Effective faithful tropicalizations and embeddings for abelian varieties

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.