Determining $t$-motives and dual $t$-motives in Anderson's theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908780210421760 |
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| author | Maurischat, Andreas |
| author_facet | Maurischat, Andreas |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_12779 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Determining $t$-motives and dual $t$-motives in Anderson's theory Maurischat, Andreas Number Theory Rings and Algebras 11G09 (Primary) 13P10, 16S36 (Secondary) 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. |
| title | Determining $t$-motives and dual $t$-motives in Anderson's theory |
| topic | Number Theory Rings and Algebras 11G09 (Primary) 13P10, 16S36 (Secondary) |
| url | https://arxiv.org/abs/2505.12779 |