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 occur 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. Reversely, 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 stem from a single algorithm in non-commutative algebra, and hence the first part of this article doesn't deal with Anderson's objects at all, but are results on finitely generated modules over skew polynomial rings.

28 pages, several tikz-figures, v1->v2: fixed typos and improved presentation suggested by the referee, accepted for publication in Acta Arithmetica