The tropical $j$-invariant

《The tropical $j$-invariant》由會員上傳分享，免費在線閱讀，更多相關(guān)內(nèi)容在行業(yè)資料-天天文庫。

1、THETROPICALj-INVARIANTERICKATZ,HANNAHMARKWIG,THOMASMARKWIGAbstract.If(Q,A)isamarkedpolygonwithoneinteriorpoint,thenageneralpolynomialf∈K[x,y]withsupportAde?nesanellipticcurveCfonthetoricsurfaceXA.IfKhasanon-archimedeanvaluationintoRwecantropicalizeCfto

2、getatropicalcurveTrop(Cf).IftheNewtonsubdivisioninducedbyfisatriangulation,thenTrop(Cf)willbeagraphofgenusoneandweshowthatthelatticelengthofthecycleofthatgraphisthenegativeofthevaluationofthej-invariantofCf.1.IntroductionPreviousworkbyGrishaMikhalkin[1

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

4、hedsuchadirectlinkforplanecubicsbyshowingthatthetropicalizationofthej-invariantisingeneralthenegativeofthecyclelength.Inthepresentpaperwegeneralizethisresulttoellipticcurvesonothertoricsurfacesusingthesamemethods.Moreprecisely,if(Q,A)isamarkedpolygonwi

5、thoneinteriorpoint,thenageneralpolynomialf∈K[x,y]withsupportAde?nesanellipticcurveCfonthetoricsurfaceXA.IfKhasanon-archimedeanvaluationwecantropicalizeCftogetatropicalcurveTrop(Cf).IftheNewtonsubdivisioninducedbyfisatriangulation,thenTrop(Cf)willbeagra

6、phofgenusoneandweshowinourmainresultinTheorem6.4thatthelatticelengthofthecycleofthegraphisthenegativeofthevaluationofthej-invariant.arXiv:0803.4021v2[math.CO]31Mar2008Inthecasewherethetriangulationisunimodular,i.e.allthetriangleshavearea1,2thisresultwa

7、sindependentlyderivedbyDavidSpeyer[17,Proposition9.2]usingTateuniformizationofellipticcurves.DavidSpeyer’sresultismoregeneralthoughinthesensethatitappliestocurvesinarbitrarytoricvarieties.Thispaperisorganizedasfollows.InSection2weconsidertoricsurfacesd

8、e?nedbyamarkedlatticepolygonwithoneinteriorpoint,werecalltheclassi?cationofthesepolygonsandweconsidertheimpactonthej-invariantforthecorrespondingellipticcurves.Section3recallsthenotionoftropicalizationandofplanetropicalcurves.Wethenintr