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The Tropical j-Invariant

Published online by Cambridge University Press:  01 February 2010

Eric Katz
Affiliation:
Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712, eekatz@math.utexas.edu
Hannah Markwig
Affiliation:
CRC “Higher Order Structures for Mathematics”, Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße 3–5, 37073 Göttingen, Germany, hannah@uni-math.gwdg.de, http://www.uni-math.gwdg.de/hannah
Thomas Markwig
Affiliation:
Fachbereich Mathematik, Technische Universität Kaiserslautern, 67653 Kaiserslautern, Germany, keilen@mathematik.uni-kl.de, http://www.mathematik.uni-kl.de/~keilen

Abstract

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If (Q, A) is a marked polygon with one interior point, then a general polynomial f belonging to K[x,y] with support A defines an elliptic curve Cf on the toric surface XA. If K has a non-archimedean valuation into R we can tropicalize Cf to get a tropical curve Trop(Cf). If in the Newton subdivision induced by f is a triangulation and the interior point occurs as the vertex of a triangle, then Trop(Cf) will be a graph of genus one and we show that the lattice length of the cycle of that graph is the negative of the valuation of the j-invariant of Cf.

Type
Research Article
Copyright
Copyright © London Mathematical Society 2009

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