeeotrmiceatlrliy- ber 13, 2011
connected). Assume also that the type λ ∈ Hom ((cid:3)F,PicY) of the torsor p : Z → Y is in-
Γ
jectivewithH(k¯)-invariantimage(e.g.,thetorsorisuniversal).Thencondition(1)ofthe
theoremissatisfied.Condition(2)issatisfiedaslongasFisfinite,orFisatorus,orY is
proper. (cid:1)
Proof. TheactionofHonYdefinesanaturalΓ-equivariantactionofH(k¯)onPicYandthe
assumptionwemadeaboutthetypeλimpliesthatthisactiongivesrisetoanaturalΓ-
equivariantactionτ(cid:3)ofH(k¯)on(cid:3)F;wehaveλ◦τ(cid:3) =h∗◦λforeachh∈H(k¯).SinceY satisfies
h
theconditionk¯[Y]∗ =k¯∗,aY-torsorunderFisuniquelydetermineduptoisomorphismby
itstype.NowtheformulaaboveshowsthattheY-torsorunderFobtainedfromZ→Yby
thebasechangeh:Y →YhasthesametypeastheY-torsorunderFobtainedfromZ→Y
3210 D.HarariandA.Skorobogatov
bythetransformationofstructuregroupτ :F→F.Hence,thesetorsorsareisomorphic,
h
by(2.1).Therefore,anyh ∈ H(k¯)liftstoanautomorphismϕ ofthek¯-varietyZ,which
h
meansthatcondition(1)ofTheorem2.2issatisfied.
Sinceλisinjective,thevarietyZisgeometricallyconnected.ThusforFfinite,the
condition(2)ofthetheoremisobvious.
If F is a torus, then the injectivity of the map λ implies that k¯[Z]∗ = k¯∗, whence
Mor¯(Z,F)=F(k¯).
k
Finally,weconsiderthecasewhenYisproper.Thek-groupFisanextensionofa
finitecommutativek-groupF byatorusT.LetY =Z/T;thispropervarietyisaY-torsor
1 1
underF .Thefunctorialityof(2.1)withrespecttothechangeofthestructuregroupF→
1
F impliesthatthetypeofY →Yisthecomposition
1 1
(cid:3)F1 (cid:1)→(cid:3)F(cid:1)→PicY, (2.14) Dow
henceisinjective.ThereforeY isgeometricallyconnected,hencek¯[Y ]∗ =k¯∗.Next,Zisa nloaded from
1 1 im
Y1-torsor0underT(cid:3)F.1Thefollo(cid:3)FwingdiagraT(cid:3)mcommut0es: at Imrn.oxfordjournals.org
(2.15) perial C
osTofhrYeYt1o→p→Yl0iY.nT.eThhheeelremef(cid:3)Fti1i-sdhdoablnevdvioesuPrqtsiuic,caaYarnledacrotrmhoewmbPuioistctettYhso1embytyotpnheeeoifsfuZtnhc→etosrYeiq,aaulinetydncoreifg(th2yt.p3-eh)awdneidtfihnorenedespbisyetcththeteottytohpree- ollege London Library on Septem
sthtraut1ctthueretogrrsoourpobchtaainngeedFfr→omF1Z.T→oYprobvyetthheacththanegreigohftt-hheansdtrsuqcutuarreegcroomumpuTt→es,Fw(epunsohte- ber 13, 2011
1
forward)givesthesameF-torsorasthepullbackoftheF-torsorZ → Y toY .Indeed,the
1
push-forwardisthequotient(Z× F)/T,whereT actsbysending(z,f)to(t−1z,tf).Here,
k
the structure of an F-torsor is obtained from the action of F on the second factor. The
canonicalisomorphismZ× F=Z× ZtranslatestheactionofTinto(z ,z )(cid:5)→(t−1z ,z ).
k Y 1 2 1 2
ThequotientbythisactionisY × Z,thepullbackofZ → Y toY .Wenowseethatthe
1 Y 1
twodifferentwaystobuildamap(cid:3)F→PicY inthediagramcoincidesincebothareequal
1
tothetypeoftheF-torsorY × Z→Y .Thisestablishesthecommutativity.
