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COMPUTINGLOW-DEGREEISOGENIESINGENUS2 WITHTHEDOLGACHEV–LEHAVIMETHOD BENJAMINSMITH ABSTRACT. Letℓbeaprime,andHacurveofgenus2overafieldkofcharacteristicnot 2 2orℓ.IfSisamaximalWeil-isotropicsubgroupofJH[ℓ],thenJH/Sisisomorphicto 1 theJacobianJX ofsome(possiblyreducible)curveX.WeinvestigatetheDolgachev– 0 LehavimethodforconstructingthecurveX,simplifyingtheirapproachandmaking 2 itmoreexplicit. Theresult,atleastforℓ=3,isanefficientandeasilyprogrammable algorithmsuitablefornumber-theoreticcalculations. n a J 6 1. INTRODUCTION 2 Letℓ 3beprime,andletH beacurveofgenus2overaperfectfieldkofcharacter- ] ≥ T isticnot2orℓ. LetJH betheJacobianofH,andletSbeamaximalℓ-Weilisotropic N subgroupofJH[ℓ]; sinceℓisprime,S ∼(Z/ℓZ)2. ThequotientJH/S isisomorphic = (asaprincipallypolarizedabelianvariety)toaJacobianJ ,whereX issomecurveof . X h genus2(see[14]);hence,thereexistsanisogeny t a φ:J J m H → X withkernelS (notethatX maybereducible,inwhichcaseJ isaproductofelliptic [ X curves).OuraimistocomputeanexplicitformforX givenH andS. 2 Inthecaseℓ 2,theproblemisresolvedbythewell-knownRichelotconstruction v (see[3]and[5,C=hapter9]). Moregenerally,ifkisfinite,thenwecanapplytheexplicit 3 thetafunction-basedalgorithmsofLubiczandRobert[10],implementedinthefreely- 6 9 availableavIsogeniespackage[1]. 2 Alternatively,thereisthealgebraic-geometricapproachdescribedbyDolgachevand . Lehavi[7],whichcomputestheimageofthethetadivisoronJ intheKummersurface 0 H 1 ofJX.Aspresentedin[7],thisapproachhastwodrawbacks: 1 (1) itisnoteffectiveforℓ 3,and 1 (2) forℓ 3,wherethetas6=tructuresareinvolved,itassumesk C. : = ⊂ v InthisworkwerenderthekerneloftheDolgachev–Lehavi method completely ex- i X plicit,withaviewtocomputationsinnumbertheory.Ourintentionistoprovideasort of“user’sguide”tothealgorithmanditsconcreteimplementation. Forℓ 3weobtain r = a asimple, efficient, andeasily-programmablealgorithm(thatdoesnotrequirek C). ⊂ Ouralgorithmretainsthepleasinggeometricflavouroftheoriginal,butisbetter-suited toeverydaycalculations. 2. ANOVERVIEWOFTHEDOLGACHEV–LEHAVICONSTRUCTION WebeginbybrieflyrecallingtheDolgachev–Lehaviconstruction,beforetreatingitin detailinthefollowingsections.SupposeH/k,S,φ,andX areasabove;weassumewe aregivenanexplicitformforH andS,andwewanttocomputeanexplicitformforX. DolgachevandLehaviobservethatifΘ andΘ arethetadivisorsonJ andJ , H X H X respectively,thenφ(Θ )isin ℓΘ (see[7,Proposition2.4]);andassuch,theimageof H X | | φ(Θ )intheKummersurfaceK J / 1 isadegree-2ℓrationalcurve1inP3 of H X X = 〈± 〉 2010MathematicsSubjectClassification. 11Y99;14Q05,14H45. 1By“rationalcurve”wemeanacurveofgenus0.Inallothercontexts,“rational”means“definedoverk”. 