Determining $t$-motives and dual $t$-motives in Anderson's theory

TL;DR

This paper develops algorithms to determine and classify t-motives and dual t-motives in Anderson's theory, translating complex algebraic questions into computational procedures.

Contribution

It introduces algorithms for identifying and constructing t-motives and dual t-motives, based on non-commutative algebra techniques, applicable to Anderson's t-modules.

Findings

Algorithms for recognizing finitely generated free t-motives

Procedures for reconstructing t-modules from t-motives

Reduction of main algorithms to non-commutative algebra problems

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.