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Let K be an algebraically closed field which is complete with respect to anontrivial, non-Archimedean valuation and let \Lambda be its value group. Givena smooth, proper, connected K-curve X and a skeleton \Gamma of the Berkovichanalytification X^\an, there are two natural real tori which one can consider:the tropical Jacobian Jac(\Gamma) and the skeleton of the Berkovichanalytification Jac(X)^\an. We show that the skeleton of the Jacobian iscanonically isomorphic to the Jacobian of the skeleton as principally polarizedtropical abelian varieties. In addition, we show that the tropicalization of aclassical Abel-Jacobi map is a tropical Abel-Jacobi map. As a consequence ofthese results, we deduce that \Lambda-rational principal divisors on \Gamma, inthe sense of tropical geometry, are exactly the retractions of principaldivisors on X. We actually prove a more precise result which says that,although zeros and poles of divisors can cancel under the retraction map, inorder to lift a \Lambda-rational principal divisor on \Gamma to a principaldivisor on X it is never necessary to add more than g extra zeros and g extrapoles. Our results imply that a continuous function F:\Gamma -> R is therestriction to \Gamma of -log|f| for some nonzero meromorphic function f on Xif and only if F is a \Lambda-rational tropical meromorphic function, and weuse this fact to prove that there is a rational map f : X --> P^3 whosetropicalization, when restricted to \Gamma, is an isometry onto its image.

The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves