The tropical j-invariant
Eric Katz, Hannah Markwig and Thomas Markwig
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 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.
|
