ABELIAN VARIETIES ASSOCIATED TO GAUSSIANLATTICES ARNAUDBEAUVILLE 2 1 ABSTRACT. Weassociatetoaunimodularlattice Γ,endowedwithanautomorphismofsquare 0 −1, a principally polarized abelian variety AΓ = ΓR/Γ. We show that the configuration of 2 i-invariantthetadivisorsofAΓ followsapatternverysimilartotheclassicaltheoryofthetachar- n acteristics;asaconsequencewefindthat AΓ hasahighnumberofvanishingthetanulls. When a Γ=E8 werecoverthe10vanishingthetanullsoftheabelianfourfolddiscoveredbyR.Varley. J 0 2 INTRODUCTION ] G AGaussianlatticeisafree,finitelygenerated Z[i]-module Γ withapositivehermitianform A Γ×Γ→Z[i]. Equivalently,wecanview Γ asalatticeover Z endowedwithanautomorphism . th i ofsquare −1Γ. Thisgivesacomplexstructureonthevectorspace ΓR :=Γ⊗ZR;weassociate a m to Γ thecomplextorus AΓ :=ΓR/Γ. Asacomplex torus A isisomorphic to Eg, where E isthe complexelliptic curve C/Z[i] [ Γ 2 and g = 12rkZΓ. Moreinterestingly,thehermitianformprovidesapolarizationon AΓ (see(1.3) v below); inparticular,if Γ isunimodular, A isaprincipallypolarizedabelianvariety(p.p.a.v. 3 forshort),whichisindecomposableif Γ isindecomposable. 4 8 The first non-trivial case is g = 4, with Γ the root lattice of type E8 (Example1.2.1). The 2 resulting p.p.a.v. is the abelian fourfold discovered by Varley [V] with a different(and more . 2 geometric)description; ithas10“vanishingthetanulls” (eventhetafunctions vanishingat0), 1 1 themaximumpossiblefora4-dimensionalindecomposablep.p.a.v. Infactthispropertychar- 1 acterizestheVarleyfourfoldoutsidethehyperellipticJacobianlocus[D]. : v Our aim is to explain this property from the lattice point of view, and to extend it to all i X unimodularlattices. Itturnsoutthatwecanmimictheclassicaltheoryofthetacharacteristics, r replacingtheautomorphism (−1) by i. Wewillshow: a • Thegroup A of i-invariantpointsof A isa F -vectorspaceofdimension g;itadmitsa i Γ 2 naturalnon-degeneratebilinearsymmetricform b. • The set of i-invarianttheta divisors of A is anaffine space over A , isomorphic to the Γ i spaceofquadraticformson A associatedto b (see(2.1)). i • Let Θ be an i-invariant theta divisor, and Q the corresponding quadratic form. The multiplicity m (Θ) of Θ at 0 satisfies 0 2m (Θ)≡σ(Q)+g (mod.8), 0 where σ istheBrowninvariantoftheform Q (2.1). Date:January23,2012. 1 2 ARNAUDBEAUVILLE As a consequence, we obtain a high number of i-invariant divisors Θ with m (Θ) ≡ 2 0 (mod.4); eachofthemcorrespondstoavanishingthetanull. When Γ iseven,thisnumberis 22g−1(2g2 −(−1)g4);for g =4 werecoverthe10vanishingthetanullsoftheVarleyfourfold. 1. GAUSSIANLATTICES 1.1. Lattices. As recalled in the Introduction, a Gaussian lattice is a free finitely generated Z[i]-module Γ endowedwithapositivehermitianform1 H :Γ×Γ→Z[i]. Wewrite H(x,y)= S(x,y)+iE(x,y); S and E are Z-bilinearformson Γ, S issymmetric, E isskew-symmetric, andwehave S(ix,iy)=S(x,y) , E(ix,iy)=E(x,y) , E(x,y)=S(ix,y). WewillratherviewaGaussianlatticeasanordinarylattice(over Z)withanautomorphism i suchthat i2 =−1 : thelastformulaabovedefines E,andwehave H =S+iE. Γ Wehave detS = detE = (detH)2; thelatticeisunimodularwhenthesenumbersareequal to 1. It is even if S(x,x) is even for all x ∈ Γ. We say that Γ is indecomposable over Z[i] if it cannotbewrittenastheorthogonalsumoftwononzeroGaussianlattices;thisisofcoursethe caseif Γ isindecomposableover Z,buttheconverseisfalse(Example3below). 1.2. Examples. 1)For g even,thelattice Γ is 2g 1 Γ :={(x )∈R2g |x ∈ Z, x −x ∈Z, x ∈2Z}. 2g j j j k j 2 X Theinnerproductisinheritedfromtheeuclideanstructureof R2g,andtheautomorphism i is giveninthestandardbasis (e ) by j ie2j−1 =e2j ie2j =−e2j−1 for 1≤j ≤g. Thelattice Γ isunimodular,indecomposablewhen g >2,andevenif g isdivisibleby4.The 2g firstcase g =4 givestherootlattice E . 8 Theautomorphism i isuniqueuptoconjugacy: for g =4 thisisclassical[C],andfor g ≥ 6 this follows easily from the fact that Aut(Γ ) is the semi-direct product (Z/2)2g−1 ⋊ S , 2g 2g actingbypermutationandevenchangesofsignofthebasisvectors (e ). j 2) The Leech lattice Λ admits an automorphism of square −1 [C-S], also unique up to 24 conjugacy. 3)Let Γ0 bealattice,and Γ:=Γ0⊗ZZ[i]. Theinnerproductof Γ0 extendstoanhermitian inner product on Γ, which is then a gaussian lattice. If Γ is unimodular, resp. even, resp. 0 indecomposable, Γ isunimodular,resp.even,resp.indecomposableover Z[i]. 1OurconventionisthatH(x,y) isC-linearin y. ABELIANVARIETIESASSOCIATEDTOGAUSSIANLATTICES 3 1.3. TheabelianvarietyAΓ. LetΓbeaGaussianlattice,ofrank2g overZ. WeputΓR :=Γ⊗ZR and AΓ := ΓR/Γ. The automorphism i defines a complex structure on ΓR, so that AΓ is a complextorus. Since Γ isafree Z[i]-module, A isisomorphicto Eg,where E isthecomplex Γ ellipticcurve C/Z[i]. ThepositivehermitianformH extendstoΓR,anditsimaginarypartE takesintegralvalues on Γ: thisisbydefinitionapolarizationon A . Thepolarizationisprincipalifandonlyif Γ is Γ unimodular;thep.p.a.v. A isindecomposable(i.e. isnotaproductoftwonontrivialp.p.a.v.) Γ ifandonlyif Γ isindecomposableover Z[i]. The multiplication by i on ΓR inducesanautomorphism of AΓ, that we simply denote i. Conversely,let A=V/Γ beacomplextorus,ofdimension g,withanautomorphisminducing on T (A) = V the multiplication by i. Then Γ is a Z[i]-module, thus isomorphic to Z[i]g, 0 sothat A isisomorphicto Eg;polarizationsof A correspondbijectivelytopositivehermitian formson Γ. 2. LINEARALGEBRAOVER