Computing isogenies between Abelian Varieties

49 Pages·2017·0.94 MB·English

Computing isogenies between Abelian Varieties
David Lubicz, Damien Robert
To cite this version:
David Lubicz, Damien Robert. Computing isogenies between Abelian Varieties. Compositio Mathe-
matica, 2012, 2012, Online publication. ￿10.1112/S0010437X12000243￿. ￿hal-00446062v2￿
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Submitted on 13 Jan 2010 (v2), last revised 21 Sep 2012 (v3)
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Computing isogenies between abelian
varieties
David Lubicz1,2, Damien Robert3
1CÉLAR,
BP7419,F-35174Bruz
2IRMAR,UniverstédeRennes1,
CampusdeBeaulieu,F-35042Rennes
3LORIA,CampusScientifique,BP239,
F-54506Vandœuvre-lès-Nancy
Abstract
Wedescribean efficientalgorithm forthecomputation ofisogeniesbetween abelian
varieties representedin the coordinate system providedby algebraic theta functions.
Weexplainhowtocomputealltheisogeniesfromanabelianvarietywhosekernelis
isomorphicto a given abstractgroup. We also describe an analog ofVélu’s formulas
tocomputeanisogeniswithprescribedkernels.Allouralgorithmsrelyinanessential
manneronageneralizationoftheRiemannformulas.
Inordertoimprovetheefficiencyofouralgorithms,weintroduceapointcompression
algorithmthatrepresentsapointoflevel4(cid:96)ofa g dimensionalabelianvarietyusingonly
g(g+1)/2·4g coordinates.WealsogiveformulastocomputetheWeilandcommutator
pairinggiveninputpointsinthetacoordinates.Allthealgorithmspresentedinthispaper
workingeneralforanyabelianvarietydefinedoverafieldofoddcharacteristic.
Contents
1 Introduction 5
2 Computing Isogenies 6
2.1 EllipticcurvesandVélu’sformulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Isogeniesonabelianvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Modular correspondences and theta null points 10
3.1 Thetastructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Isogeniescompatiblewithathetastructure . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Modularcorrespondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Theactionofthethetagroupontheaffineconeandisogenies. . . . . . . . . . 14
1
Contents
4 The addition relations 16
4.1 Evaluationofalgebraicthetafunctionsatpointsof(cid:96)-torsion . . . . . . . . . . 17
4.2 ThegeneralRiemannrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2.1 Thecasen=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Thetagroupandadditionrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Application of the addition relations to isogenies 29
5.1 Pointcompression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.1.1 Additionchainswithcompressedcoordinates . . . . . . . . . . . . . . . 31
5.2 Computingthedualisogeny. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.3 Computationofthekerneloftheisogeny . . . . . . . . . . . . . . . . . . . . . . . 33
6 The computation of a modular point 34
6.1 AnanalogofVélu’sformulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.2 Thetagroupand(cid:96)-torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.3 Improvingthecomputationofamodularpoint . . . . . . . . . . . . . . . . . . . 37
7 Pairing computations 41
7.1 Weilpairingandcommutatorpairing. . . . . . . . . . . . . . . . . . . . . . . . . . 41
7.2 Commutatorpairingandadditionchains . . . . . . . . . . . . . . . . . . . . . . . 43
8 Conclusion 46
References 46
List of Algorithms
4.3 Additionchain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.6 Multiplicationchain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.5 Pointcompression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.6 Pointdecompression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.9 Theimageofapointbytheisogeny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.1 Vélu’slikeformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.7 Computingallmodularpoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.5 Pairingcomputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2
Glossary
List of Notations
Notation Description Page
List
Z(n) (cid:90)g/n(cid:90)g 9
(cid:77) Themodulispaceofthetanullpointsofleveln. 7
n
A (A ,(cid:76),Θ )isapolarizedabelianvarietywithatheta 9
k k A
structureofklevel(cid:96)n.
B (B ,(cid:76) ,Θ ) is an abelian variety (cid:96)-isogenous to A 10
k k 0 B k
k
withathetastructureofleveln.
ϑ (ϑ ) arethecanonicalprojectivecoordinateson 9
i i i∈Z((cid:96)n)
A givenbythethetastructure.
k
0 Thethetanull0 =ϑ (0 ) . 10
Ak Ak i Ak i∈Z((cid:96)n)
0 Thethetanull0 =ϑ (0 ) . 10
B B i B i∈Z(n)
G(k(cid:76)) TheThetagroupkof(A ,(cid:76)k) 9
k
K((cid:76)) K((cid:76))=K ((cid:76))⊕K ((cid:76))isthedecompositionofthe 9
1 2
kernelofthepolarization(cid:76) inducedbytheThetastruc-
tureΘ
A
H(δ) TheHeisenberggroupoftypeδ. 9
s The natural section K ((cid:76)) → G((cid:76)) induced by the 9
K((cid:76)) 1
1
Thetastructure.
ρ(cid:101)∗(cid:76) TheaffineactionofG((cid:76))onA(cid:101)k. 10
ρ∗ TheprojectiveactionofK((cid:76))onA . 10
(cid:76) k
A(cid:101) TheaffineconeofA . 10
k k
B(cid:101) TheaffineconeofB . 11
k k
ϑ(cid:101) (ϑ(cid:101)) aretheaffinecoordinatesonA(cid:101) . 10
i i i∈Z((cid:96)n) k
(cid:101)0 Anaffineliftof0 . 15
B B
k k
(cid:101)0A Theaffineliftof0A suchthatπ(cid:101)((cid:101)0A )=(cid:101)0B . 15
πk The(cid:96)-isogenyπ:Ak →B . k k 10
k k
π π(ϑ(cid:101)(x) ) = ϑ(cid:101)(x) is the affine lift of π to 11
(cid:101) (cid:101) i (cid:101) i∈Z((cid:96)n) i (cid:101) i∈Z(n)
A(cid:101) →B(cid:101) .
k k
π π =π◦(1,i,0)=ϑ(cid:101) (·) . 14
(cid:101)i (cid:101)i (cid:101) i+j j∈Z(n)
P(cid:101)i P(cid:101)i =(1,i,0).(cid:101)0Ak =(ϑi+j((cid:101)0Ak))j∈Z((cid:96)n). 16
R(cid:101)i R(cid:101)i =π(cid:101)i((cid:101)0A )=π(cid:101)(P(cid:101)i). 16
k
(e ,...,e ) AbasisofZ((cid:96)n) 28
1 g
(d ,...,d ) d =ne 28
1 g i i
(cid:83) (cid:83) =Z((cid:96))(When(cid:96)∧n=1) 14
S S={d ,d ,...,d ,d +d ,...d +d ,d +d ,...d + 28
1 2 g 1 2 1 g 2 3 g−1
d }(When(cid:96)∧n=1)
g
e(cid:48) TheextendedcommutatorpairingonB [(cid:96)] 16
(cid:96) k
3
Glossary
Notation Description Page
List
e TheWeilpairing. 41
W ˆ
e ThecanonicalpairingonZ(n)×Z(n). 9
c
(cid:48)
B(cid:102)k theaffineconeof(Bk,(cid:77)0,ΘB ,(cid:77) )where(cid:77)0=[(cid:96)]∗(cid:76)0 16
andΘ isathetastructurkeon0 (B ,(cid:77) )compatible
B ,(cid:77) k 0
withΘk . 0
B
(cid:48)k
[(cid:221)(cid:96)] [(cid:221)(cid:96)]:B(cid:102) →B(cid:101) isthemorphismlifting[(cid:96)]:B →B . 16
k k k k
chaine_add Anadditionchain 18
chaine_multadd Amultiplicationchain 21
4
1 Introduction
1 Introduction
Inthispaper,weareinterestedinsomealgorithmicaspectsofisogenycomputationsbetween
abelianvarieties.Computingisogeniesbetweenabelianvarietiesmaybeseenasdifferentkind
ofcomputationalproblemsdependingontheexpectedinputandoutputofthealgorithm.
