# Isogenies of abelian Anderson A-modules and A-motives

## TL;DR

This paper explores the theory of abelian Anderson A-modules and A-motives, establishing dual isogenies, equivalences of isogenies, and analyzing torsion submodules and their relation to local shtukas in the function field setting.

## Contribution

It introduces a duality theory for isogenies of abelian t-modules and connects torsion submodules to local shtukas, extending the understanding of Anderson modules and t-motives over rings.

## Key findings

- Every isogeny has a dual isogeny in the opposite direction.
- A morphism is an isogeny if and only if the associated t-motives are isogenous.
- Torsion submodules relate to local shtukas and p-divisible groups analogs.

## Abstract

As a generalization of Drinfeld modules, Greg Anderson introduced abelian t-modules and t-motives over a perfect field. In this article we study relative versions of these over rings. We investigate isogenies among them. Our main results state that every isogeny possesses a dual isogeny in the opposite direction, and that a morphism between abelian t-modules is an isogeny if and only if the corresponding morphism between their associated t-motives is an isogeny. We also study torsion submodules of abelian t-modules which in general are non-reduced group schemes. They can be obtained from the associated t-motive via the finite shtuka correspondence of Drinfeld and Abrashkin. The inductive limits of torsion submodules are the function field analogs of p-divisible groups. These limits correspond to the local shtukas attached to the t-motives associated with the abelian t-modules. In this sense the theory of abelian t-modules is captured by the theory of t-motives.