Polarizations on abelian subvarieties of principally polarized abelian varieties with dihedral group

24 Pages·2014·0.3 MB·English

Math.Z.(2014)276:397–420 MathematischeZeitschrift
DOI10.1007/s00209-013-1206-1
Polarizations on abelian subvarieties of principally
polarized abelian varieties with dihedral group actions
HerbertLange · RubíE.Rodríguez · AnitaM.Rojas
Received:27September2012/Accepted:1July2013/Publishedonline:3August2013
©Springer-VerlagBerlinHeidelberg2013
Abstract For any n ≥ 2 we study the group algebra decomposition of an ([n] + 1)-
2
dimensionalfamilyofprincipallypolarizedabelianvarietiesofdimensionnwithanaction
ofthedihedralgroupoforder2n.Foranyoddprime p,n = pandn =2pwecomputethe
inducedpolarizationontheisotypicalcomponentsofthesevarietiesandsomeotherdistin-
guishedsubvarieties.Inthecaseofn = pthefamilycontainsaone-dimensionalfamilyof
Jacobians.WeusethistocomputeaperiodmatrixforKlein’sicosahedralcurveofgenus5.
Keywords Principallypolarizedabelianvariety·Groupalgebradecomposition·
Inducedpolarization
MathematicsSubjectClassification(2010) 14K02·14K12·32G20
1 Introduction
Inthenineteenthcenturythedecompositionofanabelianvarietywasexpressedintermsof
reducibleabelianintegralsandtheirthetafunctions.Howevermostauthors,startingperhaps
withAbel,weremainlylooking(inourterminology)forellipticfactorsofJacobianvarieties
(seethelastchapterofKrazer’sbook[7]).Itwasonlyrelativelyrecentlythatfurtherdecom-
ThesecondauthorwassupportedbyFondecytGrant1100767,thethirdauthorbyFondecytGrant1100113.
H.Lange
MathematischesInstitut,UniversitätErlangen-Nürnberg,Erlangen,Germany
e-mail:[email protected]
R.E.Rodríguez
FacultaddeMatemáticas,PontificiaUniversidadCatólicadeChile,Santiago,Chile
e-mail:[email protected]
B
A.M.Rojas( )
DepartamentodeMatemáticasFacultaddeCiencias,UniversidaddeChile,Santiago,Chile
e-mail:[email protected]
123
398 H.Langeetal.
positionswerestudied,mostlyforpolarizedabelianvarietieswithanactionbyafinitegroup.
However,apartfromPrymvarietiesandPrym-Tyurinvarieties,thereareonlyfewexamples
forwhichthepolarizationonthesubvarietyinducedfromthegivenpolarizationwasdeter-
mined.Inthepresentpaperweproposetodothisforafamilyofprincipallypolarizedabelian
varietieswithanactionofthedihedralgroup D oforder2n forn anoddprimeaswellas
n
twiceaprime.
Let(A,L)beapolarizedabelianvarietyandB ⊂ Aanabeliansubvariety.Thepolarization
L on Arestrictstoapolarizationon B whichwecalltheinducedpolarization.Aftersome
preliminariesweoutlineinSect.2amethodtocomputetheinducedpolarization;thatis,to
findanadequatebasisforthelatticedefiningthesubvarietyandtocomputethematrixfor
theinducedpolarizationwithrespecttothisbasis.Moreover,weshowthat,givenanyfinite
groupG andanyfaithfulintegralrepresentationofG ofdegreen,thereexists,foranytype
D = (d ,...,d ),aone-dimensionalfamilyofpolarizedabelianvarietiesofdimensionn
1 n
andtype DsuchthatGactsfaithfullyonitwiththegivenrepresentation.
