Algorithms for determination of t-module structures on some extension groups

TL;DR

This work develops and systematizes an algorithmic framework for determining the $t$-module structure on extension groups $ ext{Ext}^1_{ au}(oldsymbol{ ewphi},oldsymbol{ ewpsi})$ between Anderson–$t$-modules and Drinfeld modules. Central to the approach is representing extensions by biderivations and exploiting the six-term Hom–Ext sequence, together with a duality principle, to translate extension data into computable matrices via $t$-reduction. The authors provide general reductions, two-case executability results based on leading-term invertibility, and an extension framework via $ au$-composition series, alongside exact formulas in the two–Drinfeld-module case and integrality criteria. Collectively, the work broadens the class of $(oldsymbol{ ewphi},oldsymbol{ ewpsi})$ for which $ ext{Ext}^1_{ au}$-structures can be computed, with explicit algorithmic pseudocode and Mathematica implementations. These results deepen the computational toolbox for the category of $t$-modules and have implications for function field arithmetic and related Langlands-type frameworks.

Abstract

In \cite{kk04} the second and third author extended the methods of \cite{pr} and determined the \tm module structure on $\Ext^1(Φ,Ψ)$ where $Φ$ and $Ψ$ were Anderson \tm modules over $A={\mathbf F}_q[t]$ of some specific types. This approach involved the concept of biderivation and certain reduction algorithm. In this paper we generalize the results of \cite{pr} and \cite{kk04} and present complete algorithm for computation of \tm module structure on $\Ext^1(Φ,Ψ)$ for \tm modules $Φ$ and $Ψ$ such that $\rk Φ> \rk Ψ.$ The last condition is not sufficient for our algorithm to be executable. We show that it can be applied when the matrix at the biggest power of $τ$ in $Φ_t$ is invertible. We also introduce a notion of $τ$-composition series which we find suitable for the additive category of \tm modules and show that under certain assumptions on the composition series of $Φ$ and $Ψ$ our algorithm is also executable.

Theorems & Definitions (45)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Remark 2.8
• Theorem 2.1
• Definition 2.9