The Skeleton of the Jacobian, the Jacobian of the Skeleton, and Lifting Meromorphic Functions From Tropical to Algebraic Curves

By Matthew Baker, Joseph Rabinoff. 2013

77 citations recorded by the source corpus.

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

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/609c3558171992d1d19d3d89a59debda6db6d50f

Citations in the corpus

• Uniform bounds for the number of rational points on curves of small Mordell–Weil rank (2015, unknown): “: Div0(X) τ∗ //Jan ¯τ Div0(Γ) //J (Γ) Σoo∼ In other words, for D∈ Div0(X), the point ¯τ([D]) ∈ Σ is identified with [τ∗D] ∈ J(Γ). Proof. This is the content of Theorem 2.9 and Proposition 5.3 in [BR14]. From now on we will implicitly identify J(Γ) with Σ. In [BR14, Proposition 6.2] it is shown that”