F2[i] 2.1. Linearalgebra over F . We consider a vector space V over F , of dimension g, with a 2 2 non-degeneratesymmetricbilinearform b on V . Twodifferentsituationsmayoccur: • b(x,x)=0 forall x∈V ;inthatcase b isasymplecticform. • b(x,x) isnotidenticallyzero;itistheneasy(usinginductionon g)toprovethat V admits anorthonormalbasiswithrespectto b. Aquadraticformassociatedto b isafunction q :V →Z/4 suchthat q(x+y)=q(x)+q(y)+2b(x,y) forx,y ∈V , wheremultiplicationby2standsfortheisomorphism Z/2 −∼→ 2Z/4Z⊂Z/4Z. Observethatthisimplies q(0)=0 and q(x)≡b(x,x)(mod.2). Wedenoteby Q thesetof b quadraticformsassociatedto b;itisanaffinespaceover V ,theactionof V on Q beinggiven b by (α+q)(x)=q(x)+2b(α,x) for q ∈Q , α,x∈V . b Whenbissymplectic, q takesitvaluesin2Z/4Z∼=Z/2;thecorrespondingformq′ :V →Z/2 isaquadraticformassociatedto b intheusualsense,thatissatisfies q′(x+y)=q′(x)+q′(y)+ b(x,y) for x,y ∈V . TheBrowninvariant σ(q)∈Z/8 ofaform q ∈Q hasbeenintroducedin[B]asageneraliza- b tionoftheArfinvariant;itcanbedefinedasfollows.Ifbissymplectic,weputσ(q):=4Arf(q′), where q′ : V → Z/2 is the form defined above. Otherwise b admits an orthonormal basis (e ,...,e ); we have q(e ) = ±1,andwelet g+ (resp. g−)bethenumber ofbasisvectors e 1 g i i suchthat q(e )=1 (resp. −1).Then σ(q)=g+−g−(mod.8). i 2.2. Linearalgebraover F2[i]. Let Γ beaunimodularGaussianlatticeofrank 2g over Z. We put A := Γ/2Γ; this is naturallyidentified with the 2-torsion subgroup of A . We have the 2 Γ followingstructureson A : 2 4 ARNAUDBEAUVILLE a) A isafree F [i]-moduleofrank g. Weput ε := 1+i in F [i];then F [i] = F [ε],with 2 2 2 2 2 ε2 = 0. The subgroup A of i-invariant elements is Kerε = εA ; it is a F -vector space of i 2 2 dimension g. b)TheformE inducesonA asymplecticforme(theWeilpairingforA ).SinceE(x,iy)= 2 Γ −E(ix,y),wehave,for α,β ∈A , 2 e(α,εβ)=e(εα,β) hence e(εα,εβ)=0; thus A isaLagrangiansubspaceof A . i 2 c) Theform x7→S(x,x) inducesaquadraticform Q: A →Z/4 associatedwiththebilin- 2 earsymmetricform (α,β)7→e(α,iβ) (2.1).Inparticularwehave Q(α)≡e(α,iα)(mod.2). Since S((1+i)x,(1+i)x)=2S(x,x),wehave Q(εα)=2Q(α)=2e(α,iα). Lemma1. Let q :A →Z/4 bean i-invariantquadraticformassociatedto e. Theformulas 2 b(εα,εβ)=e(α,εβ) , Q (εα)=q(α)−Q(α) for α,β ∈A , q 2 define on A = εA a non-degenerate symmetric form b and a quadratic form Q : A → Z/4 i 2 q i associatedwith b. Proof : Since A =Kerε isisotropicfor e,theexpression e(α,εβ) isabilinearfunction b of εα i and εβ; itissymmetric by b). If e(α,εβ) = 0 forall β in A we have α ∈ A because A is 2 i i Lagrangian,hence εα=0,so b isnon-degenerate. Put Q˜ (α)=q(α)−Q(α)∈Z/4 for α∈A . Wehave q 2 Q˜ (α+β)=Q˜ (α)+Q˜ (β)+2e(α,εβ). q q