InSect.3wedefineforanyn ≥ 2,an([n]+1)-dimensionalfamilyA ofprincipally
2 n
polarizedabelianvarietiesofdimensionnwhichadmitanactionofthedihedralgroupD of
n
order2n.Theabelianvarietiesareexplicitelygivenbyperiodmatrices.Thenweconsiderthe
isotypicalandgroupalgebradecompositions(forthedefinitionsseeSect.2.1)anddetermine
theinducedpolarizationsofthecomponents,forn anoddprimeinSect.4,forn twicean
oddprimeinSect.6andforn =4inSect.7.Themethodsofproofareslightlydifferentin
Sects.4and6.Forn = pweusetwoabeliansubvarietieswhichonecandirectlyreadoffthe
periodmatrices,whereasforn =2pweuseabeliansubvarietiesassociatedtosubgroupsof
D whichadmitaprincipalpolarization.
2p
InSect.5westudytheJacobianscontainedinA :weshowthatforanoddprime pthere
p
isexactlyoneirreduciblefamilyofJacobianscontainedinourfamily(Corollary5.3).We
usethistocomputeexplicitlyaperiodmatrixfortheJacobianofKlein’sicosahedralcurve
ofgenus5(Theorem5.8).
Notation LetV/(cid:2)beacomplextorusofdimensiongand{α ,...,α }abasisofthelattice
1 2g
(cid:2).Foranyrational2g×2g-matrixM =(m )weidentifythe j-thcolumnwiththeelement
(cid:2) ij
i2=g1mijαi.Thenwedenoteby(cid:4)M(cid:5)ZthelatticegeneratedbythecolumnsofManddefine
(cid:4)M(cid:5)(cid:2) :=((cid:4)M(cid:5)Z⊗Q)∩(cid:2)
and
(cid:4)M(cid:5)C :=thecomplexvectorspacegeneratedbythecolumnsof M.
2 Preliminaries
Werecallsomenotionsandresultsfromotherpaperswhichweneedinthesequelandadd
someadditionalmaterial.
2.1 Theisotypicaldecomposition
(see[2,Section13.6]).Let A beacomplexabelianvarietyofdimension g withafaithful
actionbyafinitegroupG.TheactioninducesahomomorphismofQ-algebras
ρ :Q[G]→EndQ(A).
123
Abelianvarietieswithdihedralgroupactions 399
WedenoteanelementoftherationalgroupalgebraanditsimagesinEndQ(A)bythesame
letter.
Anyelementα ∈Q[G]definesanabeliansubvariety
Aα :=Im(mα)⊂ A
wherem issomepositiveintegersuchthatmα ∈ End(A).Thisdefinitiondoesnotdepend
onthechosenintegerm.
Let
Q[G]= Q ×···×Q
1 r
denote the decomposition of Q[G] as a product of simple Q-algebras Q . The factors Q
i i
correspondcanonicallytothefinitedimensionalirreduciblerationalrepresentations W of
i
the group G, in the sense that G acts on Q via (a multiple of) W . The corresponding
i i
decompositionof1∈Q[G],
1=e +···+e
1 r
withcentralidempotentse inducesanisogeny
i
Ae1 ×···× Aer → A (2.1)
whichisgivenbyaddition.Notethatthecomponents Aei areG-stablecomplexsubtoriof A
withHomG(Aei,Aej)=0fori (cid:10)= j.
IfW istheirreduciblerationalrepresentationofGcorrespondingtoe ,wealsodenote
i i
AWi := Aei.
The decomposition (2.1) is called the isotypical decomposition of the complex G-abelian
variety A.Theidempotentse aredeterminedasfollows:Letχ bethecharacterofoneof
i i
theirreducibleC-representationsassociatedtoW andK thefield
i i
K =Q(χ (g),g∈G).
i i
Then
ei = de|Ggχ|i (cid:3)trKi|Q(χi(g−1))g. (2.2)
g∈G
Theisotypicalcomponents A canbedecomposedfurther.AccordingtoSchur’sLemma
Wi
D :=End (W )
i G i
isaskew-fieldoffinitedimension
degχ
n = i (2.3)
i
m
i
overQ,wherem denotestheSchurindexofχ (see[4]).Itiseasytoseethatthereisaset
i i
ofprimitiveidempotents{q ,...,q }in Q ⊂Q[G]suchthat
i1 ini i
e =q +···+q .
i i1 ini
Moreover,theabeliansubvarietiesAqij aremutuallyisogenousforfixediand j =1,...,ni.
