The tropical $j$-invariant

TL;DR

This paper establishes a relationship between the tropicalization of elliptic curves defined by polynomials supported on a marked polygon and the valuation of their j-invariant, linking algebraic and tropical geometry.

Contribution

It introduces a tropical j-invariant for elliptic curves on toric surfaces, connecting the lattice length of the tropical cycle to the valuation of the classical j-invariant.

Findings

Lattice length of the tropical cycle equals the negative valuation of the j-invariant.

Tropicalization preserves genus one structure under certain Newton subdivision conditions.

Provides a method to compute the tropical j-invariant from polynomial support and valuation.

Abstract

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.