Theseproblemsare:
• GivenanabelianvarietyA andanabstractfiniteabeliangroupK computeallthe
k
abelianvarietiesB suchthatthereexistsanisogenyA →B whosekernelisisomor-
k k k
phictoK,andcomputetheseisogenies.
• GivenanabelianvarietyA andafinitesubgroupK ofA ,recoverthequotientabelian
k k
varietyB =A /K aswellastheisogenyA →B .
k k k k
• Giventwoisogenousabelianvarieties,A andB ,computeexplicitequationsforan
k k
isogenymapA →B .
k k
Here,weareconcernedwiththefirsttwoproblems.Inthecasethattheabelianvarietyis
anellipticcurve,efficientalgorithmshavebeendescribedthatsolvealltheaforementioned
problems [Ler]. Forhigher-dimensionalabelian varieties muchless is known. Richelot’s
formulas[Mes01,Mes02]canbeusedtocompute(2,2)-isogeniesbetweenabelianvarietiesof
dimension2.Thepaper[Smi09]alsointroducedamethodtocomputecertainisogeniesof
degree8betweenJacobianofcurvesofgenusthree.Inthispaper,wepresentanalgorithm
tocompute((cid:96),...,(cid:96))-isogeniesbetweenabelianvarietiesofdimension g forany(cid:96)(cid:190)2and
g (cid:190)1.Possibleapplicationsofouralgorithmincludes:
• Thetransferthediscretelogarithmfromanabelianvarietytoanotherabelianvariety
wherethediscretelogarithmiseasytosolve[Smi08]
• Thecomputationofisogenygraphtoobtainadescriptiontheendomorphismringof
anAbelianvariety.
ByTorelli’stheoremthereisaoneononecorrespondencebetweenprincipallypolarized
abelian varieties of dimension 2 and Jacobians of genus 2 hyperelliptic curves. Thus the
modularspaceofprincipallypolarizedabelianvarietiesofdimension2isparametrizedby
thethreeIgusainvariants,andonecandefinemodularpolynomialsbetweentheseinvariants
muchinthesamewayasinthegenus-onecase[BL09].Howevertheheightofthesemodular
polynomialsexplodeswiththeorder,makingtheircomputationsimpractical:onlythose
areknown[Plo29].Inordertocircumventthisproblem,inthearticle[FLR09],wehave
definedamodularcorrespondencebetweenabelianvarietiesinthemoduliofmarkedabelian
varieties.Thismodulispaceiswell-suitedforcomputatingmodularcorrespondencessincethe
associatedmodularpolynomialshavetheircoefficientsin{1,−1},andthereisnoexplosion
asbefore.
Inthispaper,weexplainhow,givenasolutiontothismodularcorrespondence(provided
forinstancebythealgorithmdescribedin[FLR09]),onecancomputetheassociatedisogeny.
Oncesuchamodularpointisobtained,theisogenycanbecomputedusingonlysimpleaddi-
tionformulasofalgebraicthetafunctions,soinpractice,thecomputationoftheisogenytakes
5
2 ComputingIsogenies
muchlesstimethanthecomputationofapointprovidedbythemodularcorrespondence.
Notethatthisissimilartothegenus-onecase.Forellipticcurves,thecomputationofaroot
ofthemodularpolynomialisnotmandatoryifthepointsinthekerneloftheisogenyare
given,sincethisistheinputtakenbyVélu’sformulas.Here,weexplainhowtorecoverthe
equations ofan isogenygiven the points ofits kernel,yielding a generalization ofVélu’s
formulas.
Ourgeneralizationintroduceshoweveradifferencecomparedtotheusualgenus-1case.
