Realization of tropical curves in abelian surfaces

Abstract: We construct algebraic curves in abelian surfaces starting from tropical curves in real tori. We give a necessary and sufficient condition for a tropical curve in a real torus to be realizable by an algebraic curve in an abelian surface. When the condition is satisfied, the number of algebraic curves can be computed by a combinatorial formula. This gives us an algebraic-tropical correspondence theorem for abelian surfaces analogous to Mikhalkin's correspondence theorem for toric surfaces. In other words, the number of algebraic curves passing through generic points in an abelian surface can be computed purely combinatorially via tropical curves.