q Take β =εγ.Since q is i-invariantwehave q(εγ)=2e(γ,iγ)=Q(εγ)byc),hence Q˜ (εγ)=0 q and Q˜ (α+εγ)=Q˜ (α). Thus Q˜ definesaquadraticform Q on A associatedto b. q q q q i Let Q(ei) be the set of i-invariantsquadratic forms on A2 associated to e. If q ∈ Qe(i) and α ∈ A2,wehave α+q ∈ Q(ei) ifandonlyif α belongsto A⊥i = Ai;inotherwords, Qe(i) isan affinesubspaceof Q ,withdirection A . e i Lemma2. Themap q 7→Qq isanaffineisomorphismof Qe(i) onto Qb. Proof : Wejusthavetoprovetheequality Qα+q =α+Qq for q ∈Qe(i), α∈Ai. Let β ∈Ai;we write β =εβ′ forsome β′ ∈A . Then 2 ′ ′ ′ Q (β)=2e(α,β )+q(β )−Q(β )=2b(α,β)+Q (β). α+q q Remark1. Let α∈A ;wehave b(εα,εα)=e(α,εα)=e(α,iα)≡Q(α)(mod.2),hencetheform 2 b issymplecticifandonlyif Γ iseven. Inthiscasewehave e(α,iα)=0 forall α∈A ;itfollows 2 that Q(ei) is the set of forms vanishing on Ai. Since Ai is Lagrangianfor e, this implies that theseforms, viewedasquadraticforms A → Z/2, arealleven(thatis, theirArfinvariantis 2 0). ABELIANVARIETIESASSOCIATEDTOGAUSSIANLATTICES 5 3. i-INVARIANT THETADIVISORS 3.1. Reminderonthetacharacteristics. Wefirstrecalltheclassicaltheoryofthetacharacteris- ticsonanarbitraryp.p.a.v. A = V/Γ. Let A ∼= Γ/2Γ bethe2-torsionsubgroupof A, T the 2 setof symmetric theta divisors on A, and Q the set of quadraticforms on A associated to e 2 theWeylpairing e. The F -vectorspace A actson T bytranslation,andon Q bytheaction 2 2 e definedin(2.1);bothsetsareaffinespacesover A ,andthereisacanonicalaffineisomorphism 2 q 7→Θ of Q onto T . Itcanbedefinedasfollows([M],§2). Let γ ∈Γ,andlet γ¯ beitsclassin q e A . For z ∈V ,weput 2 eγ(z)=iq(γ¯)eπH(γ,z+γ2) . We define an action of Γ on the trivial bundle V ×C by γ.(z,t) = (z +γ,e (z)t); then the γ quotientof V ×C bythisactionisthelinebundle O (Θ ) on A. A q 3.2. The main results. We go back to the abelianvariety A associated to a Gaussian lattice Γ Γ. Weassumethat Γ isunimodular. Weusethenotationof(2.2). Theisomorphism Q −∼→ T e iscompatiblewiththeactionof i,so i-invariantthetadivisorscorrespondtoforms q ∈Qe(i). Let q ∈ Q(ei), and let L be the line bundle OAΓ(Θq). We have i∗L ∼= L; we denote by ι : i∗L → L theuniqueisomorphisminducingtheidentityof L . Foreach α ∈ A , ι induces 0 i anisomorphism ι(α):L →L . α α Proposition1. ι(α) isthehomothetyofratio iQq(α). Proof : Theisomorphism ι−1 : L −∼→ i∗L correspondstoalinearautomorphism j of L above i: j // L L (cid:15)(cid:15) (cid:15)(cid:15) i // A A . Γ Γ Consider the automorphism ˜j : (z,t) 7→ (iz,t) of ΓR × C. Since eiγ(iz) = eγ(z), we have ˜j(γ.(z,t)) = (iγ).