Genus one correspondence between tropical and algebraic curves

Abstract

We show that the genuinely enumerative count of algebraic elliptic curves in any toric variety agrees with the count of the corresponding well-spaced tropical curves, weighted by explicit combinatorial multiplicities. This provides a complete genus-$1$ generalization of the celebrated Nishinou--Siebert correspondence theorem in genus $0$. The proof is algebro-geometric and relies on logarithmic deformation theory together with an explicit enumeration of logarithmic maps with fixed tropicalization.