# The geometric group law on a tropical elliptic curve

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The geometric group law on a tropical elliptic curve
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The geometric group law on a tropical ellipticcurve Bachelor Thesis written by Nina Otterunder the supervision of Prof. Richard Pink.Fall 2012 ETH Z¨urich.  Contents Acknoledgements 40. Introduction 51. Tropical curves 72. Tropical intersection theory 153. Tropical elliptic curves and the algebraic group law 174. The geometric group law 19Bibliography 29 3  4 Acknoledgements I wish to express my thanks to Professor Pink for supervising my thesis. I have beenextremely lucky to have had a supervisor who dedicated so much time and care to my workand could provide such critical insight into the art of mathematical writing. Thanks toProfessor Pink I have learned that one of the causes of bad mathematical writing is thelack of a clear idea about what is to be written. I think that this is the most importantthing that I learned while writing this thesis; for this I am very grateful to Professor Pink.  “Les pays exotiques m’apparaissaient comme le contrepied des nˆotres, leterme d’antipodes trouvait dans ma pens´ee un sens plus riche et plus na¨ıf que son contenu litt´eral. On m’eˆut fort ´etonn´e en disant qu’une esp`eceanimale ou v´eg´etale pouvait avoir le mˆeme aspect des deux cˆot´es du globe.Chaque animal, chaque arbre, chaque brin d’herbe, devait ˆetre radicale-ment di ﬀ  ´erent, a ﬃ cher au premier coup d’œil sa nature tropicale. LeBr´esil s’esquissait dans mon imagination comme des gerbes de palmierscontourn´es, dissimulant des architectures bizarres, le tout baign´e dansune odeur [...] (de) parfum brˆul´e.”  1 Claude L´evi-Strauss,  Tristes tropiques 0. Introduction Tropical mathematics is an area of theoretical computer science which saw its beginningin the 1970s. It was concerned with the study of min-plus semirings - semirings in whichthe operations are given by taking addition and minimum on certain sets as the set of natural numbers or the ordinal numbers smaller than a certain cardinal [ 10 ]. Among itspioneers was the Brazilian mathematician Imre Simon, in honour of whom several Frenchmathematicians - Dominique Perrin [ 10 ] and Christian Cho ﬀ  rut [ 12 ] among others - beganto call these semirings “tropical semirings”. To use the words of Sturmfels and Speyer, theadjective tropical  ‘simply stands for the French view of Brazil.’   [ 13 ]Tropical geometry is an area of algebraic geometry which is concerned with the studyof varieties over the tropical semiring of real numbers. Tropical varieties are rational poly-hedral complexes satisfying a certain equilibrium condition on the vertices.Given an algebraically closed ﬁeld  K   with valuation  v  and a non-zero polynomial in twovariables over  K  , it is possible to assign to an algebraic curve  C   =  { ( x,y ) ∈ K  2 |  f  ( x,y ) =0 }  a tropical variety  { ( v ( x ) ,v ( y ))  ∈  R 2 |  ( x,y )  ∈  C  }  which preserves many propertiesof the algebraic curve. Since tropical varieties are combinatorial objects, this method iswidely used to translate algebraic-geometric problems into combinatorial ones, for whicha solution may be easier to ﬁnd.A lot of work is being done to translate the language of algebraic geometry into tropicalgeometry. Often a translation is justiﬁed by its correct use in the tropical setting ratherthan by why it is the correct translation. 1 “Tropical countries, as it seemed to me, must be the exact opposite of our own, and the name of Antipodes had for me a sense at once richer and more ingenuous than its literal derivation. I shouldhave been astonished to hear it said that any species, whether animal or vegetable, could have the sameappearance on both sides of the globe. Every animal, every tree, every blade of grass, must be completelydi ﬀ  erent and give immediate notice, as it were, of its tropical character. I imagined Brazil as a tangledmass of palm-leaves, with glimpses of strange architecture in the middle distance, and an all-permeatingsmell of burning perfume.”(Translation by John Russell) 5
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