1 COMPUTINGLOW-DEGREEISOGENIESINGENUS2WITHTHEDOLGACHEV–LEHAVIMETHOD 2 arithmeticgenus(ℓ2 1)/2andwith(ℓ2 1)/2ordinarydoublepointscorrespondingto − − thenonzeroelementsofS,uptosign[7,Proposition3.1]. Wecancomputethiscurve withoutknowingφbyexpressingthemapΦ:H Θ J J K P3asthe ∼= H ⊂ H → X → X ⊂ compositionofadoublecoverρ ofarationalnormalcurveinP2ℓ withaprojection 2ℓ π:P2ℓ P3 whosecentredepends oncertainsecants corresponding tothenonzero → elementsofS,uptosign. TheimagesunderΦoftheWeierstrasspointsofH lieona conicQcontainedinahyperplaneofP3;thatis,atropeofK .ThedoublecoverofQ X ramifiedovertheWeierstrasspointimagesisthen(aquadratictwistof)X. 3. THEDOMAINCURVE WesupposethatH/kispresentedasanonsingularprojectivemodel 6 (1) H :Y2 F(X,Z) FiXiZ6−i P(1,3,1), = = ⊂ i 0 X= whereF isasquarefreehomogeneoussexticoverk(suchamodelalwaysexistswhenk isperfectandhascharacteristicnot2:see[5,§1.3]).ThehyperellipticinvolutionofH is ι :(X :Y :Z) (X : Y :Z). H 7−→ − ThedivisoratinfinityonH is D 1: F :0 1: F :0 ; 6 6 ∞= + − weobservethatD isdefinedov¡erkp,fixed¢byι¡H,anpdequa¢lto2(1:0:0)ifF6 0. ∞ = ThesixWeierstrass pointsofH arethefixedpoints ofι ; they correspondtothe H projectiverootsofthesexticF.TheWeierstrassdivisorW ofH istheeffectivedivisor H cutoutbyY 0;ifF(X,Z) 6 (z X x Z)overk,then = = i 1 i − i = WHQ (x1:0:z1) (x6:0:z6). = +···+ NotethatW isdefinedoverk.Finally,wefixacanonicaldivisoronH,defining H K W 2D . H H = − ∞ 4. THEKERNELOFTHEISOGENY Whendefiningtheirmethodforℓ 3,DolgachevandLehavistate“unfortunately,we = donotknowhowtoinputexplicitlythepair(H,S).InsteadweconsiderH withanodd thetastructure.” Wewilltakearathermoremiddlebrowapproachtotheproblem: we supposethatH ispresentedintheform(1),andthatSisgivenasacollectionofdivisor classesonH expressedusinganextendedMumfordrepresentation(detailedbelow). Our motivation for this choice is simple: this is precisely how one computes with hyperelliptic Jacobians in computational algebra systems such as Magma [11, 2] and Sage [13]. This choice also radically simplifies the algorithm: we can omit the theta structurecalculationsandpassdirectlytothesecantcomputations(short-circuitingthe firstfourstepsofthealgorithmin[7,§5.1]). Points on J correspond to divisor classes of degreezeroon H. The Riemann– H Rochtheoremtellsusthateverynontrivialdegree-0classhasauniquerepresentativein theformP Q D (thisrepresentationfailstobeuniqueforthetrivialclass,because [P ι (P)+D−] ∞0foreveryP inH(k)). H + − ∞ = Let e bea point of J , corresponding toa divisor class [P Q D ]. The effec- H + − ∞ tivedivisorP Q iscutoutbyanidealintheform(A(X,Z),Y B(X,Z)),whereB isa + − homogeneouscubicandAahomogeneouspolynomialofdegreed 2.Thetriple ≤ a(x),b(x),d : A(x,1),B(x,1),d 〈 〉 =〈 〉 thenencodesthepointe (withtheconventionthat Aischosensuchthata ismonic). Notethatifk′isanextensionofk,thene a,b,d isinJH(k′)ifandonlyifaandb =〈 〉 havecoefficientsink. ′ COMPUTINGLOW-DEGREEISOGENIESINGENUS2WITHTHEDOLGACHEV–LEHAVIMETHOD 3 Conversely, given a triple a,b,d , we recover the corresponding point of J by H 〈 〉 computingtheeffectivedivisorcutoutby(A(X,Z),Y B(X,Z)),whereBisthedegree-3 − homogenizationofband Aisthedegree-d homogenizationofa,andthensubtracting (d/2)D . IfH hastwopointsatinfinity(thatis,ifF6 0)thend mustbeeither2or0. ∞ 6= InthecasewhereH hasasinglepointatinfinity(thatis,whenF6 0)wealwayshave = d dega,andthepair a,bmoda isthestandardMumfordrepresentation. Thead- = 〈 〉 vantageoftheextendedrepresentationaboveisthatitgracefully handlesthegeneral