Examples of abelian surfaces with everywhere good reduction

28 Pages·2015·0.55 MB·English

by  Lassina Dembélé

Math.Ann.(2016)364:1365–1392 MathematischeAnnalen
DOI10.1007/s00208-015-1252-6
Examples of abelian surfaces with everywhere good
reduction
LassinaDembélé1 · AbhinavKumar2
Received:8October2013/Revised:7February2015/Publishedonline:23July2015
©Springer-VerlagBerlinHeidelberg2015
Abstract Wedescribeseveralexplicitexamplesofsimpleabeliansurfacesoverreal
quadraticfieldswithrealmultiplicationandeverywheregoodreduction.Theseexam-
plesprovideevidencefortheEichler–ShimuraconjectureforHilbertmodularforms
overarealquadraticfield.SeveraloftheexamplesalsosupportaconjectureofBrumer
andKrameronabelianvarietiesassociatedtoSiegelmodularformswithparamodular
levelstructures.
MathematicsSubjectClassification 11F41·11F46·11F67·11G10
1 Introduction
AcelebratedresultofFontaine[20](seealsoAbrashkin[1])assertsthatthereisno
abelianschemeoverZ.Inotherwords,thereisnoabelianvarietyoverQwithevery-
wheregoodreduction.However,longbeforethisresult,therewereafewexamplesof
ellipticcurves ofunit conductor over quadratic fields inthe literature.For example,
AKwassupportedinpartbyNationalScienceFoundationGrantDMS-0952486andbyagrantfromthe
SolomonBuchsbaumResearchFund.LDwassupportedbytheGrantEPSRCEP/J002658/1.
B
LassinaDembélé
[email protected]
AbhinavKumar
[email protected]
1 WarwickMathematicsInstitute,UniversityofWarwick,CoventryCV47AL,UK
2 DepartmentofMathematics,MassachusettsInstituteofTechnology,77MassachusettsAvenue,
Cambridge,MA02139,USA
123
1366 L.Dembélé,A.Kumar
thecurve
E : y2+xy+(cid:2)2y = x3,
√ √
where (cid:2) = 5+ 29 is the fundamental unit in F = Q( 29), was known to Tate,
2
and to Serre who extensively studied it in [38]; it is also alluded to in [42]. Since
then, there has been much work on finding elliptic curves with everywhere good
reductionovernumberfields,withaparticularemphasisonquadraticfields;seefor
example [8,12,27,36,40,48]. For real quadratic fields, the database of such curves
hasbeenconsiderablyexpandedbyElkies[16].In[14],itisshownthatthisdatabase
iscompleteforallfundamentaldiscriminants≤1000ofnarrowclassnumberone,if
oneassumesmodularity.Amoresystematicalgorithmwhich,givenanumberfieldF
andafinitesetofprimes S ofitsringofintegers,returnsthesetofallellipticcurves
over F with good reduction outside of S is given in [11]. However, this algorithm
reliesonalgorithmsfor S-integralpointsforellipticcurves,andhasnotyetbeenfull
implementedforthisreason.Analternate(andperhapsmoreefficient)approachwhich
usesS-unitequationsiscurrentlybeingexploredbyCremonaandElkies[10].Infact,
asimilarmethodhasalreadybeenusedbySmart[45]tofindhyperellipticcurvesof
genus2withgoodreductionoutside Swhen F =Q.
Incontrast,tothebestofourknowledgethereisnotasingleexampleofanabelian
surfacewitheverywheregoodreductionintheliterature(exceptinthecasewhenthe
abeliansurfacehascomplexmultiplication[13],orisaQ-surface[6,41]oraproduct
ofellipticcurves).Thiscouldpossiblybeexplainedbythefactthatallthealgorithms
we mentioned above do not readily generalize to the genus 2 situation. The goal of
thispaperistoremedythatsituationbyprovidingthefirstequationsforsuchsurfaces
overrealquadraticfields.