If B denotesoneofthem,wegetanisogeny
Wi
Bni → A
Wi Wi
foreveryi =1,...,r.Combiningwith(2.1)wegetanisogeny
123
400 H.Langeetal.
Bn1 ×···×Bnr → A, (2.4)
W1 Wr
which is called the group algebra decomposition of the G-abelian variety A. Note that,
whereas(2.1)isuniquelydetermined,(2.4)isnot.Itdependsonthechoiceoftheq aswell
ij
asthechoiceofthe B .
Wi
2.2 Theabeliansubvarietyassociatedtoasubgroup
Anysubgroup H ofGdefinesanidempotentofQ[G]
(cid:3)
1
e = h
H |H|
h∈H
whichinturndefinesanabeliansubvariety
AH := AeH
that we call the abelian subvariety associated to H (see [5]). We also need the following
remarkforwhichwereferto[4,Theorem4.4].
Remark2.1 IfW isarationalirreduciblerepresentationofagroupGsuchthat
i
(cid:4)W ,ρ (cid:5)=1,
i H
where ρ is the representation of G induced by the trivial representation of H, then the
H
idempotent
e e =e e
Wi H H Wi
isprimitiveinQ[G].
2.3 Polarizedabelianvarieties
NowsupposethatalinebundleLon Adefinesapolarizationon A,i.e.(A,L)isapolarized
abelian variety. The polarization induces the Rosati involution (cid:11) on EndQ(A) in the usual
way.Accordingto[2,Theorem5.3.1]thesymmetricidempotents(withrespecttoRosati)
arein1–1correspondencewiththeabeliansubvarietiesofAandaccordingto[2,Proposition
13.6.5]theidempotentse aresymmetricwheneverthegroup G respectsthepolarization,
i
i.e.g∗L isalgebraicallyequivalenttoL foranyg∈G.Moreoverwehave
Proposition2.2 SupposethattheactionofthegroupGon Arespectsthepolarization.For
anysubgroup H ofGtheidempotente issymmetricwithrespecttotheRosatiinvolution.
H
Proof Letφ : A → A(cid:4),a (cid:13)→ t∗L ⊗L−1 denotetheisogenyontothedualabelianvariety
L a
(cid:4)
AassociatedtothelinebundleL.Theassumptionimpliesthat
φL =φg∗L =(cid:4)gφLg
foreveryg∈G.Thisgives
g(cid:11) =φ−1(cid:4)gφ =g−1
L L
andhence
(cid:3) (cid:3)
e(cid:11) = 1 h(cid:11) = 1 h−1 =e .
H |H| |H| H
h∈H h∈H
(cid:14)(cid:15)
123
Abelianvarietieswithdihedralgroupactions 401
Clearly,ifeisasymmetricidempotentofEndQ(A),thensois f =1−e.Thefactthat
e+ f =1impliesforthecorrespondingabeliansubvarietiesthattheadditionmapinduces
anisogeny
Ae× Af → A.
Thesubvariety Af iscalledthecomplementaryabeliansubvarietyof Aewithrespecttothe
polarizationL.Thefactthattheidempotentse ande ofabovearesymmetricwithrespect
i H
toanypolarizationon Aimmediatelyimpliesthefollowingproposition.
Proposition2.3 The complementary abelian subvarieties of the abelian subvarieties A
Wi
and A of AareindependentofthepolarizationL.
H
2.4 Theinducedpolarizationonanabeliansubvariety
Let(A,L)beapolarizedabelianvarietyofdimensiongwithassociatedisogenyφ : A→
L
A(cid:4). Recall that there are uniquely determined positive integers d1,...,dg with di|di+1 for
i =1,...,g−1suchthat
Kerφ (cid:16)(Z/d Z×···×Z/d Z)2.
L 1 g
Thetuple(d ,...,d )iscalledthetypeofthepolarizationL,andtheexponentd ofthegroup
1 g g
Kerφ iscalledtheexponentofthepolarization L.Thepolarizationinducesapolarization
L
L| on every abelian subvariety B of A which we call the induced polarization (without
B
further mentioning the given polarization L). In this section we outline an algorithm to
computethetypeoftheinducedpolarization,developedin[11]forthecaseofJacobians.