casewheretherearetwopointsatinfinity. Example4.1. ConsiderthefollowingpointsontheJacobianofH :Y2 X6 Z6. = − 0isrepresentedby 1,0,0 . • 〈 〉 [(1:0:1) ( 1:0:1) D ]isrepresentedby x2 1,0,2 . • [(1:1:0)+(1−: 1:0)]− [2∞(1:1:0) D ]isrep〈res−entedb〉y 1,x3,2 . • − − = − ∞ 〈 〉 Inthisarticle,wewillassumethatthepointsofS areallk-rational. Thissimplifies theexpositionandthecomputations;however,allofourcalculationsaresymmetricin theelementsofS.ThealgorithmshouldthereforebeeasilyadaptedtothecasewhereS isrationalbutitselementsarenot. 5. THERATIONALNORMALCURVE TheRiemann–RochspaceL(2ℓK )isadirectsumofsubspaces H L(2ℓKH) L(2ℓKH)+ L(2ℓKH)−, = ⊕ whereι actsas 1ontheelementsofL(2ℓK ) and 1ontheelementsofL(2ℓK ) . H H + H − + − Writingx X/Z andy Y/Z3,wehave = = L(2ℓKH)+= xi/y2ℓ 2iℓ0 and L(2ℓKH)−= xi/y2ℓ−1 2iℓ−03. = = ThespaceL(2ℓKH)+ c orrespo®ndstothelinearsystem 2ℓKH 〈ιH®〉;weseeimmedi- | | atelythatitis(2ℓ 1)-dimensional,andthereforedefinesamap + ρ :H R P2ℓ 2ℓ 2ℓ −→ ⊂ ontoacurveR inP2ℓ.FixingcoordinatesonP2ℓ,wetakeρ tobedefinedby 2ℓ 2ℓ ρ2ℓ:(X :Y :Z) (U0: :U2ℓ) (X0Z2ℓ:XZ2ℓ−1: :X2ℓ−1Z:X2ℓ). 7−→ ··· = ··· WeseethatR isarationalnormalcurveofdegree2ℓinP2ℓ,andρ isadoublecover: 2ℓ 2ℓ (2) ρ (P) ρ (Q) P Q or P ι (Q) . 2ℓ 2ℓ H = ⇐⇒ = = (Essentially,ρ2ℓisacompositionofthecan¡onicalmapofH anda¢nℓ-upleembedding.) 6. THESECANTLINES Weadoptthefollowingconvention:ifSisasetofpointsinsomeprojectivespacePn, then S denotesthelinearsubspaceofPngeneratedbyS. 〈 〉 ForanypairofpointsPandQonH,wedefineLP,QtobethelineinP2ℓintersecting R inρ (P) ρ (Q);thatis, 2ℓ 2ℓ 2ℓ + ρ (P),ρ (Q) ifP {Q,ι (Q)} LP,Q:=½ Tρ22ℓℓ(P)(R22ℓℓ) ® othe∉rwiseH. WecanalsodefinesecantlinescorrespondingtononzeroJacobianelements: ife isa nonzeropointonJ ,thenwedefine H Le: LP,Q wheree [P Q D ]. = = + − ∞ ObservethatLP,Q=LP,ιH(Q)=LιH(P),Q=LιH(P),ιH(Q)forallP andQonH,so Le L e = − foralleinJ \{0}. H COMPUTINGLOW-DEGREEISOGENIESINGENUS2WITHTHEDOLGACHEV–LEHAVIMETHOD 4 Remark6.1. DolgachevandLehavidefinesecantsl ρ (P ),ρ (P ) foreachnon- e 2ℓ 1 2ℓ 2 = trivialpointe=[P1−P2]inJH (see[7,Theorem1.1]). OurLe isequa®ltole,because [P1−P2]=[P1+ιH(P2)−D∞]andLP1,P2=LP1,ιH(P2). ThefollowinglemmagivesexplicitandrationalformulæforthesecantsLeandtheir intersectionswitharbitraryhyperplanesinP2.Theseformulæarecentraltotheexplicit Dolgachev–Lehavimethod. Lemma6.2. Lete=〈a,b,d〉beanonzeropointofJH.LetH: 2iℓ0HiUi=0beahyper- planeinP2ℓ,andwriteh(x): 2ℓ H xi. = = i 0 i P = (1) Ifa 1andd 2,then = = P Le (0: :0:1:0),(0: :0:0:1) . = ··· ··· (a) IfH2ℓ H2ℓ 1 0 ,thenLe H. ® = − = ⊂ (b) Otherwise,Le H (0: :0:H2ℓ: H2ℓ 1). ∩ = ··· − − (2) Ifa(x) x α,then = − Le (0: :0:1),(1: :α2ℓ) . = ··· ··· (a) Ifh(α) 0andH2ℓ 0,thenLe H. ® = = ⊂ (b) Otherwise,Le H H2ℓ:H2ℓα: :H2ℓα2ℓ−1:H2ℓα2ℓ h(α) . ∩ = ··· − (3) Ifa(x) x2 a x a witha2 4a ,then = + 1 + 0 ¡ 16= 0 ¢ Le (π0: :π2ℓ),( a1:a2π0:a2π1: :a2π2ℓ 1) = ··· − ··· − whereπ0 2,π 1 a1,andπi a1πi 1 a2πi 2fori 1. ® = =− =− − − − > (a) Ifa(x)dividesh(x),thenLe H. ⊂ (b) Otherwise,Le H (γ0: :γ2ℓ)where ∩ = ··· γ H (ajσ aiσ ) i= j 2 i−j− 2 j−i 0 j 2ℓ ≤X≤ withσ 0fork 1,σ 1,andσ a σ a σ fork 1. k 1 k 1 k 1 2 k 2 (4) Ifa(x) x2 =a x a <witha2= 4a ,then=w−ritingα− f−or