Forellipticcurves,themodularpolynomialoforder(cid:96)givethemodulispaceof(cid:96)-isogenous
ellipticcurves. In ourgeneralizedsetting,the modularcorrespondence in the coordinate
system of theta null points gives (cid:96)g-isogenous abelian varieties with a theta structure of
differentlevel.Asaconsequence,apointinthismodularspacecorrespondstoan(cid:96)g-isogeny,
togetherwith a symplectic structure of level (cid:96). Anothermethod would be to describe a
modularcorrespondencebetweenabelianvarietieswiththetastructuresofthesamelevel,
see[BGL09]foranexamplewith(cid:96)=3and g =2.
Thepaperisorganizedasfollow.InSection2werecallVélu’sformulasandoutlineour
algorithms.InSection3,werecallthedefinitionofthemodularcorrespondencegivenin
[FLR09],andwestudytherelationshipbetweenisogeniesandtheactionofthethetagroup.
We recall the addition relations,which play a central role in this paperin Section 4. We
then explain how to compute the isogeny associated to a modular point in Section 5. If
the isogenyis given bytheta functions oflevel4(cid:96),itrequires (4(cid:96))g coordinates. We give
a point compression algorithm in Section 5.1,showing how to express such an isogeny
withonly g(g +1)/2·4g coordinates.InSection6wegiveafullgeneralizationofVélu’s
formulasthatconstructsanisogenousmodularpointwithprescribedkernel.Thisalgorithm
ismoreefficientthanthespecialGröbnerbasisalgorithmfrom[FLR09].Thereisastrong
connectionbetweenisogeniesandpairings,andweusetheaboveworktoexplainhowone
cancomputethecommutatorpairingandhowitrelatestotheusualWeilpairinginSection7.
2 Computing Isogenies
Inthissection,werecallhowonecancomputeisogeniesbetweenellipticcurves.Wethen
outlineouralgorithmtocomputeisogeniesbetweenabelianvarieties.
2.1 Elliptic curves and Vélu’s formulas
Let(E ,(cid:101)0 )beanellipticcurvegivenbyaWeierstrassequationy2= f(x)with f adegree-3
k E
k
monicpolynomial.Vélu’sformulasrelyontheintrinsiccharacterizationofthecoordinate
system(x,y)givingtheWeierstrassmodelofE as:
k
v (x)=−3 v (x)(cid:190)0 ifP (cid:54)=(cid:101)0
(cid:101)0Ek P Ek
v (y)=−2 v (y)(cid:190)0 ifP (cid:54)=(cid:101)0 (1)
(cid:101)0Ek P Ek
y2/x3((cid:101)0 )=1,
E
k
wherev denotesthevaluationofthelocalringofE intheclosedpointQ.
Q k
6
2 ComputingIsogenies
Theorem2.1(Vélu):
LetG⊂E (k)beafinitesubgroup.ThenE /GisgivenbyY2=g(X)with g adegree3monic
k k
polynomialwhere
(cid:88)
X(P)=x(P)+ x(P+Q)−x(Q)
(cid:110) (cid:111)
Q∈G\ (cid:101)0Ek
(cid:88)
Y(P)=y(P)+ y(P+Q)−y(Q)
(cid:110) (cid:111)
Q∈G\ (cid:101)0Ek
Proof: Indeed,X andY areink(E )G,anditiseasilyseenthattheysatisfytherelations(1).(cid:132)
k
Aconsequenceofthethattheoremisthat,givenafinitesubgroupGofcardinality(cid:96)ofan
ellipticcurveE anequationy2= f(x)with f adegre3polynomial,itispossibletocompute
k
theWeierstrassequationofthequotientE /GatthecostofO((cid:96))additionsinE .
k k
ThemodularcurveX ((cid:96))parametrizesthesetofisomorphismclassesofellipticcurves
0
togetherwitha(cid:96)-torsionsubgroup.ForinstanceX (1)isjustthelineof j-invariants.Let
0
Φ (x,y)∈(cid:90)[x,y]betheorder(cid:96)modularpolynomial.Itiswellknownthattherootsof
(cid:96)
Φ (j(E ),·)givethe j-invariantsoftheellipticcurves(cid:96)-isogenoustoE .Sincean(cid:96)-isogenyis
(cid:96) k k
givenbyafinitesubgroupofE oforder(cid:96),weseethatΦ (x,y)cutsoutacurveisomorphic
k (cid:96)
toX ((cid:96))inX (1)×X (1).
