The Tropical<i>j</i>-Invariant

Abstract

Abstract 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 C f on the toric surface X A . If K has a non-archimedean valuation into R we can tropicalize C f to get a tropical curve Trop( C f ). If in the Newton subdivision induced by f is a triangulation and the interior point occurs as the vertex of a triangle, then Trop( C f ) 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 C f .