Effective faithful tropicalizations and embeddings for abelian varieties

TL;DR

This paper constructs faithful tropicalizations of the skeleton of abelian varieties over nonarchimedean fields using theta functions, bridging classical and tropical geometry with new embedding and lifting techniques.

Contribution

It introduces a method to faithfully embed tropical abelian varieties via theta functions and lifts these to nonarchimedean theta functions, advancing tropicalization theory.

Findings

Faithful tropicalization of the skeleton using nonarchimedean theta functions

Construction of faithful embeddings of tropical abelian varieties

Lifting of tropical theta functions to nonarchimedean theta functions

Abstract

Let $A$ be an abelian variety over an algebraically closed field $k$ that is complete with respect to a nontrivial nonarchimedean absolute value. Let $A^{\mathrm{an}}$ denote the analytification of $A$ in the sense of Berkovich, and let $\Sigma$ be the canonical skeleton of $A^{\mathrm{an}}$. In this paper, we obtain a faithful tropicalization of $\Sigma$ by nonarchimedean theta functions, giving a tropical version of the classical theorem of Lefschetz on abelian varieties. Key ingredients of the proof are (1) faithful embeddings of tropical abelian varieties by tropical theta functions and (2) lifting of tropical theta functions to nonarchimedean theta functions, and they will be of independent interest. For (1), we use some arguments similar to the case of complex abelian varieties as well as Voronoi cells of lattices. For (2), we use Fourier expansions of nonarchimedean theta functions over the Raynaud extensions of abelian varieties.