# Abelian varieties with finite abelian group action

## TL;DR

This paper explores the relationship between two decomposition theorems for abelian varieties under automorphisms and extends these results to varieties with actions of arbitrary finite abelian groups.

## Contribution

It demonstrates the equivalence of the isotypical and Roan's decomposition theorems and generalizes these results to broader group actions.

## Key findings

- The two decomposition theorems are essentially the same.
- Generalization to arbitrary finite abelian group actions.
- Provides a unified framework for understanding automorphism-induced decompositions.

## Abstract

An automorphism of an abelian variety induces a decomposition of the variety up to isogeny. There are two such results, namely the isotypical decomposition and Roan's decomposition theorem. We show that they are essentially the same. Moreover, we generalize in a sense this result to abelian varieties with action of an arbitrary finite abelian group.