The module action induced by a Drinfeld module¶

This module provides the class sage.rings.function_field.drinfeld_module.action.DrinfeldModuleAction.

AUTHORS:

Antoine Leudière (2022-04)

class sage.rings.function_field.drinfeld_modules.action.DrinfeldModuleAction(drinfeld_module)[source]¶

Bases: Action

This class implements the module action induced by a Drinfeld \(\mathbb{F}_q[T]\)-module.

Let \(\phi\) be a Drinfeld \(\mathbb{F}_q[T]\)-module over a field \(K\) and let \(L/K\) be a field extension. Let \(x \in L\) and let \(a\) be a function ring element; the action is defined as \((a, x) \mapsto \phi_a(x)\).

Note

In this implementation, \(L\) is \(K\).

Note

The user should never explicitly instantiate the class \(DrinfeldModuleAction\).

Warning

This class may be replaced later on. See issues #34833 and #34834.

INPUT: the Drinfeld module

EXAMPLES:

Sage

sage: Fq.<z2> = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(2)
sage: phi = DrinfeldModule(A, [z, 0, 0, 1])
sage: action = phi.action()
sage: action
Action on Finite Field in z of size 11^2 over its base
 induced by Drinfeld module defined by T |--> t^3 + z

Python

>>> from sage.all import *
>>> Fq = GF(Integer(11), names=('z2',)); (z2,) = Fq._first_ngens(1)
>>> A = Fq['T']; (T,) = A._first_ngens(1)
>>> K = Fq.extension(Integer(2), names=('z',)); (z,) = K._first_ngens(1)
>>> phi = DrinfeldModule(A, [z, Integer(0), Integer(0), Integer(1)])
>>> action = phi.action()
>>> action
Action on Finite Field in z of size 11^2 over its base
 induced by Drinfeld module defined by T |--> t^3 + z

The action on elements is computed as follows:

Sage

sage: P = T + 1
sage: a = z
sage: action(P, a)
...
4*z + 2
sage: action(0, K.random_element())
0
sage: action(A.random_element(), 0)
0

Python

>>> from sage.all import *
>>> P = T + Integer(1)
>>> a = z
>>> action(P, a)
...
4*z + 2
>>> action(Integer(0), K.random_element())
0
>>> action(A.random_element(), Integer(0))
0

Finally, given a Drinfeld module action, it is easy to recover the corresponding Drinfeld module:

Sage

sage: action.drinfeld_module() is phi
True

Python

>>> from sage.all import *
>>> action.drinfeld_module() is phi
True

drinfeld_module()[source]¶

Return the Drinfeld module defining the action.

OUTPUT: a Drinfeld module

EXAMPLES:

Sage

sage: Fq.<z2> = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(2)
sage: phi = DrinfeldModule(A, [z, 0, 0, 1])
sage: action = phi.action()
sage: action.drinfeld_module() is phi
True

Python

>>> from sage.all import *
>>> Fq = GF(Integer(11), names=('z2',)); (z2,) = Fq._first_ngens(1)
>>> A = Fq['T']; (T,) = A._first_ngens(1)
>>> K = Fq.extension(Integer(2), names=('z',)); (z,) = K._first_ngens(1)
>>> phi = DrinfeldModule(A, [z, Integer(0), Integer(0), Integer(1)])
>>> action = phi.action()
>>> action.drinfeld_module() is phi
True