Suppose A=V/(cid:2)withV acomplexvectorspaceand(cid:2)alatticeofmaximalrankinV.
ThefirstChernclassofthelinebundleL canbeconsideredasanintegervaluedalternating
formon(cid:2)whoseelementarydivisorsgivethetypeofthepolarization L.Let(d ,...,d )
1 g
bethetypeofthepolarizationL.Asymplecticbasisforthispolarizationisabasisof(cid:2)with
respecttowhichthealternatingformisgivenbythematrix
(cid:5) (cid:6)
0 D
J :=
D −D 0
with D =diag(d ,...,d ).
1 g
Let ρ : G → GL(V) and ρ : G → GL((cid:2)⊗Q) denote the analytic and rational
a r
representationsoftheactionofGonAaswellastheirextensionstoQ[G].Foranyα ∈Q[G]
thesublatticeof(cid:2)definingtheabeliansubvariety Aα isgivenby
(cid:2)α :=(cid:4)ρ (α)(cid:5) .
r (cid:2)
Givenasymplecticbasisof(cid:2),wedenotethematrixofρ (g)withrespecttothisbasisby
r
thesamesymbol.SincetheactionofGrespectsthepolarization,wehave
ρ (g)t ·J ·ρ (g)= J .
r D r D
Thisjustmeansthatρ (g)∈SpD(Z).
r 2g
Nowchooseanybasisofthelattice(cid:2)α.Ifh =dimAα,thenexpressingtheelementsof
thisbasisintermsofthesymplecticbasisof(cid:2),wegeta(2g×2h)-integermatrixPαwhich
definesthecanonicalembedding
iα : Aα (cid:7)→ A.
Withthesenotationswehave,
123
402 H.Langeetal.
Proposition2.4 Supposewearegivenasymplecticbasisof(cid:2),thenforanyα ∈Q[G]the
α
type of the induced polarization on the abelian subvariety A is given by the elementary
divisorsofthealternatingmatrix
Pαt ·JD·Pα.
Proof Theinducedpolarizationon Aα isgivenbythelinebundle L|Aα.Itscorresponding
isogenyφL|Aα : Aα → A(cid:4)α isthecomposition
φL|Aα =(cid:4)iα◦φL ◦iα.
Theproduct Pαt · JD · Pα isjustthematrixversionofthiscompositionwithrespecttothe
chosenbases.Thisgivestheassertion. (cid:14)(cid:15)
α
WesummarizethemethodtocomputetheinducedrepresentationofA inthefollowingfive
steps:
(1) Computetherationalrepresentationρ :G →GL((cid:2)⊗Q).
r
(2) Determine a symplectic basis β of (cid:2). As outlined above, ρ is a representation with
r
valuesinSpD(Z)withrespecttothisbasis.
2g
(3) Determineabasisβα ofthelattice(cid:2)α.Forthistaketherationalvectorspacegenerated
bythecolumnsofρ (α)andintersectitwiththeZ-modulegeneratedbythecolumnsof
r
ρ (1).Theelementsofβα willbegivenaslinearcombinationsoftheelementsofβ.
r
(4) Computetheproduct Pαt · JD·Pα where Pα isthematrixwhosecolumnsarethecoor-
dinatesoftheelementsofβα withrespecttothebasisβ.
(5) ApplytheFrobeniusalgorithm([9,VI.3.Lemma1])tocomputetheelementarydivisors
ofthisalternatingmatrix.
Steps(1)and(2)arecertainlythedifficultpartofthecomputation.Inthenextproposition
weoutlineaclassofabelianvarietiesoftype D withgroupactionwherethiscanbedone.
Inthecaseofaprincipalpolarizationthiswasgivenin[3].DenotebyHn theSiegelupper
half-spaceofdegreen,H:=H1andforanyτ ∈Hby Eτ theellipticcurvedefinedbyτ.
