Determining $t$-motives and dual $t$-motives in Anderson’s theory
Abstract
Anderson $t$-modules are analogs of abelian varieties in positive characteristic. Associated to such a $t$-module, there are its $t$-motive and its dual $t$-motive. When dealing with these objects, several questions arise which one would like to solve algorithmically. For example, for a given $t$-module one would like to decide whether its $t$-motive is indeed finitely generated free, and determine a basis. Conversely, for a given object in the category of $t$-motives one would like to decide whether it is the $t$-motive associated to a $t$-module, and determine that $t$-module.
In this article, we positively answer such questions by providing the corresponding algorithms.
As it turned out, the main part of all these algorithms stems from a single algorithm in non-commutative algebra, and hence the first part of this article does not deal with Anderson’s objects at all, but with results on finitely generated modules over skew polynomial rings.
