Automorphism Groups on Tropical Curves: Some Cohomology Calculations

TL;DR

This paper proves that for any finite automorphism group acting on an abstract tropical curve, there always exists a G-invariant divisor within a G-invariant divisor class, using a tropical analogue of Hilbert's Theorem 90.

Contribution

It establishes the existence of G-invariant divisors in G-invariant classes on tropical curves, introducing a tropical version of Hilbert's Theorem 90.

Findings

Existence of G-invariant divisors on tropical curves proven.

Development of a tropical analogue of Hilbert's Theorem 90.

Applicable to all abstract tropical curves.

Abstract

Let $X$ be an abstract tropical curve and let $G$ be a finite subgroup of the automorphism group of $X$. Let $D$ be a divisor on $X$ whose equivalence class is $G$-invariant. We address the following question: is there always a divisor $D'$ in the equivalence class of $D$ which is $G$-invariant? Our main result is that the answer is "yes" for all abstract tropical curves. A key step in our proof is a tropical analogue of Hilbert's Theorem 90.