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The Tropical j-Invariant

Published online by Cambridge University Press:  01 February 2010

Eric Katz ,
Hannah Markwig  and
Thomas Markwig
Eric Katz
Affiliation:
Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712, [email protected]
Hannah Markwig
Affiliation:
CRC “Higher Order Structures for Mathematics”, Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße 3–5, 37073 Göttingen, Germany, [email protected], http://www.uni-math.gwdg.de/hannah
Thomas Markwig
Affiliation:
Fachbereich Mathematik, Technische Universität Kaiserslautern, 67653 Kaiserslautern, Germany, [email protected], 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
Information
LMS Journal of Computation and Mathematics , Volume 12 , 2009 , pp. 275 - 294
Copyright
Copyright © London Mathematical Society 2009

References

1.Allermann, Lars and Rau, Johannes, ‘First steps in tropical intersection theory.’ ArXiv:0709.3705.Google Scholar
2.Einsiedler, Manfred, Kapranov, Mikhail and Lind, Douglas, ‘Non-archimedean amoebas and tropical varieties.’ J. Reine Angew. Math. 601 (2006) 139–157.Google Scholar
3.Fulton, William, Introduction to toric varieties (Princeton University Press, 1993).CrossRefGoogle Scholar
4.Gawrilow, Ewgenij and Joswig, Michael, ‘polymake 2.3.’Tech. rep., TU Berlin and TU Darmstadt, (1997). http://www.math.tu-berlin.de/polymake.Google Scholar
5.Gelfand, Israel M., Kapranov, Mikhail M. and Zelevinsky, Andrei V., Discriminants, resultants, and multidimensional determinants (Birkhäuser, 1994).CrossRefGoogle Scholar
6.Greuel, G.-M., Pfister, G. and Schönemann, H., ‘Singular 3.0.’ A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern, (2005). http://www.singular.uni-kl.de.Google Scholar
7.Hartshorne, Robin, Algebraic geometry (Springer, 1977).CrossRefGoogle Scholar
8.Jensen, Anders Nedergaard, Markwig, Hannah and Markwig, Thomas, ‘tropical.lib.’A singular 3.0 library for computations in tropical geometry, (2007). http://www.singular.uni-kl.de/~keilen/de/tropical.html.Google Scholar
9.Katz, Eric, Markwig, Hannah and Markwig, Thomas, ‘The j-invariant of a plane tropical cubic.’ J. Algebra 320 (2008) 3832–3848.CrossRefGoogle Scholar
10.Katz, Eric, Markwig, Hannah and Markwig, Thomas, ‘jinvariant.lib.’ A singular 3.0 library for computations with j-invariants in tropical geometry, (2007). http://www.singular.uni-kl.de/~keilen/de/jinvariant.html.Google Scholar
11.Kerber, Michael and Markwig, Hannah, ‘Counting tropical elliptic curves with fixed j-invariant.’ To appear in Commentarii Mathematici Helvetici. Math.AG/0608472.Google Scholar
12.Mikhalkin, Grigory, ‘Enumerative tropical geometry in R2.’ J. Amer. Math. Soc. 18 (2005) 313–377. Math.AG/0312530.CrossRefGoogle Scholar
13.Mikhalkin, Grigory, ‘Tropical geometry and its applications.’ International Congress of Mathematicians, Eur. Math. Soc., vol. II (2006) pp. 827–852.Google Scholar
14.Poonen, Bjorn and Rodriguez-Villegas, Fernando, ‘Lattice polygons and the number 12.’ Amer. Math. Monthly 107 (2000) 238–250.CrossRefGoogle Scholar
15.Rabinowitz, Stanley, ‘A census of convex lattice polygons with at most one interior lattice point.’ Ars Combin. 28 (1989) 83–96.Google Scholar
16.Rambau, Jörg, ‘TOPCOM: Triangulations of point configurations and oriented matroids.’ ‘Mathematical Software - ICMS 2002,’ (eds Cohen, Xiao-Shan Gao Arjeh M. and Takayama, Nobuki). (2002) pp. 330–340.Google Scholar
17.Speyer, David,‘Uniformizing tropical curves I:Genus zero and one.’ ArXiv:0711.2677.Google Scholar
18.Vigeland, Magnus Dehli, ‘The group law on a tropical elliptic curve.’ Math.AG/0411485.Google Scholar