# Non-Archimedean and tropical theta functions

## TL;DR

This paper develops a tropicalization process for non-Archimedean theta functions on abelian varieties, showing their tropical counterparts retain key properties and enabling new constructions on the skeleton of these varieties.

## Contribution

It introduces a tropicalization procedure for non-Archimedean theta functions and demonstrates their correspondence with tropical theta functions, extending the theory to semi-abelian reductions.

## Key findings

- Tropicalization of non-Archimedean theta functions yields tropical theta functions.
- Tropicalization preserves key properties of Riemann theta functions up to scaling and constants.
- Constructs rational functions with prescribed behavior on the skeleton of abelian varieties.

## Abstract

We define a tropicalization procedure for theta functions on abelian varieties over a non-Archimedean field. We show that the tropicalization of a non-Archimedean theta function is a tropical theta function, and that the tropicalization of a non-Archimedean Riemann theta function is a tropical Riemann theta function, up to scaling and an additive constant. We apply these results to the construction of rational functions with prescribed behavior on the skeleton of a principally polarized abelian variety. We work with the Raynaud--Bosch--L\"utkebohmert theory of non-Archimedean theta functions for abelian varieties with semi-abelian reduction.