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Nonarchimedean geometry, tropicalization, and metrics on curves
We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and aExpand
Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta
Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certainExpand
Uniform bounds for the number of rational points on curves of small Mordell–Weil rank
Let $X$ be a curve of genus $g\geq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $\#X(F)$ of $F$-rational points of $X$ is anExpand
Skeletons and tropicalizations
Let $K$ be a complete, algebraically closed non-archimedean field with ring of integers $K^\circ$ and let $X$ be a $K$-variety. We associate to the data of a strictly semistable $K^\circ$-modelExpand
Tropical analytic geometry, Newton polygons, and tropical intersections
In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the NewtonExpand
The Skeleton of the Jacobian, the Jacobian of the Skeleton, and Lifting Meromorphic Functions From Tropical to Algebraic Curves
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 aExpand
On the structure of non-archimedean analytic curves
Let K be an algebraically closed, complete non-Archimedean field and let X be a smooth K-curve. In this paper we elaborate on several aspects of the structure of the Berkovich analytic space Xan. WeExpand
Lifting harmonic morphisms II: tropical curves and metrized complexes
In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism ofExpand
On the structure of nonarchimedean analytic curves
Let K be an algebraically closed, complete nonarchimedean field and let X be a smooth K-curve. In this paper we elaborate on several aspects of the structure of the Berkovich analytic space X^an. WeExpand
Lifting non-proper tropical intersections
We prove that if X, X' are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along aExpand
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