˜j(z,t). Thus ˜j factorsthroughanisomorphism L → L above i whichisthe identityon L ,henceequalto j;thatis,wehaveacommutativediagram: 0 ΓR×C ˜j // ΓR×C π π (cid:15)(cid:15) (cid:15)(cid:15) j // L L where π isthequotientmap. Let α ∈ A , and let γ be an element of Γ whose class (mod. 2Γ) is α. Then δ := iγ − γ i 2 2 belongsto Γ. Wehave γ iγ γ γ j(π( ,t))=π( ,t)=π( ,e ( )−1t), δ 2 2 2 2 6 ARNAUDBEAUVILLE hence ι(α) = j(α)−1 is the homothety of ratio e (γ). Let β be the class of δ in A . Since δ 2 2 γ =−(1+i)δ,wehave α=εβ,hence γ eδ( )=iq(β)eπ2H(δ,γ+δ) =iq(β)−H(δ,δ) =iQq(α) . 2 From ι : i∗L → L we deduceanisomorphism ι♭ : L −∼→ i∗L, inducing onglobalsections anautomorphismof H0(A ,L). Γ Proposition2. ι♭ actson H0(AΓ,L) bymultiplicationby ei4π(σ(Qq)+g). Notethat σ(Q )≡g(mod.2) ([B],Thm. 1.20,(vi)),sothisnumberisapowerof i. q Proof : Since dimH0(A ,L)= 1 itsufficestocompute Trι♭. Thisisgivenbytheholomorphic Γ Lefschetzformula[A-B]appliedto (i,ι). Since Hi(A ,L)=0 for i>0,wefind Γ Trι(α) Trι♭ = =(1−i)−g iQq(α). (1−i)g αX∈Ai αX∈Ai We have (1−i)−g = 2−g2eiπ4g and α∈AiiQq(α) = 2g2ei4πσ(Qq) ([B],Thm. 1.20,(xi)),hence theresult. P Proposition3. Let α∈A ,andlet m (Θ ) bethemultiplicityof Θ at α. Wehave i α q q 2m (Θ )≡σ(Q )+g−2Q (α) (mod.8). α q q q Proof : Let θ beanonzerosectionof H0(A ,L). Choose alocalnon-vanishing section s of L Γ around α. Wecanwrite θ =fs inaneighborhoodof α,with f ∈O . Wehave ι♭(θ)=ikθ AΓ,α with 2k≡σ(Q )+g(mod.8) (Proposition2),hence q (i∗f)ι♭(s)=ikfs. Welookatthisequalityin mmL/mm+1L,where m isthe maximalidealof O and m := α α α AΓ,α m (Θ).Wehavei∗f =imf (mod.mm+1),andι♭(s)=ι(α)s(mod.m L).Weobtainimι(α)=ik, α α α hencetheresultinviewofProposition1. Corollary. Thenumberof i-invariantthetadivisors Θ with m (Θ)≡2(mod.4) is 0 2g2−1(2g2 −(−1)g4) if Γ iseven, and 2g−2−22g−1cosπg if Γ isodd; 4 eachofthesedivisorscorrespondstoavanishingthetanull. Proof :AccordingtotheProposition,wehave m (Θ )≡2 (mod.4) ifandonlyif σ(Q )≡4−g 0 q q (i) (mod.8). When q runsover Qe , Qq runsover Qb (Lemma2.2),sowemustfindhowmany elements Q of Q satisfy σ(Q)≡4−g (mod.8). b If Γ is even (so that g is divisible by 4), we identify Q with the set of quadratic forms b Q : A → Z/2 associated with the symplectic form b; the previous congruence becomes 2 Arf(Q) ≡ 1 + g (mod. 2). There are 2g2−1(2g2 + 1) such forms with Arf invariant 0 and 4 2g2−1(2g2 −1) withArfinvariant1,hencetheresult. ABELIANVARIETIESASSOCIATEDTOGAUSSIANLATTICES 7 Assumethat Γ isodd;wechooseanorthonormalbasis (e ,...,e ) for b.Theforms Q∈Q 1 g b aredeterminedbytheirvalues Q(e )= ±1;theconditionisthatthenumber g+ of +1 values i satisfies 2g+−g ≡4−g(mod.8), hence g+ ≡2(mod.4). Thenumberofformswiththerequiredpropertyisthusthenumberofsubsets E ⊂ {1,...,g} with Card(E)≡2(mod.4),thatis g + g +...