Isogenies of abelian Anderson A-modules and A-motives

Abstract

As a generalization of Drinfeld modules, Greg Anderson introduced Abelian t -modules and t -motives over a perfect base field. In this article we study relative versions of these defined over base 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.