a /−2,weha>ve = + 1 + 0 1= 0 − 1 Le (1:α: :α2ℓ),(0:1:2α: :2ℓα2ℓ−1) . = ··· ··· (a) Ifa(x)divides h(x),thenLe H. ® ⊂ (b) Otherwise,Le H (γ0: :γ2ℓ)whereγi iαi−1h(α) αih′(α). ∩ = ··· = − Proof. Ingeneral,givenpointsα (α : :α )andβ (β : :β )inP2ℓ,wehave 0 2ℓ 0 2ℓ = ··· = ··· H Lα,β (Aβ0 Bα0: :Aβ2ℓ Bα2ℓ) ∩ = − ··· − where A= 2iℓ0Hiαi and B = i2ℓ0Hiβi; if A=B =0, then Lα,β ⊂H (andthepoint aboveisnotd=efined). Inthefollow=ing,wesupposee [P Q D ];wehavee 0,so P P = + − ∞ 6= wecansupposeP ι (Q). H 6= Incase(1)bothP andQ areatinfinity,soρ (P) ρ (Q) (0: :0:1); ourex- 2ℓ 2ℓ = = ··· pressionforLe givesgeneratorsforthetangenttoR2ℓat(0: :0:1).Theintersection ··· formulafollowsimmediately. Incase(2),wehaveP (1:0:0)andQ (α: pF(α,1):1),soρ (P) (0: :0:1) 2ℓ = = ± = ··· andρ (Q) (1:α: :α2ℓ).Theintersectionformulafollowsimmediately. 2ℓ = ··· Incase(3), wehaveP (α: pF(α,1):1) andQ (β: F(β,1):1) withα β, = ± = ± 6= α β a ,andαβ a ;soρ (P) (1:α: :α2ℓ)andρ (Q) (1:β: :β2ℓ).Ifwe + =− 1 = 2 2ℓ = ··· 2ℓ p= ··· take T (2:α β: :α2ℓ β2ℓ) and S (α β:2βα: :αβ2ℓ βα2ℓ), = + ··· + = + ··· + thenweeasilyverifythatLP,Q LT,S; itisastraightforwardexercisewithsymmetric = polynomialstoshowthatαi βi π for0 i 2ℓandαβi βαi a π fori 0, i 2 i 1 + = ≤ ≤ + = − > whenceourformulaforLe.TheintersectionH Le is ∩ H LP,Q (h(α) h(β):h(α)β h(β)α: :h(α)β2ℓ h(β)α2ℓ); ∩ = − − ··· − COMPUTINGLOW-DEGREEISOGENIESINGENUS2WITHTHEDOLGACHEV–LEHAVIMETHOD 5 itisanotherstraightforwardexercisetoshowthat αjβi βjαi (β α)(ajσ aiσ ), − = − 2 i−j− 2 j−i soh(α)βi h(β)αi 2ℓ H (β α)(ajσ aiσ ) (β α)γ for0 i 2ℓ,andthus H∩Le=−(γ0:···:γ=2ℓP).j=0 j − 2 i−j− 2 j−i = − i ≤ ≤ Incase(4),wehaveP Q (α: pF(α,1):1);ourexpressionforLegivesgenerators forthetangenttoR at=ρ (=P) (±1:α: :α2ℓ).Theintersectionformulafollows. (cid:3) 2ℓ 2ℓ = ··· 7. THEWEIERSTRASSSUBSPACE SinceR isarationalnormalcurveofdegree2ℓ,any2ℓ 1distinctpointsonR 2ℓ 2ℓ + arelinearlyindependent. Inparticular, theimagesofthesixWeierstrasspointsofH underρ arelinearlyindependentbecauseℓ 3.Inviewof(2)theimagesaredistinct, 2ℓ ≥ sothesubspace W : ρ (W ) P2ℓ 2ℓ H = ⊂ isfive-dimensional. ® Foreach0 i 2ℓ 6,wedefinealinearform ≤ ≤ − 6 W : F U . i j i j = + j 0 X= Lemma7.1. ThespaceW is 2ℓ 6 − W V(W ) V({W :0 i 2ℓ 6}). i i = = ≤ ≤ − i 0 \= Proof. Each hyperplane V(Wi) contains W, since Wi ρ2ℓ XiZ2ℓ−6−iF(X,Z). But ◦ = theWi arelinearlyindependent,sotheintersection 2iℓ−06V(Wi)is5-dimensional,and henceequaltoW. = (cid:3) T 8. THETHEOREMOFDOLGACHEVANDLEHAVI WearenowreadytostatethemaintheorembehindtheDolgachev–Lehavimethod. Theorem8.1([7,Theorem1.1]). ThereexistsauniquehyperplaneH P2ℓsuchthat ⊂ (1) HcontainsW,and (2) theintersectionpointsofH withthesecantsLe foreachnonzeroeinSarecon- tainedinasubspaceN ofcodimension3inH. TheimageoftheWeierstrassdivisorundertheprojectionP2ℓ P3withcentreN lieson → aconicQ(whichmaybereducible),andthedoublecoverofQramifiedoverthisdivisor isastablecurveX ofarithmeticgenus2suchthatJ J /S. X ∼ H = ItiscrucialtonotethatTheorem8.1isnotconstructive:itdoesnotinitselfyieldthe hyperplaneH,northecentreN oftheprojectiontoP3. Itisnotedin[7,§3.4]thatH is definedbyφ (Θ ),butinoursituationwedonotyethaveanexpressionforφorΘ . ∗ X X Inthecaseℓ 3, wearesavedbyahappy coincidence: 2ℓ 1 5, so H W (we = − = = returntothiscasein§11below). For ℓ 3, wemust compute H insomeotherway; > Lemma8.2,aneasycorollaryofLemma7.1,characterizesthepossiblehyperplanes. Lemma8.2. ThelinearsystemofallhyperplanesinP2ℓcontainingW isgeneratedbythe 2ℓ 5hyperplanesV(W )for0 i 2ℓ 6.Thatis,ifH W isahyperplaneinP2ℓ,then i − ≤ ≤ − ⊃ H V(α0W0 α2ℓ 6W2ℓ 6) = +···+ − − forsome(α : :α )inP2ℓ 6(k). 0 2ℓ 6 − ··· − COMPUTINGLOW-DEGREEISOGENIESINGENUS2WITHTHEDOLGACHEV–LEHAVIMETHOD 6 InviewofLemma8.2,onenaïveapproachtocomputingHforℓ 3wouldbetotake > ageneric H =V 2iℓ−06αiWi andcomputeitsintersectionwiththesecantsLe. This yields(ℓ2 1)/2 poi=nts whosecoordinatesarelinear expressions intheα . Wecould − ¡P ¢ i thensolveforthevaluesofα bycomputingthezerolocusofthe(2ℓ 2) (2ℓ 2)minors i − × − ofthematrixformedbytheintersectionsH Le;buteachminorisstilladegree-(2ℓ 2) ∩ − polynomialin2ℓ 5variables,andthenumberofminorsisexponentialinℓ. − Alternatively,wecouldtakeagenericsetoflinearequationsdeterminingN insidethe genericH;requiringthatthiscentreintersectsanyoneofthe(ℓ2 1)/2secantsimposes − O(ℓ4)quarticpolynomialconditionsontheO(ℓ)unknowns. Ineachapproachthesystemishighlyoverdetermined,andwithacleverchoiceof minorswemighthopetogetluckyandfindsolutionsfortoyexamples. However,both approachesalreadyrepresentasignificantundertakingforℓ 5,evenoverfinitefields; = theyaretotallyimpracticalforlargerℓandforinfinitefields. Wecontinuethetreatmentforgeneralℓin§9and§10,supposingthatanequationfor H hasbeenfound;withoutsuchanequation,theavIsogeniespackage[1]represents amuchmoresensibleapproachforℓ 5(ifkisfinite).Forℓ 3,theDolgachev–Lehavi ≥ = methodisaspracticalasitisinteresting;wespecializetothiscasein§11and§12. 9. FROMTHEORYTOPRACTICE TocomputeX viaTheorem8.1,wemustcomputethemap Φ: π ρ :H P3, 2ℓ = ◦ → whereπ:P2ℓ P3istheprojectionwithcentreN.Supposethatwehaveanequation → H: α W 0 i i = i X for H. WecanthenapplyLemma 6.2tocomputethecentre N Le H:e S\{0} . =〈 ∩ ∈ 〉 SinceN H,wemaycomputeν ,...,ν ,ν ,...,ν ,ν ,...,ν inksuchthat 0,0 0,2ℓ 1,0 1,2ℓ 2,0 2,2ℓ ⊂ 2ℓ 2ℓ 2ℓ 2ℓ 6 − N V ν U , ν U , ν U , α W . 0,i i 1,i i 2,i i i i = Ãi 0 i 0 i 0 i 0 ! X= X= X= X= (This amounts to computing the kernel of the matrix whose rows are formed by the coordinatesoftheLe∩H;thechoiceof 6i 0αiWi forthefourthdefiningequationwill beconvenientlaterintheprocedure.) = P FixingcoordinatesonP3,theprojectionπwithcentreN isdefinedby 2ℓ 2ℓ 2ℓ 2ℓ 6 − π:(U : :U ) (V :V :V :V ) ν U , ν U , ν U , α W ; 0 2ℓ 0 1 2 3 0,i i 1,i i 2,i i i i ··· 7−→ =Ãi 0 i 0 i 0 i 0 ! X= X= X= X= thecomposedmapΦ π ρ isthen 2ℓ = ◦ Φ:(X :Y :Z) (V :V :V :V ) Φ (X,Z):Φ (X,Z):Φ (X,Z):Φ (X,Z) , 0 1 2 3 0 1 2 3 7−→ = where ¡ ¢ 2ℓ 2ℓ 2ℓ Φ0: ν0,iXiZ2ℓ−i, Φ2: ν1,iXiZ2ℓ−i, Φ2: ν2,iXiZ2ℓ−i, = = = i 0 i 0 i 0 X= X= X= and 2ℓ 6 Φ3: − αiXiZ2ℓ−6−iF(X,Z). = i 0 X= The image of Φ is a rational curve of degree 2ℓ in P3. It lies on the Kummer sur- faceK oftheunknowncodomainJacobianJ ,andisthereforetheintersectionofa X X quadricandacubichypersurfaceinP3(see[9,ChapterXIII]): Φ(P1) Q C where Q V Q˜(V0,V1,V2,V3) and C V C˜(V0,V1,V2,V3) = ∩ = = ¡ ¢ ¡ ¢ e e e e COMPUTINGLOW-DEGREEISOGENIESINGENUS2WITHTHEDOLGACHEV–LEHAVIMETHOD 7 forsomeformsQãndCõfdegree2and3,respectively.TheformsQãndC˜generatethe eliminationideal Q˜,C˜ (V Φ ,V Φ ,V Φ ,V Φ ) k[V ,V ,V ,V ]; 0 0 1 1 2 2 3 3 0 1 2 3 = − − − − ∩ notethatQ˜is¡uniqu¢elydetermined,andC˜isdeterminedmodulo(V Q,V Q,V Q,V Q). 