We note that the non-existence of abelian varieties over Q with good reduction
everywhereisinstrumentalintheKhare–WintenbergerproofoftheSerreconjecture
forGaloisrepresentationsofGal(Q/Q).Asdescribedin[28],theproofoftheSerre
conjectureinretrospectcanbeviewedasamethodtoexploitanaccidentwhichoccurs
inthreedifferentguises:
(a) (Fontaine,Abrashkin)Therearenonon-zeroabelianvarietiesoverZ.
(b) (Serre,Tate)Therearenoirreduciblerepresentations
ρ :Gal(Q/Q)→GL (F),
2
whereFisthealgebraicclosureofF orF ,thatareunramifiedoutsideof2and
2 3
3respectively.
(c) S (SL (Z))=0,i.e.,therearenocuspformsoflevelSL (Z)andweight2.
2 2 2
Thefailureofthishappyaccidentovergeneralnumberfields,suchasrealquadratic
fields,meansthatnewtechniquesareneededforanalogousmodularityresults.
Ourapproachtotheconstructionofabeliansurfaceswitheverywheregoodreduc-
tion combines three key elements: (a) recent advances in the computation of Hecke
eigenvaluesofHilbertmodularforms,(b)newrationalmodelsofHilbertmodularsur-
faces,and(c)theEichler–ShimuraconjectureforHilbertmodularforms.Asaresult
123
Examplesofabeliansurfaceswitheverywheregoodreduction 1367
ofourinvestigation,weproducefurtherevidencefortheEichler–Shimuraconjecture,
aswellasforaconjectureofBrumerandKramer[5]associatingabelianvarietiesto
paramodularSiegelmodularformsonSp(4).
Theoutlineofthepaperisasfollows:inSect.2,webrieflyrecallthebasicfacts
regardingthesethreeingredients.InSect.3,wedescribeourstrategytopredictand
findexamplesofgoodreductionabeliansurfaces,assumingtheEichler–Shimuracon-
jecture.Section4providesseveralillustrativeexamplesofourmethods,givingexplicit
abeliansurfaceswithgoodreductioneverywhere,andconnectingthemtoappropriate
Hilbertmodularforms.WeconcludewithalistofallourexamplesinSect.5.
2 Background
2.1 Hilbertmodularforms
Let F beatotallyrealfieldofnarrowclassnumberoneanddegreed.WeletO be
F
theringofintegersof F,d thedifferentof F.Foreachi = 1,...,d,leta (cid:5)→ a(i)
F
denotethei-thembeddingofFintoR,sothatwehaveanidentificationF⊗R(cid:7)Rd.
WeletF+bethesetoftotallypositiveelementsinF,i.e.theinverseimageof(R+)d,
andOF,+ = F+∩OF.Wefixatotallypositivegeneratorδ ofdF.(Notethatevery
idealhassuchageneratorsince F hasnarrowclassnumberone.)
LetHbethePoincaréupperhalfplane.TheHilbertmodulargroupSL (O )acts
2 F
onHd byfractionallineartransformations:
(cid:2) (cid:3) (cid:4) (cid:5)
a b ·(z ,...,z )= a(i)zi +b(i) .
c d 1 d c(i)z +d(i)
i i=1,...,d
LetNbeanintegralideal,andset
(cid:6)(cid:2) (cid:3) (cid:7)
a b
Γ (N)= ∈SL (O ):c∈N .
0 c d 2 F
Letk ≥ 2beaneveninteger.AHilbertmodularformofweight1 k andlevelNisa
holomorphicfunction f :Hd →Csuchthat
(cid:4)(cid:8)d (cid:5)k (cid:2) (cid:3)
f(γz)= (c(i)z +d(i)) f(z) forall γ = a b ∈Γ (N).
i c d 0
i=1
Let f bea(cid:9)Hilb(cid:10)ertmodularformofweightkandlevelN.Then f isinvariantunder
