Set of morphisms between two Drinfeld modules

This module provides the class sage.rings.function_field.drinfeld_module.homset.DrinfeldModuleHomset.

AUTHORS:

• Antoine Leudière (2022-04)

class sage.rings.function_field.drinfeld_modules.homset.DrinfeldModuleHomset(X, Y, category=None, check=True)

Bases: Homset

This class implements the set of morphisms between two Drinfeld \(\mathbb{F}_q[T]\)-modules.

INPUT:

• X – the domain

• Y – the codomain

EXAMPLES:

sage: Fq = GF(27)
sage: A.<T> = Fq[]
sage: K.<z6> = Fq.extension(2)
sage: phi = DrinfeldModule(A, [z6, z6, 2])
sage: psi = DrinfeldModule(A, [z6, 2*z6^5 + 2*z6^4 + 2*z6 + 1, 2])
sage: H = Hom(phi, psi)
sage: H
Set of Drinfeld module morphisms
 from (gen) 2*t^2 + z6*t + z6
   to (gen) 2*t^2 + (2*z6^5 + 2*z6^4 + 2*z6 + 1)*t + z6

sage: from sage.rings.function_field.drinfeld_modules.homset import DrinfeldModuleHomset
sage: isinstance(H, DrinfeldModuleHomset)
True

There is a simpler syntax for endomorphisms sets:

sage: E = End(phi)
sage: E
Set of Drinfeld module morphisms from (gen) 2*t^2 + z6*t + z6 to (gen) 2*t^2 + z6*t + z6
sage: E is Hom(phi, phi)
True

The domain and codomain must have the same Drinfeld modules category:

sage: rho = DrinfeldModule(A, [Frac(A)(T), 1])
sage: Hom(phi, rho)
Traceback (most recent call last):
...
ValueError: Drinfeld modules must be in the same category

sage: sigma = DrinfeldModule(A, [1, z6, 2])
sage: Hom(phi, sigma)
Traceback (most recent call last):
...
ValueError: Drinfeld modules must be in the same category

One can create morphism objects by calling the homset:

sage: identity_morphism = E(1)
sage: identity_morphism
Identity morphism of Drinfeld module defined by T |--> 2*t^2 + z6*t + z6

sage: t = phi.ore_polring().gen()
sage: frobenius_endomorphism = E(t^6)
sage: frobenius_endomorphism
Endomorphism of Drinfeld module defined by T |--> 2*t^2 + z6*t + z6
  Defn: t^6

sage: isogeny = H(t + 1)
sage: isogeny
Drinfeld Module morphism:
  From: Drinfeld module defined by T |--> 2*t^2 + z6*t + z6
  To:   Drinfeld module defined by T |--> 2*t^2 + (2*z6^5 + 2*z6^4 + 2*z6 + 1)*t + z6
  Defn: t + 1

And one can test if an Ore polynomial defines a morphism using the in syntax:

sage: 1 in H
False
sage: t^6 in H
False
sage: t + 1 in H
True
sage: 1 in E
True
sage: t^6 in E
True
sage: t + 1 in E
False

This also works if the candidate is a morphism object:

sage: isogeny in H
True
sage: E(0) in E
True
sage: identity_morphism in H
False
sage: frobenius_endomorphism in H
False

Element

alias of DrinfeldModuleMorphism

class sage.rings.function_field.drinfeld_modules.homset.DrinfeldModuleMorphismAction(A, H, is_left, op)

Bases: Action

Action of the function ring on the homset of a Drinfeld module.

EXAMPLES:

sage: Fq = GF(5)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(3)
sage: phi = DrinfeldModule(A, [z, 1, z])
sage: psi = DrinfeldModule(A, [z, z^2 + 4*z + 3, 2*z^2 + 4*z + 4])
sage: H = Hom(phi, psi)
sage: t = phi.ore_variable()
sage: f = H(t + 2)

Left action:

sage: (T + 1) * f
Drinfeld Module morphism:
  From: Drinfeld module defined by T |--> z*t^2 + t + z
  To:   Drinfeld module defined by T |--> (2*z^2 + 4*z + 4)*t^2 + (z^2 + 4*z + 3)*t + z
  Defn: (2*z^2 + 4*z + 4)*t^3 + (2*z + 1)*t^2 + (2*z^2 + 4*z + 2)*t + 2*z + 2

Right action currently does not work (it is a known bug, due to an incompatibility between multiplication of morphisms and the coercion system):

sage: f * (T + 1)
Traceback (most recent call last):
...
TypeError: right (=T + 1) must be a map to multiply it by Drinfeld Module morphism:
  From: Drinfeld module defined by T |--> z*t^2 + t + z
  To:   Drinfeld module defined by T |--> (2*z^2 + 4*z + 4)*t^2 + (z^2 + 4*z + 3)*t + z
  Defn: t + 2