0 1 2 3 The Weierstrass points of H map into the hyperplane V3 0, which we identify = withP2. (Thissimplification motivatesourchoiceofΦ .) Theorem8.1assertsthata 3 conicQpassesthroughthesiximages,andindeed Q V(Q(V0,V1,V2)) P2, where Q(V0,V1,V2) Q˜(V0,V1,V2,0). = ⊂ = TheimageoftheWeierstrassdivisorunderΦisthereforeQ C,where ∩ C V(C(V0,V1,V2)) P2 with C(V0,V1,V2) C˜(V0,V1,V2,0). = ⊂ = WearemoreinterestedintheformsQandCthaninQãndC˜,anditisasimplematterto interpolatethem.ForQ,wecomputethesixquinticpolynomialsΦ Φ (x,1)modF(x,1) i j for0 i j 2; theuniquelinearrelationbetweenthem(andbetweentheν ν if i,0 j,0 ≤ ≤ ≤ F 0) yields the coefficients ofQ. Similarly, to findC wecompute the ten quintics 6 = Φ Φ Φ (x,1)modF(x,1)for0 i j k 2;anyoneofthelinearrelationsbetween i j k ≤ ≤ ≤ ≤ them(andtheν ν ν ifF 0)givesanequationforavalidcubicC. i,0 j,0 k,0 6 = 10. THECODOMAINCURVE Thedata(Q,Q C)specifiesagenus2curveX (uptoaquadratictwist)asadouble ∩ coverofQramifiedoverthesixpointsofQ C. ThisistheoutputoftheDolgachev– ∩ LehavialgorithmandofTheorem8.1, anditissufficientforcomputing isomorphism invariantsofX (see,forexample,[4]and[12]). Insomesituations,however,wewouldliketoderiveadefiningequationforX itself. WhenQisnonsingular,werecoverahyperellipticcurve;inthedegeneratecasewhere Qissingular,werecoveraunionoftwoellipticcurvesX andX ,whicharegenerally definedoveraquadraticextensionofk(inwhichcaseth+eyareGa−loisconjugates). The procedureisessentiallystandard(cf.[4,§2]),butwerecallithereforcompleteness. Algorithm10.1. Computesa(possiblyreducible)genus2curverepresentingadouble coverofagivenplaneconicramifiedovertheintersectionwithaplanecubic. Input: AplaneconicQ:Q(V0,V1,V2) 0andcubicC :C(V0,V1,V2) 0. = = Output: Agenus2curveX formingadoublecoverofQramifiedoverQ C. If ∩ Q issingular,thenX willbeaone-pointunionofellipticcurvesX andX , + − withX ramifiedoverP0andC L whereQ L L andP0 L L . ± ∩ ± = ++ − = +∩ − 1: LetMbethematrixdefinedby 2q q q 0,0 0,1 0,2 M:= q0,1 2q1,1 q1,2 , where qi,jViVj =Q(V0,V1,V2). q q 2q 0 i j 2 0,2 1,2 2,2 ≤X≤ ≤   2: Ifdet(M) 0,thenQissingular. = 2a: Compute adiagonalmatrixD diag(a,b,0) andaninvertible matrixT = suchthatM TDT 1. − = 2b: Setδ p a/b,anddefinehomogeneouscubicsC (X,Z)andC (X,Z) = − + − by C : C (t δt )Z t X,(t δt )Z t X,(t δt )Z t X 00 01 02 10 11 12 20 21 22 ± = ± + ± + ± + where ¡ ¢ t t t 00 01 02 t10 t11 t12 T.  = t t t 20 21 22   COMPUTINGLOW-DEGREEISOGENIESINGENUS2WITHTHEDOLGACHEV–LEHAVIMETHOD 8 2c: DefineellipticcurvesX andX overk(δ)inP(2,3,2)by + − X :Y2 C (X,Z) and X :Y2 C (X,Z), + = + − = − andreturntheunionofX andX identifyingthepointsatinfinity. + − 3: Otherwise,Qisnonsingular. 