1 Y 1
An easy diagram chase now shows that the map T(cid:3) → PicY is injective. By the
1
remarksafter(2.2),wehavek¯[Z]∗ =k¯∗.SinceFisofmultiplicativetype,thisimpliesthat
Mor¯(Z,F)=F(k¯). (cid:2)
k
Non-AbelianDescentandtheArithmeticofEnriquesSurfaces 3211
2.3 Examples
Example2.6. Mumford’stheta-groups.LetLbealinebundleonanabelianvarietyA.The
complementtothezerosectionofLisanA-torsorunderthemultiplicativegroupG .Let
m
K(L) ⊂ Abetheclosedsubschemewhoseclosedpointsaretheelementsa ∈ A(k¯)such
thatLisisomorphictoa∗L,thetranslationofLbya.NotethatK(L)isfiniteifandonly
ifLisample(see[10,II.6,Proposition1]).TheassumptionsofTheorem2.2aresatisfied
becauseofProposition2.5.TheresultingextensionofK(L)byG isatheta-group;these
m
groupshavenumerousbeautifulapplications,see[10,VI.23].
Example2.7. LetAbeanabelianvarietywithanactionofafinitegroupschemeH.Letus
assumethatthegroupschemeAH(thepointsofAfixedbyH)isfinite.LetYbeaprincipal
homogeneousspaceofAsuchthattheclass[Y] ∈ H1(k,A)comesfromH1(k,AH).Then,
HnaturallyactsonY.LetDbeaprojectivevarietywithafreeactionofH.Thediagonal D
ow
actionoLfeHtoαn:YA×→kDBis=fAre/eA.LHebteXt=he(Yna×tkurDa)l/iHso.geny.Chooseapositiveintegermsuch nloaded from
im
mbothfya=BKteβsAru◦H(cαβh,)⊂wt=hhAae[Atrme[[Ym]β.]]T:/=hABeHβ→m)∗.[AuZTl]hit.siiTpsahlnimecinaastpotihgomeenrnaebyky.ieSssmuaZpmnp→aoatspueYrftaahiclnatpttoouZrssaihstt-haofrorporsruwoignrahcruidαpnad,mlseahorpottmhhZeaot→ggewrnoYeueocp(qausnuscoswhtpeiraemicnteeet at Imrn.oxfordjournals.org
A[m]/AH. perial C
msoufartphjeecttiooLrnestoAArtZt[mb×e]k→tDhe→A(cid:11)dHuY.a×LleaktbDFelubianenditevsrakArie[ermtny]e/olA;ftHAh.i(swTihhseictdhhueaaCcltaosrftttirheivreidianuljlaeyclotonifoDnA)A[miHs]/t→AheHAc.o[Tmmh]episotystehpdee ollege London Library on S
eptem
F−→At[m]−→At(cid:1)k¯(cid:2)=Pic0Y −→PicY −→Pic(cid:1)Y×k¯ D(cid:2). (2.16) ber 13, 2011
FromTheorem2.2,weobtainthatZ× DisanX-torsorunderafinitek-groupGwhichis
k
anextensionofHbyF.
ThesimplestcaseiswhenH = Z/2,andthenontrivialelementofHactsonAas
multiplicationby−1.ThenAH = A[2],Y isanyprincipalhomogeneousspaceofAsuch
that2[Y] = 0,misanypositiveevennumber,B = A,αisthemultiplicationby2map.If
Aisanellipticcurve, andDisacurveofgenus1onwhichthenontrivialelementofH
actsasatranslation,thenXisabiellipticsurface.Itscuriousarithmeticpropertieswere
studiedin[3,14].Animportantrolein[14]wasplayedbyatorsorZ× D→Xwithm=8
k
(thenGisnon-abelian).ThegroupH = µ leadstobiellipticsurfacesofadifferenttype;
3
theirarithmeticwasstudiedin[1].