the matrices 1μ for μ ∈ O , which act as z (cid:5)→ z +μ. Hence, by the Koecher
01 F
principle[4], f admitsaq-expansionoftheform
1 Moreprecisely,thisdefinesaHilbertmodularformofparallelweightk.
123
1368 L.Dembélé,A.Kumar
(cid:11)
f(z)=a0+ aμe2πiTr(μδz),
μ∈OF,+
where Tr(νz) = ν(1)z1 +···+ν(d)zd, for ν ∈ F+. We say that f is a cusp form
if a = 0. Since f is invariant under the action of the matrices diag((cid:2),(cid:2)−1) for
0
(cid:2) ∈ O×F inSL2(OF),whichactas z (cid:5)→ (cid:2)2z,wehavea(cid:2)2μ = aμ forallμ ∈ OF,+
and (cid:2) ∈ O×. Let f be a cusp form of weight k and level N. Then, for every ideal
F
m ⊆ OF, the quantity am(f) = aμ, where μ is a totally positive generator of m,
is well-defined and depends only on m. We call it the m-th Fourier coefficient of
f. When f is a normalized eigenform for the Hecke operators (i.e. a(1)(f) = 1),
the eigenvalue of the Hecke operator Tm is am(f) for each m (cid:2) N. It is a theorem
of Shimura [44] that in this situation, the am(f) are algebraic integers and the Z-
subalgebraOf =Z[am(f):m⊆OF]⊂Chasfiniterankandisthereforeanorder
in some number field K (called the field of Fourier coefficients of f). For more
f
backgroundonHilbertmodularforms,see[4,14,19].
Here,wewishtopointoutsomenewtechniquesinthecomputationofHilbertmod-
ularforms,whicharisefromtheEichler–Jacquet–Langlands–Shimizucorrespondence
betweenHilbertmodularformsandquaternionicmodularforms.Wewillnotgointo
details here, but instead refer the reader to [14] for a detailed description of these
methods. The upshot is that it is possible to efficiently compute systems of Hecke
eigenvaluesforHilbertmodularcuspformsbyinsteadcomputingmodularformson
finitespacesoronShimuracurves.Thiswillbecrucialtoourmethodsinthispaper.
ThecorrespondingalgorithmshavebeenimplementedintheHilbertModularForms
PackageinMagma[3]).
2.2 Hilbertmodularsurfaces
LetK bearealquadraticfieldofdiscriminantD(cid:13).TheHilbertmodularsurfaceY−(D(cid:13))
isacompactificationofthecoarsemodulispacewhichparametrizesprincipallypolar-
ized abelian surfaces with real multiplication by the ring of integers O of K, i.e.
K
pairs (A,ι), where ι : O → End (A) is a homomorphism. The complex points
K Q
Y−(D(cid:13))(C)ofthisspaceareobtainedbycompactifyingSL2(OK)\(H+×H−),where
H+andH−aretheupperandlowerhalf-planesrespectively,byaddingfinitelymany
cuspsandresolvingthesingularitiesoftheresultingspace.TheHilbertmodularsurface
mapstothemodulispaceA ofprincipallypolarizedabeliansurfaces,byforgetting
2
theactionofOK.ItsimageistheHumbertsurfaceHD(cid:13),andthemapY−(D(cid:13))→HD(cid:13)
isadoublecover,ramifiedalongaunionofmodularcurves.ThesurfacesY−(D(cid:13))have
(cid:13)
modelsovertheintegers,withgoodreductionawayfromprimesdividing D.
Recently, Elkies and the second author [17] computed explicit birational models
(cid:13)
over Q for these Hilbert modular surfaces for all the fundamental discriminants D
lessthan100,byidentifyingtheHumbertsurfaceHD(cid:13) withamodulispaceofelliptic
K3 surfaces, which may be computed explicitly. For the fundamental discriminants
intherange1 < D(cid:13) < 100,theHumbertsurfaceisarationalsurface,i.e.birational
to P2 over Q (and in fact, even over Q). Therefore, they are able to exhibit HD(cid:13) as
a double cover of P2, with equation z2 = f(r,s), wherer,s are parameters on P2.
123
Examplesofabeliansurfaceswitheverywheregoodreduction 1369
TheyalsogetthemaptoA ,whichisbirationaltoM ,themodulispaceofgenus
2 2
2curves.ItisgivenbyproducingtheIgusa–Clebschinvariantsoftheimagepointas
rationalfunctionsofr ands.
Remark1 Hilbertmodularsurfaceshavebeenanobjectofextensivestudyinnumber
theoryandarithmeticgeometry,especiallyinthelatterhalfofthetwentiethcentury.
