From Curves to Tropical Jacobians and Back

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by  Barbara Bolognese

From Curves to Tropical Jacobians and Back
BarbaraBolognese,MadelineBrandt,LynnChua
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Abstract Givenacurvedefinedoveranalgebraicallyclosedfieldwhichiscomplete
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with respect to a nontrivial valuation, we study its tropical Jacobian. This is done
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by first tropicalizing the curve, and then computing the Jacobian of the resulting

] weightedmetricgraph.Ingeneral,itisnotknownhowtofindtheabstracttropical-
G
izationofacurvedefinedbypolynomialequations,sinceanembeddedtropicaliza-
A tionmaynotbefaithful,andthereisnoknownalgorithmforcarryingoutsemistable
. reduction in practice. We solve this problem in the case of hyperelliptic curves by
h
t studyingadmissiblecovers.Wealsodescribehowtotakeaweightedmetricgraph
a
andcomputeitsperiodmatrix,whichgivesitstropicalJacobianandtropicaltheta
m
divisor.Lastly,wedescribethepresentstatusofreversingthisprocess,namelyhow
[ tocomputeacurvewhichhasagivenmatrixasitsperiodmatrix.

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v
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9 1 Introduction
1
3
0 WedescribetheprocessoftakingacurveandfindingitstropicalJacobian.Weaim
. tocarryouteachstepasalgorithmicallyaspossible,however,somestepscannotyet
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be completed in such a way. We now give a brief overview of the steps involved,
0
7 whichwedepictinFigure1.
1 Let k be an algebraically closed field which is complete with respect to a non-
v: archimedean valuation v, and let X be a nonsingular curve of genus g over k. Let
i
X
BarbaraBolognese
r TheUniversityofSheffield,Sheffield,UKe-mail:[email protected]
a
MadelineBrandt
DepartmentofMathematics,UniversityofCalifornia,Berkeley,970EvansHall,Berkeley,CA,
94720,e-mail:[email protected]
LynnChua
DepartmentofElectricalEngineeringandComputerScience,UniversityofCalifornia,Berkeley,
643SodaHall,Berkeley,CA,94720,e-mail:[email protected]
1
2 BarbaraBolognese,MadelineBrandt,LynnChua
R be the valuationring of k with maximal ideal m, and let k=R/m be its residue
field.WecanassociatetoX itsabstracttropicalization,whichisthedualweighted
metricgraphΓ ofthespecialfiberofasemistablemodelofX.Inpractice,finding
the abstract tropicalization of a general curve is difficult and there is no known
algorithmtodothisingeneral[CJar,Remark3].Inthispaper,wesolvethisproblem
for hyperelliptic curves, which is new to the literature, and discuss known results
towardsfindingabstracttropicalizationsofallcurves.
Given Γ, we compute its period matrix Q . This corresponds to the tropical
Γ
JacobianofthecurveX.BytakingtheVoronoidecompositiondualtotheDelaunay
subdivisioncorrespondingtoQ ,weobtainthetropicalthetadivisor.Thisprocess
Γ
canalsobeinverted.ThesetofperiodmatricesthatariseasthetropicalJacobianof
acurveisthetropicalSchottkylocus.Startingwithaprincipallypolarizedtropical
abelianvarietywhoseperiodmatrixQisknowntolieinthetropicalSchottkylocus,
wegiveaproceduretocomputeacurvewhosetropicalJacobiancorrespondstoQ.
This process of associating a tropical Jacobian to a curve can also be carried
outbylookingatclassicalJacobiansofcurves.Jacobiansofcurvesareprincipally
polarizedabelianvarietiesinanaturalway;theyarethemostwellknownandexten-
sivelystudiedamongabelianvarieties.Bothalgebraiccurvesandabelianvarieties
have extremely rich geometries, which cannot be fully understood in many cases.