0 0 0
GivenanellipticcurveE with j-invariant j ,thecomputationofisogeniescanbedone
k E
k
intwosteps:
• First,findthesolutionsofΦ (j ,X)whereΦ (X,Y)istheorder(cid:96)modularpolynomial;
(cid:96) E (cid:96)
thenrecoverfromaroot jEk(cid:48) thkeequationofthecorrespondingcurveEk(cid:48) whichis(cid:96)-
isogenoustoE .
k
• Next,usingVélu’sformulas,computetheisogenyE →E(cid:48).
k k
Forsome applications suchas isogeny-graphcomputation,onlythe firststepis required,
whileforotherapplicationsitisnecessarytoobtaintheexplicitequationsdescribingthe
isogeny.Notethatthefirststepisunnecessaryifonealreadyknowthepointsinthekernel
oftheisogeny.
2.2 Isogenies on abelian varieties
LetA beanabelianvarietyofdimension g overafieldk anddenotebyK(A )itsfunction
k k
field. An isogeny is a finite surjective map of abelian varieties. In the following we only
considerseparableisogeniesi.e. isogeniesπ:A →B suchthatthefunctionfieldK(A )
k k k
isafiniteseparableextensionofK(B ).Aseparableisogenyisuniquelydeterminedbyits
k
kernel,whichisafinitesubgroupofA (k).Inthatcase,thecardinalityofthekernelisthe
k
degreeoftheisogeny. Intherestofthispaper,by(cid:96)-isogenyfor(cid:96)>0,wealwaysmeana
((cid:96),···,(cid:96))-isogenywhere((cid:96),···,(cid:96))∈(cid:90)g.
7
2 ComputingIsogenies
WehaveseenthatitispossibletodefineamodularcorrespondencebetweentheIgusa
invariants,parameterizingthesetofdimension2principallypolarizedabelianvarieties,but
the coefficients explosion of the related modular polynomials makes it computationally
inefficient. In orderto mitigate this problem and obtain formulas suitable forgeneral g-
dimensionalabelianvarieties,weusethemodulispaceofmarkedabelianvarieties.
Let g ∈(cid:78)∗andletn∈(cid:78)besuchthat2|n.Letn=(n,n,...,n)∈(cid:90)g,andZ(n)=(cid:90)g/n(cid:90)g.
Wedenote(cid:77) themodularspaceofmarkedabelianvarieties(A ,(cid:76),Θ )where(cid:76) isa
n k A
polarizationandΘ issymmetricthetastructureΘ oftypeZ(n)(see[kMum66]). The
A A
forgetting map (A ,(cid:76)k ,Θ ) (cid:55)→ (A ,(cid:76)) is a finite mapk from (cid:77) to the moduli space of
k A k n
abelianvarietieswithapolakrizationoftypeZ(n).
Werecall[Mum67a]thatif4|n,then(cid:77) isopenintheprojectivevarietydescribedby
n
thefollowingequationsin(cid:80)(k(Z(n))):
(cid:0) (cid:88) χ(t)a a (cid:1).(cid:0) (cid:88) χ(t)a a (cid:1)=
x+t x+t u+t u+t
t∈Z(2) t∈Z(2)
(cid:0) (cid:88) χ(t)a a (cid:1).(cid:0) (cid:88) χ(t)a a (cid:1) (2)
z−x+t z−y+t z−u+t z−v+t
t∈Z(2) t∈Z(2)
a =a
x −x
ˆ
forallx,y,u,v∈Z(n),suchthatx+y+u+v=2z andallχ ∈Z(2).
