The module associated to a Drinfeld module

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{E}(\phi) to be the field \Kbar endowed with the standard \Fq[X]-module law (P, z) \mapsto \phi_P(z). We call it the module associate to \phi. A Drinfeld module isogeny u: \phi \to \psi gives rise to an \Fq[X]-module morphism \mathcal{E}(u): \mathcal{E}(\phi) \to \mathcal{E}(\psi); \mathcal{E} is a covariant fonctor.

We propose to create a class DrinfeldModuleModule that is a parent representing \mathcal{E}(\phi) and whose elements — instances of DrinfeldModuleModuleElement — are elements of \mathcal{E}(\phi). We would define the action of \Fq[X] on those elements. We would also create a class DrinfeldModuleModuleMorphism to represent the morphism \mathcal{E}(u). String representation of elements in the module should differ to that of elements in \Kbar, to avoid confusion.

sage: # phi is a Drinfeld Fq[X]-module over K
sage: module = phi.module()  # Get the module associated to phi
sage: z = module(a)  # a is an element in Kbar
sage: a
a
sage: z
a * e_phi
sage: P = X^2 + X + 1
sage: P * z  # phi_P(z), the action of P on z
(a^(q^2) + a^q + a) * e_phi

As the algebraic closure \Kbar is not always implemented, we suggest that the method DrinfeldModule.module takes an optional K-algebra L as an argument and returns L with its \Fq[X]-module structure.

This supplements the job of DrinfeldModuleAction, which we suggest to remove (see ticket #33713). The module is a dual of the motive associated to a Drinfeld module (ticket #34834) but the two constructions are independent. We propose to work on those tickets once the main ticket (#33713) is merged.

We also wish to implement torsion spaces of a Drinfeld modules. This will be the job of a new, future ticket.

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/34833

Description changed:

--- 
+++ 
@@ -12,16 +12,16 @@
 elements in `\Kbar`, to avoid confusion.

sage: # phi is a Drinfeld Fq[X]-module over K
sage: module = phi.module() # Get the module associated to phi
sage: z = module(a) # a is an element in Kbar
sage: a
a
sage: z
a * e_phi
sage: P = X^2 + X + 1
sage: P * z # phi_P(z), the action of P on z

(a^(q^2) + a^q + a) e_phi +sage: # phi is a Drinfeld Fq[X]-module over K +sage: module = phi.module() # Get the module associated to phi +sage: z = module(a) # a is an element in Kbar +sage: a +a +sage: z +a e_phi +sage: P = X^2 + X + 1 +sage: P z # phi_P(z), the action of P on z +(a^(q^2) + a^q + a) e_phi


As the algebraic closure `\Kbar` is not always implemented, we suggest that the
@@ -30,7 +30,7 @@

This supplements the job of `DrinfeldModuleAction`, which we suggest to remove
(see ticket #33713). The module is a dual of the motive associated to a Drinfeld
-module (ticket ???) but the two constructions are independent. We propose to
+module (ticket #34834) but the two constructions are independent. We propose to
work on those tickets once the main ticket (#33713) is merged.

We also wish to implement torsion spaces of a Drinfeld modules. This will be

sagemath / sage

The module associated to a Drinfeld module #34833