Jacobiansprovidealinkbetweensuchgeometries,andtheyoftenrevealhiddenfea-
turesofalgebraiccurveswhichcannotbeuncoveredotherwise.Inordertoassociate
atropicalJacobiantoacomplexalgebraiccurveC,onefirstconstructsitsclassical
Jacobian
J(C):=H0(C,ω )∗/H (C,Z), (1)
C 1
wherewedenotebyω thecotangentbundleofthecurve.Thiscomplextorusad-
C
mits a natural principal polarizationΘ, called the theta divisor, such that the pair
(J(C),Θ)isaprincipallypolarizedabelianvariety.Wecanthenobtainthetropical
JacobianbytakingtheBerkovichskeletonoftheclassicalJacobian.Baker-Rabinoff,
and independently Viviani, proved that this alternative path gives the same result
as the procedure we describe in this paper [BR15, Viv13]. However, the classical
process hides more difficulties on the computational level and proves much more
challenging to carry out in explicit examples. Methods have been implemented in
theMaplepackagealgcurvesforcomputingJacobiansnumericallyoverC[DP11].
ThestructureofthispaperisdepictedinFigure1.InSection2wefindtheab-
stract tropicalization of all hyperelliptic curves, a result which is new to this pa-
per.Then,inSection3wediscussissueswithembeddedtropicalization,statesome
knownresultsaboutcertifyingafaithfultropicalization,andoutlinetheprocessof
semistablereduction.Thisstepoftheprocedureisfarfrombeingalgorithmic,and
this section focuses on examples which provide obstacles to doing this in general.
InSection4wedescribehowtofindtheperiodmatrixofaweightedmetricgraph.
Then we define and give examples of the tropical Jacobian and its theta divisor in
Section 5. In Section 6, we discuss the tropical Schottky problem. Finally, we de-
scribetheobstaclestoreversingthisprocessinSection7.
FromCurvestoTropicalJacobiansandBack 3
CurveX
Hyperelliptic
AllCurves
Section7.3 Curves
Section3
Section2
WeightedMetricGraphΓ
Section7.2 Section4
PeriodMatrixQ
Section7.1 Section5
TropicalJacobianandThetaDivisor(J,Θ)
Fig.1 Thestructureofthepaper,andtheprocessofassociatingatropicalJacobiantoacurve.
Dashedarrowsindicatestepswhicharenotyetachievablealgorithmically.
2 HyperellipticCurves
We now study the problem of finding the abstract tropicalization of hyperelliptic
curves.Inthecaseofellipticcurves,thetropicalizationcanbecompletelydescribed
intermsofthe j-invariant[KMM08].Similarly,tropicalizationsofgenus2curves
(allofwhicharehyperelliptic)canbedescribedbystudyingtropicalIgusainvari-
ants [Hel]. This problem was also solved in genus 2 by studying the curve as a
doublecoverofP1ramifiedat6points,asdonein[RSS14,Section5].
Inthissection,wegeneralizethelattermethodtofindtropicalizationsofallhy-
perellipticcurves,whichwaspreviouslynotknown.LetX beanonsingularhyper-
elliptic curve of genus g over k, an algebraically closed field which is complete
withrespecttoanontrivial,non-archimedeanvaluationv.OurgoalistofindΓ,the
abstracttropicalizationofX.
WedenotebyM themodulispaceofgenusgcurveswithnmarkedpoints,see
g,n
[HM98]forathoroughintroduction.ThespaceM mapssurjectivelyontothe
0,2g+2
hyperellipticlocusinsideM byidentifyingeachhyperellipticcurveofgenusgwith
g
adoublecoverofP1ramifiedat2g+2markedpoints.Whenthecharacteristicofk
4 BarbaraBolognese,MadelineBrandt,LynnChua
isnot2,thenormalformfortheequationofahyperellipticcurveisy2= f(x),where
f(x)hasdegree2g+2,andtherootsof f aredistinct.Then,thesearepreciselythe
ramificationpoints.