Proposition2.5 LetG beafinitegroupand D = (d ,...,d )atupleofpositiveintegers
1 n
with di|di+1. Given a faithful representation ρ : G → GLn(Z), a G-invariant real inner
product BonZn andanelementτ ∈H,thereisanabelianvariety
A = A (ρ,B,τ)
D D
ofdimensionnandapolarizationLoftypeDonA suchthatGactsfaithfullyon(A ,L).
D D
If BhasrationalvaluesonZn,theabelianvariety AD isisogenousto Eτn.
Proof Recall from [2, Section 8.1] that for any Z ∈ Hn the matrix (D,Z) is the period
matrixofapolarizedabelianvariety(A,H)oftypeDanddimensionn,wherethehermitian
form H isgivenbythematrix(ImZ)−1withrespecttothecanonicalbasisofCn.Infact,
(cid:5) (cid:6)
D 0
A=Cn/(cid:2) with (cid:2) := Z2n.
D D 0 Z
Thegroup
(cid:7) (cid:5) (cid:6) (cid:8)
α β
G := M = ∈Sp (Q)| Mt(cid:2) ⊂(cid:2)
D γ δ 2n D D
actsonHn by M(Z)=(α+Zγ)−1(β+Zδ).Inparticular M definesanautomorphismof
(A,H)ifandonlyifM(Z)= Z.
123
Abelianvarietieswithdihedralgroupactions 403
For(ρ,B,τ)asinthepropositionwedefine A = A (ρ,B,τ)bytheperiodmatrix
D D
(cid:12) :=(D,τB−1).
Clearlythematrix (cid:5) (cid:6)
ρ(g) 0
M(g):= 0 (ρ(g)t)−1
iscontainedinG foranyg∈G.TheG-invarianceof Bimplies
D
ρ(g)−1B−1(ρ(g)t)−1 = B−1.
Thisgives
M(g)(τB−1)=ρ(g)−1τB−1((ρ(g)t)−1)=τB−1
for all g ∈ G. Hence (A , 1 B) is a polarized abelian variety of type D. Since G acts
D Imτ
faithfully on the lattice (cid:2) , it acts faithfully on the tangent space of A and thus on A
D D D
itself.Theactionrespectsthepolarization,since BisG-invariant.
Forthelastassertionnotethat
(cid:5) (cid:6)
(D,τB−1) D−1 0 =(1 ,τ1 ).
0 B n n
If B ∈GLn(Q),chooseapositiveintegerm(cid:5)suchthat(cid:6)mB andmD−1areintegralmatrices.
D−1 0
Thentheaboveequationimpliesthatm · givesanisogenybetween A and
0 B D
Eτn. (cid:14)(cid:15)
ThefollowingdirectconsequenceofProposition2.5isperhapsworthmentioning.
Corollary2.6 For any finite group G and any faithful integral representation ρ of G of
degree n, there is a one-dimensional family of polarized abelian varieties of dimension n
ofanygiventype DsuchthatG actsfaithfullyoneachelementofthefamily,withanalytic
representationdeterminedbyρ.
3 Abelianvarietiesofdimensionnwitha D -action
n
ConsidertheRiemannmatrices Z ofthefollowingform:Forn =2m−1, m ≥2:
⎛ ⎞
z1 z2 ... zm zm zm−1 ... z2
⎜⎜ z1 z2 ... zm zm zm−1 ... z3⎟⎟
Z =⎜⎜⎜ ... ... ... ⎟⎟⎟, (3.1)
⎝ ⎠
z z
1 2
z
1
andforn =2m, m ≥1:
⎛ ⎞
z1 z2 ... zm+1 zm zm−1 ... z2
⎜⎜ z1 z2 ... zm+1 zm zm−1 ... z3⎟⎟
Z =⎜⎜⎜ ... ... ... ⎟⎟⎟. (3.2)
⎝ ⎠
z z
1 2
z
1
Inbothcases Z issymmetricandsymmetricwithrespecttotheantidiagonal.
123
404 H.Langeetal.
Proposition3.1 Theprincipallypolarizedabelianvarieties A = A(Z)withperiodmatrix
(1 ,Z)formanm-dimensional,respectively(m+1)-dimensional,familyA forn =2m−1,
n n
respectivelyn =2m.