Inparticular,theirgeometricclassificationwasdescribedbyHirzebruch,vandeVen,
Zagier, van der Geer and others. In the comprehensive reference [50], arithmetic
modelsforsomeofthemarealsodescribed.However,forourwork,weneedexplicit
equationsforthesesurfacesalongwiththemaptoA ,andthisdoesnotseemtobe
2
availableforanydiscriminantotherthan5exceptin[17](thoughitcouldbeworked
outinprincipleusingHilbertandSiegelmodularforms).Consequently,wewilluse
theequationsfrom[17]throughout.
2.3 Eichler–Shimuraconjecture
Thefollowingconjectureisinstrumentalinidentifyingtheexamplesinthispaper.
Conjecture1 (Eichler–Shimura) Let F be a totally real number field and N an
integral ideal of F. Let f be a Hilbert newform of weight 2 and level N. Let
Of = Z[am(f) : m ⊆ OF] be the order generated by the Fourier coefficients of
f,and K itsfieldoffractions.Thereexistsanabelianvariety A /F ofdimension
f f
[K :Q]withgoodreductionoutsideofNandwithO (cid:10)→End (A ),suchthat
f f F f
(cid:8)
L(A ,s)= L(fτ,s),
f
τ∈Hom(Kf,C)
where
L(fτ,s):= (cid:11) am(f)τ.
Nms
m⊆O
F
When F = Q, this conjecture is a theorem, due to Eichler for prime level and
Shimura in the general case. The Eichler–Shimura construction can be summarized
as follows. Let N > 1 be an integer, and let X (N) be the modular curve of level
1
Γ (N). This curve and its Jacobian J (N) are defined over Q. We recall that the
1 1
space S (Γ (N)) of cusp forms of weight 2 and level Γ (N) is a T-module, where
2 1 1
T is the Hecke algebra. Let f ∈ S (Γ (N))be a newform, and let I = Ann (f).
2 1 f T
Shimura[43]showedthatthequotient
A := J (N)/I J (N)
f 1 f 1
isanabelianvariety A ofdimension[K :Q]definedoverQwithendomorphisms
f f
bytheorderO =Z[a (f):n ≥1]andthat
f n
(cid:8)
L(A ,s)= L(g,s),
f
g∈[f]
where[f]denotestheGaloisorbitof f.
123
1370 L.Dembélé,A.Kumar
OneofthemainconsequencesoftheproofoftheSerreconjecture[39]byKhare–
Wintenberger[29]isthattheconversetoConjecture1istruewhen F = Q.Thatis,
anabelianvarietyofGL -typeisisogenoustoaQ-simplefactorof J (N)forsomeN
2 1
[30].Andso,thisprovidesatheoreticalconstructionofallabelianvarietiesofGL -
2
typeoverQwithaprescribedconductor.Infact,onecanmakethisexplicitinmany
cases(see[9]forellipticcurves,and[23,25]forabeliansurfaces).
For [F : Q] > 1, the known cases of Conjecture 1 exploit the cohomology of
Shimuracurves.Forinstance,theconjectureisknownwhen[F :Q]isodd,orwhen
NisexactlydivisiblebyaprimepofO [51].ThesimplestcaseinwhichConjecture1
F
isstillunknowniswhen f isanewformoflevel(1)andweight2overarealquadratic
field.Inthatcase,theconjecturepredictsthattheassociatedabelianvariety A has
f
everywheregoodreduction.
3 Thestrategy
Let F beanumberfieldofclassnumber1,and E anellipticcurveover F givenbya
(globalminimal)Weierstrassequation
E : y2+a xy+a y = x3+a x2+a x +a ,
1 3 2 4 6
with a ∈ O , the ring of integers of F. The invariants c and c of E satisfy the
i F 4 6
equationc3−c2 =1728Δ,whereΔisthediscriminantofE.Inotherwords,thepair
4 6
(c ,c )isanO -integralpointonthecurve
4 6 F
y2 = x3−1728Δ. (1)
Since E haseverywheregoodreductionifandonlyifΔisaunitinO ,wecanfind
F
alltheellipticcurvesoverFwitheverywheregoodreductionbysolving(1)asΔruns
over a finite set of representatives of O×/(O×)12. (Note that given a pair (c ,c ),
F F 4 6
we get a minimal model by using the Tate algorithm.) Most of the algorithms we
mentionedearlierrelyonthisfact.