The space Mtr is the tropicalization of M . A phylogenetic tree is a
0,2g+2 0,2g+2
metric tree with leaves labeled {1,...,m} and no vertices of degree 2. Such a tree
isuniquelyspecifiedbythedistancesd betweentheleaves.WeseethatMtr
ij 0,2g+2
parametrizes the space of phylogenetic trees with 2g+2 leaves using the Plu¨cker
embedding to map M into the Grassmannian Gr(2,k2g+2) [MS15, Chapter
0,2g+2
4.3]:
{(a :b)}2g+2(cid:55)→(p :p :···:p ) where p =ab −a b. (2)
i i i=1 12 13 2g+1,2g+2 i,j i j j i
Thedistancesarethengivenbyd =−2p +n·1forasuitableconstantn[RSS14,
ij ij
Section5].UsingtheNeighborJoiningAlgorithm[PS05,Algorithm2.41],onecan
constructtheuniquetree,alongwiththelengthsofitsinterioredges,usingonlythe
leafdistancesd asinput.Sincethelengthsofleafedgescanonlybedefinedupto
ij
addingaconstantlengthtoeachleaf,wethinkofthistreeasametricgraphwhere
theleaveshaveinfinitelengthandtheinterioredgeshavelengthsasdescribed.
ThisrealizesMtr asa2g−1dimensionalfaninsideTP(2g2+2)−1(cf.[MS15,
0,2g+2
Section 2.5]). The space Mtr can be computed as a tropical subvariety of
0,2g+2
TP(2g2+2)−1,sinceithasatropicalbasisgivenbythePlu¨ckerrelationsforGr(2,k2g+2)
[MS15, Chapter 4.4]. Each cone corresponds to a combinatorial type of tree (see
Figure 2), and the dimension of each cone corresponds to the number of interior
edgesinthetree.
The next step in finding the tropicalization of the hyperelliptic curve X is to
take the corresponding point in Mtr , as a tree on 2g+2 leaves, and compute
0,2g+2
a weighted metric graph in Mtr. Figure 2 gives this correspondence in the case
g
g=3. We now give some definitions related to metric graphs in order to describe
thiscorrespondenceforgeneralg,following[Cha13].
Definition2.1.AmetricgraphisametricspaceΓ,togetherwithagraphGanda
length function l :E(G)→R ∪{∞} such thatΓ is obtained by gluing intervals
>0
e of length l(e), or by gluing rays to their endpoints, according to how they are
connected in G. In this case, the pair (G,l) is called a model for Γ. A weighted
metric graph is a metric graph Γ together with a weight function on its points
w:Γ →Z≥0,suchthat∑v∈Γw(v)isfinite.
Wecalledgesofinfinitelengthinfiniteleaves,andtheseonlymeettherestofthe
graphinoneendpoint.Abridgeisanedgewhosedeletionincreasesthenumberof
connectedcomponents.
Thegenusofaweightedmetricgraph(Γ,w)is
∑w(v)+|E(G)|−|V(G)|+1, (3)
v∈Γ
FromCurvestoTropicalJacobiansandBack 5
Fig.2 Theposetofunlabeledtreeswith8leaves,andtropicalizationsofhyperellipticcurvesof
genus3.Bothareorderedbytherelationofcontractinganedge.
whereGisanymodelofΓ.Wesaythattwoweightedmetricgraphsofgenus≥2
areisomorphicifonecanbeobtainedfromtheotherviagraphautomorphisms,or
byremovinginfiniteleavesorleafverticesvwithw(v)=0,togetherwiththeedge
connectedtoit.Inthisway,everyweightedmetricgraphhasaminimalskeleton.
Amodelislooplessifthereisnovertexwithaloopedge.Thecanonicalloopless
modelofΓ,withgenusofΓ ≥2,isthegraphGwithvertices
V(G):={x∈Γ |val(x)(cid:54)=2orw(x)>0orxisthemidpointofaloop}. (4)
6 BarbaraBolognese,MadelineBrandt,LynnChua
If(G,l)and(G(cid:48),l(cid:48))arelooplessmodelsformetricgraphsΓ andΓ(cid:48),thenamorphism
oflooplessmodelsφ:(G,l)→(G(cid:48),l(cid:48))isamapofsetsV(G)∪E(G)→V(G(cid:48))∪E(G(cid:48))
suchthat
• AllverticesofGmaptoverticesofG(cid:48).