Proof Supposen =2m−1.Theperiodmatricesoftheform(1 ,Z)formanopensetinthe
n
m-dimensionalcomplexvectorspaceCm withcoordinatefunctionsz ,...,z .Thissetis
1 m
non-empty,sincethematriceswithz =···=z =0andwithz suchthattheimaginary
2 m 1
part(cid:18)(z )ispositivearecontainedinit.Theproofforn =2missimilar. (cid:14)(cid:15)
1
Letn ≥2beaninteger,andconsiderthen×nintegralmatrices RandSgivenby
⎛ ⎞
0 1 0 ... 0
⎜ ⎟
⎜0 0 1 0 0⎟
R =⎜⎜⎜... 0 ... 0⎟⎟⎟. (3.3)
⎝ ⎠
0 0 0 0 1
1 0 0 ... 0
⎛ ⎞
0 0 0 0 1
⎜⎜0 0 ... 1 0⎟⎟
⎜. . ⎟
S =⎜⎜⎝.. 0 ... .. 0⎟⎟⎠. (3.4)
0 1 0 0
1 0 0 0 0
andobservethat
Rn =1,S2 =1,(RS)2 =1;
thatis,thegroupgeneratedby RandSisthedihedralgroup D oforder2n.
n
Proposition3.2 TheabelianvarietiesA∈A admitanactionofthegroupD withanalytic
n n
representationgivenby(3.3)and(3.4).
Proof Definetherationalrepresentationρ : D →Sp(2n,Z)ofthegroupD =(cid:4)R,S(cid:5)by
r n n
(cid:5) (cid:6) (cid:5) (cid:6)
R 0 S 0
ρ (R)= and ρ (S)= . (3.5)
r 0 R r 0 S
NotethatthematricesarecontainedinSp(2n,Z),sincethetransposeinverseofRcoincides
with R:(Rt)−1 = RandsimilarlyforS.WehavetocheckforM = RandSthat
(cid:5) (cid:6)
M 0
M(1 ,Z)=(1 ,Z) (3.6)
n n 0 M
whichisequivalenttoMZ = ZM.Butthisisaneasycomputation. (cid:14)(cid:15)
Remark3.3 Ifwerequiretherationalrepresentationtobegivenby(3.5),thentheformof
the matrices Z is completely determined by the subgroup (cid:4)R(cid:5); that is, one can show that
thesearealltheRiemannmatricesofsizensatisfying(3.6)withM = R,andthenthatthey
alsosatisfy(3.6)withM = S.
123
Abelianvarietieswithdihedralgroupactions 405
4 Actionof D for panoddprime
p
4.1 Notationandinducedpolarizationon A and A
0 1
Letn = p=2m−1beanoddprime,anddenote
G = D =(cid:4)r,s :rp =s2 =(rs)2 =1(cid:5).
p
ObservethatGhasthreeirreduciblerationalrepresentations:thetrivialoneW ,anotherone
0
ofdegree1,andW ofdegree p−1,whichoverCdecomposesasthesumofthedegreetwo
1
representationsofGgivenby
(cid:15) (cid:16) (cid:5) (cid:6)
wj 0 0 1
Vj :r → 0p w−j , s → 1 0
p
wherew denotesaprimitivep-throotofunityand1≤ j ≤ p−1.Notethattherepresentation
p 2
ofGgivenby RandSisequivalent(overQ)toW ⊕W .
0 1
Thecorrespondingcentralidempotentsaregivenby
⎛ ⎞
(cid:3) (cid:3)p−1
e = 1 g and e = 1 ⎝(p−1)1 − rj⎠ (4.1)
0 1 G
2p p
g∈G j=1
Notethate isprimitiveinQ[G],bute isnot,beingthesumoftwo(notuniquelydetermined)
0 1
primitiveidempotentsinQ[G](by(2.3)).Soby(2.4)wehaveisogenies
A∼ A × A ∼ A ×B2
0 1 0 1
withanellipticcurve A andabeliansubvarieties A and B of Aofdimensions p−1and
0 1 1
p−21, respectively. Note that if A = V/(cid:2), then Aj = Vj/(cid:2)j, with Vj = (cid:4)ρr(ej)(cid:5)C and
(cid:2)j =(cid:4)ρr(ej)(cid:5)(cid:2)(seethenotationinSect.1).
