2020/07/31 by Takeo Nishinou, Nishinou, Takeo · 1 citation
Computer Science · Mathematics · #14N10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2007.16148
openalex publication_date 2020/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct algebraic curves in abelian surfaces by utilizing 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 this condition is satisfied, the number of algebraic curves can be computed by a combinatorial formula. This provides the algebraic-tropical correspondence theorem for abelian surfaces analogous to Mikhalkin's correspondence theorem for toric surfaces. Thus, the number of algebraic curves passing through generic points in an abelian surface can be computed purely combinatorially via tropical curves.