The Tropicalj-Invariant
Eric Katz, Hannah Markwig, Thomas Markwig · LMS Journal of Computation and Mathematics · 2009
Abstract If (Q,A) is a marked polygon with one interior point, then a general polynomialfbelonging toK[x,y] with supportAdefines an elliptic curveCfon the toric surfaceXA. IfKhas a non-archimedean valuation intoRwe can tropicalizeCfto get a tropical curve Trop(Cf). If in the Newton subdivision induced byfis 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 thej-invariant ofCf.