3a: ComputearationalpointP (α0:α1:α2)inQ(k)(seeRemark10.2). = 3b: Letπ:P1 Qbethecorrespondingparametrization,definedby → π:(X :Z) (V :V :V ) (P (X,Z):P (X,Z):P (X,Z)) 0 1 2 0 1 2 7−→ = (theP arequadraticforms). i 3c: ReturnX :Y2 C(P0(X,Z),P1(X,Z),P2(X,Z)). = Remark10.2. Step3aofAlgorithm10.1requiresustocomputeak-rationalpointP on theconicQ. IfH hasarationalWeierstrasspointW0,thenwemaytakeP Ψ(W0). = Generically,however,H hasnorationalWeierstrasspoints,andthenweareobligedto searchfor a rationalpoint onQ. Weareguaranteed thatsucha rationalpoint exists (cf.[12,Lemme1]). Overafinitefield,findingarationalpointisstraightforward;over therationals,wecanapply(forexample)theCremona–Rusinalgorithm[6]. 11. THEALGORITHMFORℓ 3 = Considernowthespecialcaseℓ 3.Themapρ6:H R6 P6isdefinedby = → ⊂ ρ :(X :Y :Z) (U :U : :U :U ) Z6:XZ5: :X5Z:X6 . 6 0 1 5 6 7−→ ··· = ··· ThehyperplaneH ofTheorem8.1containsW ρ¡ (W ) bydefinition;b¢utdimH 6 H = = dimW 5,soH W.ApplyingLemma7.1,wefind = = ® 6 (3) H V(W0) V FiUi P6; = = Ãi 0 !⊂ X= thisallowsustosimplifyLemma6.2forthecaseℓ 3. = Proposition11.1. Ife a,b,d isanonzero3-torsionpointofJ ,then H =〈 〉 H Le γ0(e): :γ6(e) , ∩ = ··· wheretheγ aredefinedasfollows: ¡ ¢ i (1) Ifa 1,thenγ (e) 0for0 i 5,withγ (e) F andγ (e) F . i 5 6 6 5 = = ≤ < = =− (2) Ifaislinear,thenγ (e) 0for0 i 6,andγ (e) 1. i 6 = ≤ < = (3) Ifa(x) x2 a x a witha2 4a ,then = + 1 + 0 16= 0 6 γ (e) F (ajσ aiσ ) for 0 i 6 i = j 2 i−j+ 2 j−i ≤ ≤ j 0 X= withσ 0fork 1,σ 1,andσ a σ a σ fork 1. k 1 k 1 k 1 2 k 2 (4) Ifa(x) =x2 a x<a wi=tha2 4a ,=th−en − − − > = + 1 + 0 1= 0 6 γi(e) (i j)Fj( a1/2)i+j−1 for 0 i 6. = − − ≤ ≤ j 0 X= Proof. ThisfollowsimmediatelyfromLemma6.2onsettingH V 6 F U andnot- = i 0 i i ingthata(x)cannotdivideh(x) 6 F xi (sinceotherwiseewouldh=aveorder2). (cid:3) = i 0 i ¡P ¢ = WenowpresentaversionofthePDolgachev–Lehavialgorithmforℓ 3basedonthe = extendedMumfordrepresentation. Thealgorithmrequiresonlyelementarymatrixal- gebraandpolynomialarithmetic,andshouldbeeasilyimplementedinmostcomputa- tionalalgebrasystems. COMPUTINGLOW-DEGREEISOGENIESINGENUS2WITHTHEDOLGACHEV–LEHAVIMETHOD 9 Algorithm11.2. AstreamlinedDolgachev–Lehavi-stylealgorithmforℓ 3. = Input: Agenus2curveH :Y2=F(X,Z)= 6i 0FiXiZ6−i overkandamaximal Weil-isotropicsubgroupSofJ [3],itselem=entsdefinedoverkandpresented H P asin§4. Output: Agenus2curveX/ksuchthatthereexistsanisogenyφ:J J with H X → kernelS(thecurveX iscomputeduptoaquadratictwist,sotheisogenymay onlybedefinedoveraquadraticextensionofk). 1: ComputeaminimalsubsetS ofSsuchthatS {e:e S } { e:e S } {0} ± ± ± = ∈ ∪ − ∈ ∪ (then{Le:e S±} {Le:e S\{0}};thisavoidsredundancyinSteps2and3). ∈ = ∈ 2: ForeacheinS±,computethevectorve (γ0(e),...,γ6(e))usingtheformulæin = Proposition11.1. 3: Compute vectors n (ν ,...,ν ) such that {n ,n ,n ,(F :0 j 6)} is a i i,0 i,6 0 1 2 j = ≤ ≤ basisforthe(left)kernelofthe7 4matrix(vt:e S ).Set × e ∈ ± 6 Φi νi,jXjZ6−j for0 i 2. = ≤ ≤ j 0 X= 4: Foreach0 i j 2,computethevectorr oflength6whosenthentryisthe i,j ≤ ≤ ≤ coefficientofxn 1in(Φ Φ )(x,1)modF(x,1). IfF 0,thentakethe6thentry − i j 6 = ofr tobeν ν :thisallowsustocorrectlyinterpolatethroughtheimageof i,j i,0 j,0 theWeierstrasspointatinfinity. 