The tropical $j$-invariant

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1、THETROPICALj-INVARIANTERICKATZ,HANNAHMARKWIG,THOMASMARKWIGAbstract.If(Q,A)isamarkedpolygonwithoneinteriorpoint,thenageneralpolynomialf∈K[x,y]withsupportAdefinesanellipticcurveCfonthetoricsurfaceXA.IfKhasanon-archimedeanvaluationintoRwecantropicalizeCftog

2、etatropicalcurveTrop(Cf).IftheNewtonsubdivisioninducedbyfisatriangulation,thenTrop(Cf)willbeagraphofgenusoneandweshowthatthelatticelengthofthecycleofthatgraphisthenegativeofthevaluationofthej-invariantofCf.1.IntroductionPreviousworkbyGrishaMikhalkin[13]

3、,byMichaelKerberandHannahMarkwig[11]andbyMagnusVigeland[18]showsthatthelengthofthecycleofatropicalcurveofgenusonehaspropertieswhichoneclassicallyattributestothej-invariantofanellipticcurvewithoutgivingadirectlinkbetweenthesetwonumbers.In[9]weestablished

4、suchadirectlinkforplanecubicsbyshowingthatthetropicalizationofthej-invariantisingeneralthenegativeofthecyclelength.Inthepresentpaperwegeneralizethisresulttoellipticcurvesonothertoricsurfacesusingthesamemethods.Moreprecisely,if(Q,A)isamarkedpolygonwithon

5、einteriorpoint,thenageneralpolynomialf∈K[x,y]withsupportAdefinesanellipticcurveCfonthetoricsurfaceXA.IfKhasanon-archimedeanvaluationwecantropicalizeCftogetatropicalcurveTrop(Cf).IftheNewtonsubdivisioninducedbyfisatriangulation,thenTrop(Cf)willbeagraphofg

6、enusoneandweshowinourmainresultinTheorem6.4thatthelatticelengthofthecycleofthegraphisthenegativeofthevaluationofthej-invariant.arXiv:0803.4021v2[math.CO]31Mar2008Inthecasewherethetriangulationisunimodular,i.e.allthetriangleshavearea1,2thisresultwasindep

7、endentlyderivedbyDavidSpeyer[17,Proposition9.2]usingTateuniformizationofellipticcurves.DavidSpeyer’sresultismoregeneralthoughinthesensethatitappliestocurvesinarbitrarytoricvarieties.Thispaperisorganizedasfollows.InSection2weconsidertoricsurfacesdefinedby

8、amarkedlatticepolygonwithoneinteriorpoint,werecalltheclassificationofthesepolygonsandweconsidertheimpactonthej-invariantforthecorrespondingellipticcurves.Section3recallsthenotionoftropicalizationandofplanetropicalcurves.Wethenintr