• Ife∈E(G)mapstov∈V(G(cid:48)),thentheendpointsofemustalsomaptov.
• Ife∈E(G)mapstoe(cid:48)∈E(G(cid:48)),thentheendpointsofemustmaptoverticesof
e(cid:48).
• InfiniteleavesinGmaptoinfiniteleavesinG(cid:48).
• Ifφ(e)=e(cid:48),thenl(cid:48)(e(cid:48))/l(e)isaninteger.Theseintegersmustbespecifiedifthe
edgesareinfiniteleaves.
We call an edge e∈E(G) vertical if φ maps e to a vertex of G(cid:48). We say that φ is
harmonicifforeveryv∈V(G),thelocaldegree
d = ∑ l(cid:48)(e(cid:48))/l(e) (5)
v
e(cid:51)v,
φ(e)=e(cid:48)
isthesameforallchoicesofe(cid:48)∈E(G(cid:48)).Ifitispositive,thenφ isnondegenerate.
Thedegreeofaharmonicmorphismisdefinedas
∑ l(cid:48)(e(cid:48))/l(e). (6)
e∈E(G),
φ(e)=e(cid:48)
Wealsosaythatφ satisfiesthelocalRiemann-Hurwitzconditionif:
2−2w(v)=d (2−2w(cid:48)(φ(v)))−∑(cid:0)l(cid:48)(φ(e))/l(e)−1(cid:1). (7)
v
e(cid:51)v
Ifφ satisfiesthisconditionateveryvertexvinthecanonicallooplessmodelofΓ,
thenφ iscalledanadmissiblecover[CMR16].
Definition2.2.[Cha13, Theorem 1.3] Let Γ be a weighted metric graph, and let
(G,l) denote its canonical loopless model. We say that Γ is hyperelliptic if there
existsanondegenerateharmonicmorphismofdegree2fromGtoatree.
A hyperelliptic curve will always tropicalize to a hyperelliptic weighted metric
graph,howevernoteveryhyperellipticweightedmetricgraphisthetropicalization
ofahyperellipticcurve.
Theorem2.3.[ABBR15,Corollary4.15]LetΓ beaminimalweightedmetricgraph
ofgenusg≥2.ThenthereisasmoothproperhyperellipticcurveX overkofgenus
g having Γ as its minimal skeleton if and only if Γ is hyperelliptic and for every
p∈Γ thenumberofbridgeedgesadjacentto pisatmost2w(p)+2.
Lemma2.4anditsproofgiveanalgorithmfortakingatreewith2g+2infinite
leavesandobtainingametricgraphwhichisanadmissiblecoverofthetree.
FromCurvestoTropicalJacobiansandBack 7
Lemma2.4.EverytreeT with2g+2infiniteleaveshasanadmissiblecoverφ by
auniquehyperellipticmetricgraphΓ ofgenusg,andφ isharmonicofdegree2.
Proof. LetT beatreewith2g+2infiniteleaves.Ifallinfiniteleavesaredeleted,
then a finite tree T(cid:48) remains. Let v ,...,v be the vertices of T(cid:48), ordered such that
1 k
thedistancefromv tov isgreaterthanorequaltothedistancefromv tov ,for
i k j k
i< j.
WeconstructΓ iterativelybybuildingthepreimageofeachvertexv,asserting
i
along the way that the local Riemann-Hurwitz condition holds. This also gives an
algorithmforfindingΓ.Webeginwithv ,whichhasapositivenumbern ofleaf
1 1
edgesinT.Sinceφ hasdegree2,itmustbelocallyofdegree1or2ateveryvertex
ofΓ.Sincethepreimageofeachinfiniteleafmustbeaninfiniteleaf,wewillattach
n infiniteleavesatthepreimageφ−1(v )inΓ.
1 1
AtanyvertexinΓ withinfiniteleaves,φ haslocaldegree2,hencewewillattach
toΓ aninfiniteleafesuchthatl(φ(e))/l(e)=2.Then,thereisauniquevertexinthe
preimageφ−1(v ).Otherwise,therewouldneedtobeanotheredgeinthepreimage
1
ofeachleaf,sothedegreeofthemorphismwouldbegreaterthan2.