Unfortunately,thereductionofabelianvarietiesofhigherdimensionisnotcharac-
terizedbyanicesinglediophantineequationsuchas(1).Forthisreason,weneedan
additionalingredientwhichwillguideoursearch.Thisextrainputisprovidedbythe
Eichler–Shimuraconjecture.
SupposewehaveaHilbertmodulareigenform f ofweight2over F,withHecke
eigenvalues am(f) in a real quadratic field Kf of discriminant D(cid:13). The Eichler–
Shimura conjecture predicts that there should be an abelian variety A over F of
dimension [K : Q] = 2, (up to isogeny) associated to this data, which has real
f
multiplicationbyanorderin K .Furthermore,theconductorof Ashoulddividethe
f
level N of f. In particular, if f has level (1), the conjectural abelian surface A has
good reduction everywhere. This observation will be the source of our examples in
thispaper,forwhichtheabeliansurfaceturnsouttobeprincipallypolarized,andalso
hasrealmultiplicationbythefullringofintegersofK .Ourstrategytoproducesuch
f
Aisasfollows:
123
Examplesofabeliansurfaceswitheverywheregoodreduction 1371
(a) FindaHilbertmodularformoflevel(1)andweight2forarealquadraticfield
(cid:13)
F,withcoefficientsinarealquadraticfield K ofdiscriminant D.
f
(b) Find an F-rational point on the Hilbert modular surface Y−(D(cid:13)), for which the
L-function of the associated abelian surface matches that of f at several Euler
factors,uptotwist.
(c) Computethecorrectquadratictwistoftheabeliansurface,orthegenus2curve.
(d) Checkthattheabeliansurfacehasgoodreductioneverywhere.
(e) Provethatthe L-functionsindeedmatchup.
Note that there is no reason one has to restrict to the case when the base field
is a real quadratic field F. The next interesting case in which the Eichler–Shimura
conjecture is not known is that of totally real quartic base fields L. So one could
look for eigenforms of weight 2 for SL (O ) whose Fourier coefficients are in a
2 L
realquadraticfield K ofdiscriminant D,andontheotherhandtrytofind L-rational
points on the Hilbert modular surface Y−(D). In this paper, we looked at quadratic
base fields F for convenience. On the other hand, if we instead want examples for
whichthefield K haslargerdegree,wemightneedexplicitrationalmodelsforthe
f
appropriate Hilbert modular varieties, which are not currently available. Hence the
choiceof K isrestricted.
Forsimplicity,weinvestigatedonlyrealquadraticfieldsFofnarrowclassnumber1
anddiscriminantlessthan1000.Wefoundtwenty-eightexamplesofHilbertnewforms,
andcorrespondingabeliansurfacesformostoftheseforms.Wewillsayafewwords
lateraboutthe“missing”examples,whichwehopewillbefoundinfuturework.
4 Theexamples
Fromnowon,F willdenotearealquadraticfieldofnarrowclassnumberone.Welet
√Dbeitsfund√amentaldiscriminant.WewilldenoteitsringofintegersbyOF.Letw =
Dor(1+ D)/2accordingasDis0or1mod4,sothat{1,w}isaZ-basisofO .
F
ForaHilbertnewform f ofweight2over F,wewillletOf =Z[am(f):m⊆OF]
and K betheorderandthefieldgeneratedbytheFouriercoefficients,respectively.
f
Wewillfocusonformssuchthat[K :Q]=2,sincewedonotyetknowhowtowrite
f
(cid:13)
simpleequationsforge√neralHilbertm√odularvarieties.Welet D bethediscriminant
of K and write e = D(cid:13) or (1+ D(cid:13))/2. We denote the non-trivial element of
f
Gal(F/Q)andGal(K /Q)byσ andτ respectively.The L-seriesoftheconjectural
f
surface A attachedto f iswrittenas
f
(cid:8)
1
L(A ,s)= L(f,s)L(fτ,s)= ,
f p Qp(N(p)−s)
where
Qp(T):=(T2−ap(f)T +N(p))(T2−ap(f)τT +N(p))
=T4−sp(f)T3+tp(f)T2−N(p)sp(f)T +N(p)2.
