# Galois actions on analytifications and tropicalizations

## TL;DR

This paper develops a framework connecting Galois actions with tropicalizations of algebraic varieties, enabling the recovery of Galois representations from combinatorial data and showing that tropicalizations encode all arithmetic information of the variety.

## Contribution

It introduces Galois-equivariant tropicalizations, linking Galois actions to tropical geometry, and proves they capture the full arithmetic structure of varieties over Henselian fields.

## Key findings

- Galois-equivariant tropicalizations are Galois-equivariant maps.
- They can recover Galois orbits of points faithfully.
- The Berkovich analytification is the inverse limit of all such tropicalizations.

## Abstract

This paper initiates a research program that seeks to recover algebro-geometric Galois representations from combinatorial data. We study tropicalizations equipped with symmetries coming from the Galois-action present on the lattice of $1$-parameter subgroups inside ambient Galois-twisted toric varieties. Over a Henselian field, the resulting tropicalization maps become Galois-equivariant. We call their images Galois-equivariant tropicalizations, and use them to construct a large supply of Galois representations in the tropical cellular cohomology groups of Itenberg, Katzarkov, Mikhalkin, and Zharkov. We also prove two results which say that under minimal hypotheses on a variety $X_{0}$ over a Henselian field $K_{0}$, Galois-equivariant tropicalizations carry all of the arithmetic structure of $X_{0}$. Namely: (1) The Galois-orbit of any point of $X_{0}$ valued in the separable closure of $K_{0}$ is reproduced faithfully as a Galois-set inside some Galois-equivariant tropicalization of our variety. (2) The Berkovich analytification of $X_{0}$ over the separable closure of $K_{0}$, equipped with its canonical Galois-action, is the inverse limit of all Galois-equivariant tropicalizations of our variety.