Lete betheedgeconnectingv tosomeotherv.Therearetwopossibilities:
1 1 i
1. Thepreimageofe istwoedgesinΓ,eachwithlengthl(e ).ThelocalRiemann-
1 1
Hurwitzequationreads
2−2w(φ−1(v ))=2(2−0)−(n +0+0). (8)
1 1
Thisisonlypossibleifn iseven,andφ−1(v )hasweight(n −2)/2.
1 1 1
2. The preimage of e is one edge inΓ, with length l(e )/2. The local Riemann-
1 1
Hurwitzequationreads
2−2w(φ−1(v ))=2(2−0)−(n +1). (9)
1 1
Thisisonlypossibleifn isodd,andφ−1(v )hasweight(n −1)/2.
1 1 1
Now, we proceed to the other vertices. As long as the order of the vertices is re-
spected,ateachvertexv therewillbeatmostoneedgee whosepreimageinΓ we
i i
do not know. Then, what happens at v can be completely determined by studying
i
the local Riemann-Hurwitz data. For i>1, let n be the number of infinite leaves
i
at v plus the number of edges e∈T such that e={v,v }, j<i, and φ−1(e) is a
i i j
bridgeinΓ.Ifn >0,theneither1or2holds.However,itispossiblethatn =0,in
i i
whichcasewehaveathirdpossibility:
3. Ifn =0,letv(cid:48)∈φ−1(v).ThelocalRiemann-Hurwitzequationreads:
i i i
2−2w(v(cid:48))=d (2−0)−(0). (10)
i vi
Then we must have d = 1 and w(v(cid:48)) = 0, which implies that there are two
vi i
verticesinφ−1(v).
i
8 BarbaraBolognese,MadelineBrandt,LynnChua
Finally, we glue the pieces ofΓ as specified by T, and contract the leaf edges on
Γ. The fact that Γ has genus g is a consequence of the local Riemann-Hurwitz
condition. (cid:116)(cid:117)
We remark that this process did not require the fact that the tree had an odd
numberofleaves.Indeed,ifonerepeatsthisprocedureforsuchatree,ahyperelliptic
metric graph will be obtained. However, this graph is not the tropicalization of a
hyperellipticcurve.
Example2.5.InFigure3,wehaveatreewithverticeslabelledv ,...,v .Beginning
1 7
withv ,weobservethatn =2,whichmeansthattheedgefromv tov hastwo
1 1 1 3
edgesinitspreimage.Thesameistrueforv .Movingontov ,weseethatn =0,
2 3 3
whichmeansthatv hastwopointsinΓ whichmaptoit.Wecanconnecttheedges
3
from φ−1(v ) and φ−1(v ) to the two points in φ−1(v ). Since φ−1(v ) has two
1 2 3 3
points, the edge from v to v corresponds to two edges in Γ, so n =2, which
3 4 4
means that the edge from v to v also splits. Next, n =1, which means that the
4 5 5
edgefromv tov correspondstoabridgeinΓ.Then,n =4,whichmeansthatthe
5 6 6
edgev tov splits,andthevertexmappingtov hasgenus1.Lastly,sincen =2,
6 7 6 7
thepointmappingtov hasgenus0.Alledgesdepictedintheimagehavethesame
7
lengthasthecorrespondingedgesinthetree,exceptforthebridge,whichhaslength
equaltohalfthelengthofthecorrespondingedgeinthetree.
Fig.3 ThetreeT with12infiniteleavesfromExample2.5andthehyperellipticweightedmetric
graphΓ ofgenus5whichadmissiblycoversT byφ.
Thefollowingtheoremshowsthatthismetricgraphisactuallythetropicalization
ofahyperellipticcurve.
FromCurvestoTropicalJacobiansandBack 9
Theorem2.6.Letg≥1beaninteger.LetX beahyperellipticcurveofgenusgover
k,givenbytakingthedoublecoverofP1 ramifiedat2g+2points p ,...,p .If
1 2g+2
T isthetreewhichcorrespondstothetropicalizationofP1 withthemarkedpoints
p ,...,p described above, and Γ is the unique hyperelliptic weighted metric
1 2g+2
graphwhichadmitsanadmissiblecovertoT,thenΓ istheabstracttropicalization
ofX.