Wewillnowexplicitlycompute A and A ,i.e.theirlatticesandinducedpolarizations.
0 1
Let {α ,...,α ,β ,...,β } denote the symplectic basis of (cid:2) with respect to which the
1 p 1 p
matricesρ (R)andρ (S)aregiven.
r r
Proposition4.1 Theabeliansubvariety A associatedtoW isoftype(p),anditslattice
0 0
(cid:2) isgivenby
0
(cid:2) =(cid:4)α +α +···+α ,β +β +···+β (cid:5) .
0 1 2 p 1 2 p Z
Proof ThesymmetricidempotentassociatedtoW ise .Thisgivestheassertionconcerning
0 0
thebasisof(cid:2) .Sowithrespecttothebasis{α ,...,β }theembedding A (cid:7)→ Aisgiven
0 1 p 0
bythematrix P with
0
(cid:5) (cid:6)
1 ··· 1 0 ···0
Pt = .
0 0 ··· 0 1 ··· 1
Thisimpliestheassertion,since
(cid:5) (cid:6) (cid:5) (cid:6)
0 1 0 p
Pt p P = .
0 −1 0 0 −p 0
p
(cid:14)(cid:15)
Proposition4.2 Theabeliansubvariety A of AassociatedtoW isoftype(1,...,1,p).
1 1
Proof A andA arecomplementaryabeliansubvarietiesintheprincipallypolarizedabelian
0 1
variety(A,L).SotheassertionfollowsfromProposition4.1and[2,Corollary12.1.5]. (cid:14)(cid:15)
123
406 H.Langeetal.
4.2 Twoabeliansubvarietiesandthelatticeof A
1
Accordingto(4.1)andthefactthatρ (R)isgivenby(3.3)wehave
a
⎛ ⎞
p−1 −1 ... −1
⎜⎜ −1 p−1 −1 ... −1 ⎟⎟
⎜⎜⎜⎜ ... 0p×p ⎟⎟⎟⎟
⎜ ⎟
⎜ ⎟
1 ⎜ −1 ... p−1 ⎟
ρr(e1)= p ⎜⎜⎜ p−1 −1 ... −1 ⎟⎟⎟
⎜ −1 p−1 −1 ... −1 ⎟
⎜ ⎟
⎜⎜⎜ 0p×p ... ⎟⎟⎟
⎝ ⎠
−1 ... p−1
Denotebyc the j-thcolumnofρ (e ),identifiedwiththecorrespondingelementof(cid:2)⊗Q.
j r 1
Forinstance,
p−1 1 1
c = α − α −···− α .
1 1 2 p
p p p
Then
(cid:2)1 =(cid:4)ρr(e1)(cid:5)(cid:2) =(cid:4)c1,...,c2p(cid:5)(cid:2).
Lemma4.3
(cid:2) =(cid:4)α −α ,α −α ,...,α −α ,β −β ,β −β ,...,β −β (cid:5) .
1 1 p 2 p p−1 p 1 p 2 p p−1 p Z
Moreover,theelementsoftherighthandsideformabasisof(cid:2) .
1
Notethat(cid:2) isnotthesameas
1
(cid:4)pc1,...,pc2p(cid:5)Z
sinceforinstancec −c =α −α belongsto(cid:2) butNOTtothelastlattice.
i j i j 1
Proof Firstnotethattheright-handsideisasublatticeof(cid:2) ,ofthesamerankas(cid:2) ,since
1 1
α −α =c −c for 1≤ j,k ≤ p,
j k j k
βj −βk =cp+j −cp+k for 1≤ j,k ≤ p.
Henceitsufficestoshowthattheintersectionmatrixfortheproposedbasishasdeterminant
p2,andthereforethetwolatticescoincide.Theintersectionproductfortheproposedbasis
isgivenby
(cid:4)α −α ,β −β (cid:5)=δ +1.
j p k p jk
Thereforetheintersectionmatrixhastheform
(cid:5) (cid:6)
0 B
−B 0
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