The tropical $j$-invariant

Details The tropical $j$-invariant The tropical $j$-invariant

2008-03-27

arxiv.org/abs/0803.4021v2

LMS J. Comput. Math. 12, 2009, 275-294.

Mathematics

Combinatorics

15 pages

Scientific paper

If (Q,A) is a marked polygon with one interior point, then a general polynomial f in K[x,y] with support A defines an elliptic curve C on the toric surface X_A. If K has a non-archimedean valuation into the real numbers we can tropicalize C to get a tropical curve Trop(C). If the Newton subdivision induced by f is a triangulation, then Trop(C) 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.

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