Proof. Thisfollowsfrom[CMR16],Remark20andTheorem4.Indeed,thehyper-
ellipticlocusofM canbeunderstoodasthespaceH ((2),...,(2))ofadmis-
g g→0,2
sible covers with 2g+2 ramification points of order 2. Its tropicalization is con-
an
structedandstudiedin[CMR16].ThespaceH ((2),...,(2))istheBerkovich
g→0,2
analytificationofH ((2),...,(2)),andthusapointX isrepresentedbyanad-
g→0,2
missiblecoveroverSpec(K)with2g+2ramificationpointsoforder2.ByTheorem
4in[CMR16],thediagram
an
H ((2),...,(2)) (11)
g→0,2
bran srcan
(cid:119)(cid:119)
(cid:39)(cid:39)
Man Man
trop
0,2g+2 g
(cid:15)(cid:15)
trop
trop H ((2),...,(2)) trop
g→0,2
brtrop srctrop
(cid:15)(cid:15) (cid:119)(cid:119) (cid:39)(cid:39) (cid:15)(cid:15)
Mtrop Mtrop
0,2g+2 g
commutes. The morphisms src take a cover to its source curve, marked at the
entire inverse image of the branch locus, and the morphisms br take a cover
to its base curve, marked at its branch points. We start with an element X of
Han ((2),...,(2)), and we wish to find trop(srcan(X))∈M trop. The unicity in
g→0,2 g
Lemma2.4enablesustofindaninverseforbrtrop.ThenT =trop(bran(X)),andso
bycommutativityofthediagram,trop(srcan(X))=srctrop((brtrop)−1(T))=Γ. (cid:116)(cid:117)
Example2.7.([Stuar,Problem2onCurves])Considerthecurve
y2=(x−1)(x−2)(x−3)(x−6)(x−7)(x−8)
withthe5-adicvaluation.InMtrop,thisgivesusthepoint
0,6
(0,0,1,0,0,0,0,1,0,0,0,1,0,0,0)=(p ,p ,...,p ).
1,2 1,3 5,6
Whennisanyinteger,thisgivesusatreemetricof
(n,n,n−2,n,n,n,n,n−2,n,n,n,n−2,n,n,n)=(d ,d ,...,d ).
1,2 1,3 5,6
10 BarbaraBolognese,MadelineBrandt,LynnChua
Then,thisisametricforthetreeontheleftofFigure4,andthemetricgraphonthe
right.
Fig.4 ThisisthetreeandmetricgraphforProblem2fromtheCurvesworksheet.
3 OtherCurves
Outside of the hyperelliptic case, finding the abstract tropicalization of a curve is
very hard. In this section, we highlight some of the difficulties and discuss two
approaches to this problem: faithful tropicalization and semistable reduction. We
offerthefollowingexampleasmotivationforwhythisisadifficultproblem.
Example3.1.([Stuar,Problem9onAbelianCombinatorics])Webeginwithacurve
inP2,givenbythezerolocusof
f(x,y,z)=41x4+1530x3y+3508x3z+1424x2y2+2490x2yz
−2274x2z2+470xy3+680xy2z−930xyz2+772xz3
+535y4−350y3z−1960y2z2−3090yz3−2047z4,
definedoverQ .Here,theinducedregularsubdivisionoftheNewtonpolygonwill
2
be trivial, since the 2-adic valuation of the coefficients on the x4, y4, z4 terms is
each0.Therefore,wecandetectnoinformationaboutthestructureoftheabstract
tropicalizationfromthisembeddedtropicalization.InExample3.4,wewillpicka
niceenoughchangeofcoordinatestoallowustofindafaithfultropicalization.
3.1 FaithfulTropicalization
WebrieflydiscusstheBerkovichskeletonofacurve.Letkbeanalgebraicallyclosed
field which is complete with respect to a nontrivial, nonarchimedean valuation v.