= 1 (1+1)g+(1−1)g−(1+i)g−(1−i)g =2g−2−22g−1cosπg . (cid:18)2(cid:19) (cid:18)6(cid:19) 4 4 (cid:2) (cid:3) Thuswe find anumber of vanishingthetanullsasymptotically equivalentto 2g−1 when Γ is even, and 2g−2 when Γ is odd. These numbers are rather modest, at least by comparison with the number of vanishing thetanulls of a hyperelliptic Jacobian, which is asymptotically equivalentto 22g−1. However, when Γ iseven, the vanishingthetanullsof A havethe par- Γ ticular property of being “syzygetic” in the classical terminology, which just means that the corresponding quadratic forms (3.1) lie in an affine subspace of Q which consists of even e forms(Remark1). Suchasubspacehasdimension ≤g,anditmightbethatthenumbergiven bytheCorollaryintheevencaseisthemaximumpossibleforasyzygeticsubsetofvanishing thetanulls. 4. COMPLEMENTS 4.1. Automorphisms. Theautomorphismgroupof A isthecentralizerof i in Aut(Γ). This Γ groupcanberatherlarge:ithasorder46080for Γ=E and2012774400for Γ=Λ [C-S]. For 8 24 thelattice Γ (Example1.2.1)with g >4,ithasorder 22g−1g!. 2g For the lattice Γ = Γ0 ⊗Z Z[i] of Example 1.2.3, Aut(AΓ) is generated by i and the group Aut(Γ ). Notethatthereareexamplesofunimodularlattices(evenorodd) Γ with Aut(Γ )= 0 0 0 {±1} [Ba],sothat Aut(A ) isreducedto {±1,±i}. Γ 4.2. Jacobians. We observe that for g > 1 the p.p.a.v. A can not be a Jacobian. Indeed, let Γ C be a curve of genus g; if JC ∼= A , Torelli theorem provides an automorphism u of C Γ inducingeither i or −i on JC,hencealsoon T (JC)= H0(C,K )∗. Then u actstriviallyon 0 C theimageofthecanonicalmap C →P(H0(C,K )∗);thisimpliesthat u istheidentityorthat C C ishyperellipticand u isthehyperellipticinvolution. Butinthesecases u actson H0(C,K ) C bymultiplicationby ±1,acontradiction. REFERENCES [A-B] M.Atiyah,R.Bott:ALefschetzFixedPointFormulaforEllipticComplexes:II.Applications.Ann.ofMath.(2) 88(1968),451–491. [Ba] R.Bacher:Unimodularlatticeswithoutnontrivialautomorphisms.Int.Math.Res.Notes2(1994),91–95. [B] E.H.Brown,Jr.:GeneralizationsoftheKervaireinvariant.Ann.ofMath.(2)95(1972),368–383. [C] R.W.Carter:ConjugacyclassesintheWeylgroup.CompositioMath.25(1972),1–59. 8 ARNAUDBEAUVILLE [C-S] J.H.Conway,N.Sloane: D4,E8,Leechandcertainotherlatticesaresymplectic.Invent.Math.117(1994),no.1, 53–55. [D] O.Debarre: Annulationdetheˆtaconstantessurlesvarie´te´sabe´liennesdedimensionquatre.C.R.Acad.Sci. ParisSe´r.IMath.305(1987),no.20,885–888. [M] D.Mumford:Abelianvarieties.OxfordUniversityPress,London,1970. [V] R.Varley: Weddle’ssurfaces,Humbert’scurves,andacertain4-dimensionalabelianvariety.Amer.J.Math. 108(1986),no.4,931–952. LABORATOIREJ.-A.DIEUDONNE´,UMR7351DUCNRS,UNIVERSITE´DENICE,PARCVALROSE,F-06108NICE CEDEX2,FRANCE E-mailaddress:[email protected]