Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Andreas Maurischat

TL;DR

This work develops a complete algorithmic framework to determine when Anderson $t$-modules, their $t$-motives, and dual objects are abelian or coabelian, and to compute explicit $K[t]$-basises and Frobenius actions. Central to the approach is adapting Janet bases for modules over the skew polynomial ring $K\{\pi,\rho\}$, allowing one to decide finite generation, extract bases, and describe $\tau$- and $\sigma$-actions in a canonical form. The paper then applies this machinery to translate between Anderson $t$-modules, $t$-motives, and $t$-comotives, providing constructive criteria and procedures to recover one object from another and to compute tensorial and dual constructions. The results yield practical, implementable algorithms for verifying abelian/coabelian properties and for obtaining explicit presentations, bases, and action matrices, thereby enabling computational exploration of function-field analogs of abelian varieties and their motives. Collectively, the contributions unify non-commutative algebra with Anderson’s theory to enable algorithmic handling of these positive-characteristic motivic objects.

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.

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

TL;DR

This work develops a complete algorithmic framework to determine when Anderson -modules, their -motives, and dual objects are abelian or coabelian, and to compute explicit -basises and Frobenius actions. Central to the approach is adapting Janet bases for modules over the skew polynomial ring , allowing one to decide finite generation, extract bases, and describe - and -actions in a canonical form. The paper then applies this machinery to translate between Anderson -modules, -motives, and -comotives, providing constructive criteria and procedures to recover one object from another and to compute tensorial and dual constructions. The results yield practical, implementable algorithms for verifying abelian/coabelian properties and for obtaining explicit presentations, bases, and action matrices, thereby enabling computational exploration of function-field analogs of abelian varieties and their motives. Collectively, the contributions unify non-commutative algebra with Anderson’s theory to enable algorithmic handling of these positive-characteristic motivic objects.

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.
Paper Structure (16 sections, 11 theorems, 82 equations, 1 figure)

This paper contains 16 sections, 11 theorems, 82 equations, 1 figure.

Table of Contents

  1. Notation and basics
  2. Janet basis and algorithm
  3. Monomial orders and normal forms
  4. Janet basis
  5. Janet algorithm
  6. Finite generation and relations
  7. Anderson modules, motives, and comotives
  8. Basic setting
  9. Anderson's objects
  10. Relation between these objects
  11. Bases of abelian $t$-motives
  12. Examples
  13. Computing the Anderson $t$-module to a $t$-motive
  14. Anderson comotives
  15. Answer to Question \ref{['item:basis-of-dual-t-motive']}
  16. ...and 1 more sections

Key Result

Proposition 1.1

Let $M$ be a $K\{\pi,\rho\}$-module, and denote by $M_{\pi\text{-tor}}$ and $M_{\rho\text{-tor}}$ the $K\{\pi\}$-torsion subspace, and the $K\{\rho\}$-torsion subspace respectively, i.e., the $K$-vector spaces Then

Figures (1)

  • Figure 1: Graphical visualization for the set $T= \{ (p_1,\{\rho\}), (p_2,\{\pi,\rho\}), (p_3,\{\rho\}), (p_4,\{\pi,\rho\})\}, \text{ where}$p_1= -\underline{\pi \rho\kappa_1}+\pi\kappa_1 + \pi \kappa_2+\rho\kappa_2,\quadp_3= -(\underline{\rho^2}-2\rho+1)\underline{\kappa_1}+ (\pi^2-\rho)\kappa_2,\quadp_2= \underline{\pi^2\kappa_1}+\rho\kappa_1-\pi\kappa_2-\kappa_2,\quadp_4= \underline{\pi^3\kappa_2}+\pi^2\rho\kappa_2 -\rho^2\kappa_2.(The underlined terms are the leading monomials.)

Theorems & Definitions (46)

  • Proposition 1.1
  • Remark 1.2
  • proof : Proof of Proposition \ref{['prop:observations-on-M']}
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7: dr:faepde
  • ...and 36 more