123
1372 L.Dembélé,A.Kumar
Table1 Asummaryofthe (cid:13)
D D Case
examples
53 8 I
61 12 I
73 5 I
193 17 II
233 17 II
277 29 II
349 21 II
353 5 III
373 93 II
389 8 II
397 24 II
409 13 II
421 5 I
421 5 III
433 12 II
461 29 II
613 21 II
677 13 II
677 29 II
677 85 II
709 5 II
797 8 II
797 29 II
809 5 II
821 44 II
853 21 II
929 13 II
997 13 II
Our examples (see Table 1) can be subdivided in the following cases, with the
majorityofexamplescomingfromCaseII.
I: Theform f isGal(F/Q)-invariant.
II: Theform f isnotGal(F/Q)-invariant,butitsGal(K /Q)-orbit{f, fτ}is.
f
III: TheGal(K /Q)-orbit{f, fτ}isnotGal(F/Q)-invariant.
f
WewillseethatCaseIissomewhatspecial:itisfrequentlypossibletoproducethe
associatedabeliansurfacethroughanalyticmethodsforclassicalmodularforms.
In[5],Brumer–Kramerproposedthefollowingconjectureasagenus2analogue
oftheEichler–Shimuraconstructionforclassicalnewformsofweight2(withinteger
coefficients).
Conjecture2 (Brumer–Kramer) Let g be a paramodular Siegel newform of genus
2,weight2andlevel N,withintegerHeckeeigenvalues,whichisnotinthespanof
123
Examplesofabeliansurfaceswitheverywheregoodreduction 1373
Gritsenkolifts.Thenthereexistsanabeliansurface BdefinedoverQofconductorN
suchthatEnd (B)=Zand L(g,s)= L(B,s).
Q
TheexamplesinCaseIIshowthatthereisastrongconnectionbetweenthiscon-
jectureandConjecture1.
4.1 CaseI
Inthiscase,theHeckeeigenvaluesoftheHilbertmodularform f satisfy
ap(f)=apσ(f).
Thisimpliesthattheform f isabasechangefromQ.LetgbeanewforminS (Γ (D))
2 1
whosebasechangeis f.Sincethelevelof f is(1),theformg ∈ S (Γ (D),χ )new
2 1 D
by[33,P√rop.2,p.263],whereχD isthefundamentalcharacterofthequadraticfield
F = Q( D). Let L be the coefficient field of g. Then, L is a quartic CM field
g g
which contains K . The non-trivial element of Gal(L /K ), which we denote by
f g f
(x (cid:5)→ x,x ∈ L ), extends to complex conjugation. Let B be the abelian variety
g g
attachedtotheformg.Then B isafourfoldsuchthatEnd (B )⊗Q(cid:7) L .Letw
g Q g g D
betheAtkin–Lehnerinvolutionon S (Γ (D),χ )new.Thisinducesaninvolutionon
2 1 D
B ,whichwestilldenotebyw .Shimura[41,§7.7]showsthefollowing:
g D
(a) w isdefinedover F,andwσ =−w ;
D D D
(b) w ·[x] = [x]·w , where [x] denotes the endomorphism induced on B by
D D g
x ∈ L .
g
(c) Theabeliansurface A :=(1+w )B isdefinedoverF,andisisogenoustoits
f D g
Galoisconjugategivenby Aσ :=(1−w )B .Moreover,wehave
f D g
B ⊗ F ∼ A × Aσ.
g Q f f
Sointhiscase,theexistenceofthesurface A isadirectconsequenceoftheclassical
f
Eichler–Shimuraconstruction.
AlthoughConjecture1isknowninthiscase,itwouldstillbedesirabletohavean
explicit equation for the surface A . We outline two methods to find it, the first of
f
whichisspecialtothiscase.
