The tropical $j$-invariant

Source: arXiv


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. Comment: 15 pages

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Available from: Hannah Markwig, Jan 16, 2015
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    ABSTRACT: We find restrictions on the topology of tropical varieties that arise from a certain natural class of varieties. We develop a theory of tropical degenerations that is a nonconstant coefficient analogue of Tevelev's theory of tropical compactifications, and use it to construct normal crossings degenerations of a subvariety X of a torus, under mild hypotheses on X. These degenerations allow us to construct a natural, "multiplicity-free" parameterization of Trop(X) by a topological space \Gamma_X. We give a geometric interpretation of the cohomology of \Gamma_X in terms of the action of a monodromy operator on the cohomology of X. This gives bounds on the Betti numbers of $\Gamma_X$ in terms of the Betti numbers of $X$. When $X$ is a sufficiently general complete intersection, this allows us to show that the cohomology of Trop(X) vanishes in degree less than dim(X). In addition, we give a description for the top power of the monodromy operator acting on middle cohomology in terms of the volume pairing on $\Gamma_X$.
    Canadian Journal of Mathematics 04/2008; 64(4). DOI:10.4153/CJM-2011-067-9 · 0.74 Impact Factor