5: Computeagenerator(q :0 i j 2)forthe(left)kernelofthe6 6matrix i,j ≤ ≤ ≤ × whoserowsarether for0 i j 2.Set i,j ≤ ≤ ≤ Q(V ,V ,V ): q V2 q V V q V V q V2 q V V q V2. 0 1 2 = 0,0 0 + 0,1 0 1+ 0,2 0 2+ 1,1 1 + 1,2 1 2+ 2,2 2 6: Foreach0 i j k 2,computethevectors oflength6whosenthentry i,j,k ≤ ≤ ≤ ≤ isthecoefficientofxn−1in(ΦiΦjΦk)(x,1)modF(x,1). IfF6 0,thentakethe = 6thentryofs tobeν ν ν . i,j,k i,0 j,0 k,0 7: Computeanynontrivialelement(c :0 i j k 2)ofthe(left)kernelof i,j,k ≤ ≤ ≤ ≤ the10 6matrixwhoserowsarethes for0 i j k 2,andset i,j,k × ≤ ≤ ≤ ≤ C(V ,V ,V ): c V V V . 0 1 2 i,j,k i j k = 0 i j k 2 ≤≤X≤ ≤ 8: ReturntheresultX ofAlgorithm10.1appliedtoQ V(Q)andC V(C). = = 12. THEALGORITHMINPRACTICE Weconcludewithanexampleforℓ 3. Toavoidavisually overwhelmingmassof = coefficients,wewillworkoverasmallfinitefield;thecurvewaschosenatrandom. Considerthegenus2curveoverF definedby 997 H :Y2 X6 113X5Z 99X4Z2 363X3Z3 64X2Z4 503XZ5 630Z6. = + + + + + + ComputingthezetafunctionofH (usingMagma),weseethatitsWeilpolynomialis P(T) T4 31T3 54T2 30907T 994009, = − + − + soJ isabsolutelysimplebytheHowe–Zhucriterion[8,Theorem6]. Theelements H D1 x2 392x 208,579x 603,2 andD2 x2 48x 527,918x 832,2 ofJH have =〈 + + + 〉 =〈 + + + 〉 order3,andS D1,D2 isamaximal3-WeilisotropicsubgroupofJH[3]. Applying =〈 〉 Algorithm11.2,wemaytake x2 392x 208,579x 603,2 , x2 48x 527,918x 832,2 , S±= 〈x2+428x+880,252x+901,2〉,〈x2+348x+ 292,596x+ 269,2〉 ½〈 + + + 〉 〈 + + + 〉¾ inStep1.Equation(3)showsthatthehyperplaneH P6isdefinedby ⊂ H:630U 503U 64U 363U 99U 113U U 0, 0 1 2 3 4 5 6 + + + + + + = COMPUTINGLOW-DEGREEISOGENIESINGENUS2WITHTHEDOLGACHEV–LEHAVIMETHOD 10 sothematrixinStep3is 234 319 906 896 780 16 29 754   500 565 703 398  680 329 823 248 ;    324 68 779 868     742 416 468 392     664 395 698 952      computingkernelvectors,wetake Φ 121X6 742X5Z 549X4Z2 XZ5, 0 = + + + Φ 285X6 642X5Z 332X4Z2 X2Z4, 1 = + + + Φ 889X6 701X5Z 454X4Z2 X3Z3. 2 = + + + ThequadraticformofStep5isthen Q(V ,V ,V ) V2 52V V 361V2 548V V 715V V 296V2, 0 1 2 = 0 + 0 1+ 1 + 0 2+ 1 2+ 2 andwemaytakethecubicforminStep7tobe C(V ,V ,V ) V3 167V3 149V V V 836V2V 885V V2 538V V2 294V3. 0 1 2 = 0 + 1 + 0 1 2+ 1 2+ 0 2 + 1 2 + 2 We now apply Algorithm 10.1 to Q :Q(V0,V1,V2) 0 and C :C(V0,V1,V2) 0. The = = conicQ isnonsingular, andH hasa rationalWeierstrasspoint ( 76:0:1)mapping − tothepoint( 36: 80:1)onQ.TheassociatedparametrizationP1 Qisdefinedby − − → (X :Z) (36X2 781XZ 109Z2:80X2 865XZ 17Z2:996X2 945XZ 636Z2); 7−→ + + + + + + substitutingitsdefiningpolynomialsintoC,wefindthatX hasamodel X :Y2 118X5Z 183X4Z2 613X3Z3 35X2Z4 174XZ5 474Z6. = + + + + + Infact,thisisthequadratictwistofthetrueX: explicitcalculationshowsthatitsWeil polynomialisP( T). − REFERENCES [1] G.Bisson,R.Cosset,andD.Robert,avIsogenies:alibraryforcomputingisogeniesbetweenabelianvari- eties. avisogenies.gforge.inria.fr [2] W.Bosma, C.Playoust, andJ. J.Cannon, The Magmaalgebrasystem.I.The userlanguage. J.Sym- bolicComput.24(1997),235–265 [3] J.-B.BostandJ.-F.Mestre, Moyennearithmético-géométriqueetpériodesdescourbesdegenre1et2. 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