In[FLR09],wehavedescribedamodularcorrespondenceϕ:(cid:77) →(cid:77) ×(cid:77) for(cid:96)∈(cid:78)∗,
(cid:96)n n n
whichcanbeseenasageneralizationofthemodularcorrespondenceX ((cid:96))→X (1)×X (1)
0 0 0
forellipticcurves.Let p and p bethecorrespondingprojections(cid:77) ×(cid:77) →(cid:77) ,andlet
1 2 n n n
ϕ = p ◦ϕ,ϕ = p ◦ϕ.Themapϕ :(cid:77) →(cid:77) issuchthat(x,ϕ (x))aremodularpoints
1 1 2 2 1 (cid:96)n n 1 (cid:16) (cid:17)
corresponding to (cid:96)-isogenous varieties We recall that ϕ is defined by ϕ (a ) =
1 1 i i∈Z((cid:96)n)
(a ) whereZ(n)isidentifiedasasubgroupofZ((cid:96)n)bythemapx(cid:55)→(cid:96)x.
i i∈Z(n)
Supposethatwearegivenamodularpoint(b ) correspondingtothemarkedabelian
i i∈Z(n)
variety(B ,(cid:76) ,Θ ).IfB istheJacobianvarietyofanhyperellipticcurve,onemayrecover
k 0 B k
theassociatedmodkularpointforn=4viaThomaeformulas[Mum84].
Supposefornowthat4|nandthat(cid:96)isprimeton.Ouralgorithmworksintwosteps:
1. ModularcomputationComputeamodularpoint(a ) ∈ϕ−1(cid:128)(b ) (cid:138).This
i i∈Z((cid:96)n) 1 i i∈Z(n)
canbedoneviathespecializedGröbnerbasisalgorithmdescribedin[FLR09],butsee
alsoSection6foramoreefficientmethod.
2. Vélu’slikeformulasUsetheadditionformulainB tocomputetheisogenyπˆ:B →
k k
A associatedtothemodularpointsolution.Here(A ,(cid:76),Θ )isthemarkedabelian
k k A
varietycorrespondingto(a ) .ThisstepisdescribedinSkection5.
i i∈Z((cid:96)n)
WecanalsocomputeanisogenygivenbyitskernelK byusingtheresultsofSection6.1
toconstructthecorrespondingmodularpoint(a ) fromK.Wethushaveacomplete
i i∈Z((cid:96)n)
generalizationofVélu’sformulasforhigherdimensionalabelianvarietiessincethereconstruc-
tionofthemodularpoint(a ) fromthekernelK onlyrequirestheadditionformulas
i i∈Z((cid:96)n)
8
2 ComputingIsogenies
inB (togetherwiththeextractionof(cid:96)th-roots).InSection6.3weexplainhowtousethisto
k
speedupStep1ofouralgorithm(wecallthisStep1’).
Ifthekerneloftheisogenyisunknown,themosttime-consumingpartofouralgorithm
is the computation ofa maximalsubgroupofrank g ofthe (cid:96)-torsion,whichmeans that
currently,with g =2wecangoupto(cid:96)=31relyingonthecurrentstate-of-the-artimplemen-
tation[GS08].InordertospeedupStep2,whichrequiresO((cid:96)g)additionstobeperformed
inB ,andcomputewithacompactrepresentation,itisimportanttoconsiderthesmallest
k
possiblen.Ifn=2,wecannotprovethatthemodularsystemtobesolvedinStep1isof
dimension0.HoweverStep1’,whichisfaster,doesnotrequireamodularsolutionbutonly
thekerneloftheisogeny,soouralgorithmworkswithn=2too.Note,however,thatsome
caremustbetakenwhencomputingadditionsonB ,sincethealgebraicthetafunctionsonly
k
giveanembeddingoftheKummervarietyofB forn=2.
k
For an actual implementation the case n = 2 is critical (it allows for a more compact
representationofthepointsthann=4:wegainafactor2g,itallowsforafasteraddition
chain,seeSection5.1.1,butmostimportantlyitreducesthemostconsumingpartofour
algorithm,thecomputationofthepointsof(cid:96)-torsion,sincetherearehalfasmuchsuchpoints
ontheKummervariety).Foreachalgorithmthatweuse,wegiveanexplanationonhowto
adaptitforthelevel2case:seeSection4.2.1andtheendofSections5.2,6.1,6.3and7.2.
