Galois actions on analytifications and 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.