4.1.1 Method1
Thismethodisanalytic,andhasanobviousconnectionwiththeOdaconjecture[35,
p.xii]forHilbertmodularformsthatarisefrombasechange.Itassumesthatboth A
f
σ
andA areprincipallypolarizable.Todescribeit,werecallthatby[7,Theorems6.2.4
f
and6.2.6],thereexistnewformsg ,g ∈ S (Γ (D),χ )new suchthatg ,g ,g and
1 2 2 1 D 1 1 2
g formabasisoftheHeckeconstituentofgand
2
w (g )=λ (g )g , w (g )=λ (g )g ,
D 1 D 1 1 D 2 D 2 2
123
1374 L.Dembélé,A.Kumar
where a (g) is the Hecke eigenvalue of g at D and λ (g) = a√D(g), the pseudo-
D D
D
eigenvalueofw .Thematrixofw inthebasis{g ,g ,g ,g }isgivenby
D D 1 1 2 2
⎡ ⎤
0 λ (g ) 0 0
D 1
WD :=⎢⎢⎣λD0(g1) 00 00 λ 0(g )⎥⎥⎦
D 2
0 0 λ (g ) 0
D 2
Fromthis,weseethat Wσ = −W .Thefollowinglemmaisasimpleadaptationof
D D
Cremona’s[8,Lemma5.6.2].
Lemma1 The set of forms h± := 1(g ±w (g )), i = 1,2, are bases for the ±-
i 2 i D i
eigenspacesofW ,actingontheHeckeconstituentofg,whichgiveadecomposition
D
ofthespaceofdifferential1-formsH0(B ⊗ F,(cid:16)1 )accordingtotheaction
g Q Bg⊗QF/F
ofGal(F/Q).
LetH (B ,Z)±denotethe±-eigenspacesofw .TheyarefreeHeckesubmodules
1 g D
of H (B ,Z)ofrank4overZ,whicharedirectsummands.
1 g
Lemma2 LetΛ±betheperiodlatticesobtainedbyintegratingtheformsinLemma1
g
against H (B ,Z)±,andsetΛ = Λ+⊕Λ−.Then,thereexistanabelianfourfold
1 g g g g
B(cid:13) definedoverQ,andanisogenyφ : B(cid:13) → B whosedegreeisapowerof2,such
g g g
that Bg(cid:13)(C)=C4/Λg.Moreover, Bg(cid:13) =ResF/Q(Af)where Af isanabeliansurface
definedover F.
Proof WefirstnotethatthecomplextoriC2/Λ±andC4/Λ havecanonicalRiemann
g g
formsobtainedbyrestrictionoftheintersectionpairing(cid:16)·,·(cid:17)on B .Therefore,they
g
are the complex points of some abelian varieties. Since h+,h+,h−,h− is a basis
1 2 1 2
of the Hecke constituent of g, [41, Theorem 7.14 and Proposition 7.19] imply that
there exist a fourfold B(cid:13) defined over Q, and an isogeny φ : B(cid:13) → B , such that
g g g
B(cid:13)(C)=C4/Λ .
g g
Next,letx ∈ H (B ,Z),thenwehave
1 g
2x =(x +wDx)+(x −wDx)= y++y− ∈ H1(Bg,Z)+⊕H1(Bg,Z)−.
Hencetheexponentof H (B ,Z)+⊕ H (B ,Z)− inside H (B ,Z)divides2.This
1 g 1 g 1 g
impliesthatthedegreeofφisapowerof2.
Sincew isdefinedoverF andwσ =−w ,thebases{h+,h+}and{h−,h−}are
D D D 1 2 1 2
Gal(F/Q)-conjugate.ThereforeC2/Λ+ andC2/Λ− arethecomplexpointsofsome
g g
abeliansurfacesdefinedover F thatareGaloisconjugate.Let A bethesurfacesuch
f
that Af(C)=C2/Λ+g.Then,weseethat Bg(cid:13) =ResF/QAf byconstruction.
(cid:13)
Inpractice,wecanreplace B by B ,andhenceassumethat
g g
H (B ,Z)= H (B ,Z)+⊕H (B ,Z)− = H (A ,Z)⊕H (Aσ,Z).
1 g 1 g 1 g 1 f 1 f
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