Theassumptionthatn isprimeto(cid:96)isnotnecessaryeitherbutthereisoneimportant
differenceinthiscase.SupposethatwearegivenB [(cid:96)].SinceB isgivenbyathetastructure
k k
ofleveln,wealsohaveB [n].If(cid:96)isprimeton,thisgivesusB [(cid:96)n],andwecanuseStep1’
k k
toreconstituteamodularpointoflevel(cid:96)n.If(cid:96)isnotprimeton,wehavetocomputeB [(cid:96)n]
k
directly.
Itis also possible to compute more generaltypes ofisogenies via ouralgorithm. With
thenotationsofSection3,letδ =(δ ,...,δ )beasequenceofintegerssuchthatδ |δ ,
0 1 g i i+1
andlet(b ) ∈(cid:77) beamodularpointcorrespondingtoanabelianvarietyB . Let
i i∈Z(δ) δ k
δ(cid:48) =((cid:96) ,...,(cid:96) )0(where0(cid:96) |(cid:96) )anddefineδ =(δ (cid:96) ,...,δ (cid:96) ).Let(a ) ∈(cid:77) be
1 g i i+1 1 1 g g i i∈Z(δ) δ
(cid:128) (cid:138)
suchthatϕ (a ) =(b ) .Thethetanullpoint(a ) correspondstoanabelian
i i∈Z(δ) i i∈Z(δ) i i∈Z(δ)
0
varietyA ,suchthatthereisa((cid:96) ,···,(cid:96) )-isogenyπ:A →B ,whichcanbecomputedby
k 1 g k k
theisogenytheorem[Mum66](seeSection3.2).TheisogenywecomputeinStep2isthe
contragredientisogenyπˆ:B →A oftype((cid:96) /(cid:96) ,(cid:96) /(cid:96) ,···,1,(cid:96) ,(cid:96) ,···,(cid:96) ).Usingthe
k k g 1 g 2 g g g
modularcorrespondenceϕtogobacktoamodularpointoflevelδ (seeSection3.3)givesan
0
isogenyoftype((cid:96) /(cid:96) ,(cid:96) /(cid:96) ,···,1,(cid:96) (cid:96) ,(cid:96) (cid:96) ,···,(cid:96) (cid:96) ).Fortheclarityoftheexposition,
g 1 g 2 1 g 2 g g g
wewillsticktothecaseδ =nandδ=(cid:96)nandweleavetothereadertheeasygeneralization.
0
Letusmakesomeremarksonouralgorithm.Firstnotethattocompute(cid:96)-isogenies,we
startfromathetanullpointoflevelntogetathetanullpointoflevel(cid:96)n.Wecanthengo
backtoapointofleveln(seeSection3.3),butinthiscasewearecomputing(cid:96)2-isogenies.A
secondremarkisthatallourcomputationsaregeometric,notarithmetic,sincetheprojective
embeddinggivenbythetafunctionsoflevel(cid:96)n isnotrational.Alastremarkisthatsince
weusedifferentmodulispaces,ourmethodisnotastraight-upgeneralizationofthegenus-1
case.Inparticular,computingamodularpointsolution(a ) isthesameaschoosingan
i i∈Z((cid:96)n)
(cid:96)-isogenyandathetastructureoflevel(cid:96),sotherearemanymoremodularsolutionsthan
thereis(cid:96)-isogenies.Hence,asnotedintheintroduction,themostefficientmethodinour
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