The motive associated to a Drinfeld module

[kryzar]

Let \Fq be a finite field and K be an \Fq[X]-field. Let \phi be a Drinfeld \Fq[X]-module over K. We denote \mathcal{M}(\phi) to be the ring K\{\tau\}endowed with the \Fq[X]-module law (P, f) \mapsto f \phi_P. We call it the motive associate to \phi. A Drinfeld module isogeny u: \phi \to \psi gives rise to an \Fq[X]-module morphism \mathcal{M}(u): \mathcal{E}(\psi) \to \mathcal{M}(\phi); \mathcal{M} is a contravariant foncctor.

We propose to create a class DrinfeldModuleMotive that is a parent representing \mathcal{M}(\phi) and whose elements — instances of DrinfeldModuleMotiveElement — are elements of \mathcal{M}(\phi). We would define the action of \Fq[X] on those elements. We would also create a class DrinfeldModuleMotiveMorphism to represent the morphism \mathcal{M}(u).

The motive \mathcal{M}(\phi) is a free \Fq[X] \otimes_\Fq K-tensor whose basis is (1, \tau, \dots, \tau^{r-1}), r being the rank of the Drinfeld module.

sage: # phi is a Drinfeld Fq[X]-module over K
sage: motive = phi.motive()  # Get the motive associated to phi
sage: f = phi(X)
sage: f
t^2 + t + a   
sage: motive(f)
(t^2 + t + a) * m_phi
sage: motive.basis()
(1, t)
sage: motive(f).coordinates()
(X, 0)

The motive is a dual of the module associated to a Drinfeld module (ticket #34833) but the two constructions are independent. We propose to work on those tickets once the main ticket (#33713) is merged.

More generally, we can define an \Fq[X]-motive independently of Drinfeld modules. The category of \Fq[X]-motives is anti-equivalent to the so-called category of abelian \Fq[X]-modules, which contains the category of Drinfeld modules as a full subcategory.

This is part of an ongoing work with Xavier Caruso.

Depends on #33713

CC: @xcaruso @DavidAyotte @spaenlehauer @emmanuelthome

Component: number theory

Author: Xavier Caruso, Antoine Leudière

Issue created by migration from https://trac.sagemath.org/ticket/34834

[Demon-Sheriff]

Hey @kryzar I would like to work on this one.

[kryzar]

Yo. Could you tell us more about yourself, how you would want to contribute, and why? I believe Xavier Caruso (@xcaruso) is preparing an implementation, which is already well advanced (even though development time in summer usually comes to a halt!). In fact, he gave a presentation at the Caipi Symposium last February, showcasing an early version!

Thanks in advance for your answers, and have a nice day :beach_umbrella:

[xcaruso]

The current version of the code is available here: https://plmlab.math.cnrs.fr/caruso/anderson-motives

In addition, I would like to add support for manipulating morphisms between Anderson motives and maybe also to improve how elements are displayed (in particular for tensorial constructions).