3212 D.HarariandA.Skorobogatov
Example2.8. LetXbeanEnriquessurfaceoverk,andletY →XbeaK3-coveringofX.Let
ZbeauniversalY-torsor.ItisatorsorundertheNe´ron-SeveritorusT ofY.Theorem2.2
thensaysthatZisanX-torsorunderak-groupGwhichisanextensionofZ/2byT.
ThegroupGiscommutativeifandonlyifthenaturalmapPicX→PicYissurjec-
tive.Indeed,theexactsequenceintheproofofLemma3.5showsthatZ/2actstrivially
onPicY ifandonlyifPicX → PicY issurjective. Inthiscase, Z/2alsoactstriviallyon
T. SinceH 2(Z/2,k¯∗) = 0 for the trivial Z/2-module structure on k¯∗, the extension is a
semidirectproduct.Becauseofthetrivialaction,wehaveG=T×Z/2.Conversely,ifGis
commutative,thenZ/2actstriviallyonPicY,hencePicX→PicYissurjective.
2.4 Non-abeliantorsorobstructionversusManinobstructiononabeliantorsors
Inthissubsection,weclarifytherelationbetweenthe(non-abelian)torsorobstruction D
ow
q[8u]aasnidprtohjeec“titiveera.tFeodrMmaonreindoetbasitlrsuocntitown”is[t1e4d].fAolrlmvsaroifetgireosuaprseaansdsutmoresdortos,bseeesm[1o6o,tChhaanpd- nloaded from
im
GrPtattLhheenrersee=odmpr2oieGelmn]fe..cnet(cid:3)taxTeaZtir2rhos(cid:3)e.fet9to→si.hksra-emoLfXmnoecotorabomfnGterutGp(cid:3)aiGrhnbadt(cid:3)iueloceooar.afifucsAnkGtossei-rosfd1dounu-ebrcmnfiomoydnfeceρeyoGtr,schfaolGaaaenntn“(cid:3)dρ.teZawSvoZ.luegif(cid:3)rspeΓyt=bpemwrodZas”ioiρetacrhpitckshtch-iaotgiohsternmffieothoutcffewpriΓeotiGnosmorttn.essZLdoGiertn,ftosoZdGGrZem→(nk¯m→o)oXtasfeXpbudZsecabhZdnayedtttfiγohoZ(cid:1)narages(cid:3)tpdoa→Gorsb(cid:3)iuonyX=nptρ.dpuGTeownhrρsdieeGtei(cid:1)dnhsr, at Imperial College London Library on Srn.oxfordjournals.org
to the standard action γg, where γ ∈ Γ, g ∈ G(k¯). Choose an isomorphism of X-torsors eptem
tuontdheersGta,Znd(cid:7)arZd(cid:3)a.cTthioennZγz(cid:3),dwehfienreesγa∈“tΓw,izst∈edZ”(k¯a)c.tTiohneopfoΓinotnsγZ(cid:1),zdaenndotγezdbbeyloγn(cid:1)gztaostohpepsoasmede ber 13, 2011
fibreofZ→X.Henceγ(cid:1)z=g(z,γ)·γz,where,forafixedγ,g(z,γ)isamorphismfromZto
G.Byourassumption,g(z,γ)doesnotdependonz.Wewriteg(z,γ)=g(γ).Itisclearthat
thisisalocallyconstant,hencecontinuousfunctionΓ →G(k¯).(RecallthatΓ hasnatural
profinitetopology,andG(k¯)hasdiscretetopology.)Letg∈G(k¯),z∈Z(k¯).Then
γ(cid:1)(gz)=g(γ)·γ(gz)=g(γ)·γg·γz. (2.17)
Ontheotherhand,
γ(cid:1)(gz)=γ(cid:1)g·γ(cid:1)(z)=γ(cid:1)g·g(γ)·γz. (2.18)