Drinfeld module morphisms

This module provides the class sage.rings.function_fields.drinfeld_module.morphism.DrinfeldModuleMorphism.

AUTHORS: - Antoine Leudière (2022-04)

class sage.rings.function_field.drinfeld_modules.morphism.DrinfeldModuleMorphism(parent, ore_pol)

Bases: Morphism, UniqueRepresentation

This class represents Drinfeld \(\mathbb{F}_q[T]\)-module morphisms.

Let \(\phi\) and \(\psi\) be two Drinfeld \(\mathbb{F}_q[T]\)-modules over a field \(K\). A morphism of Drinfeld modules \(\phi \to \psi\) is an Ore polynomial \(f \in K\{\tau\}\) such that \(f \phi_a = \psi_a f\) for every \(a \in \mathbb{F}_q[T]\). In our case, this is equivalent to \(f \phi_T = \psi_T f\). An isogeny is a nonzero morphism.

To create a morphism object, the user should never explicitly instantiate DrinfeldModuleMorphism, but rather call the parent homset with the defining Ore polynomial:

sage: Fq = GF(4)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(3)
sage: phi = DrinfeldModule(A, [z, z^2 + z, z^2 + z])
sage: t = phi.ore_polring().gen()
sage: ore_pol = t + z^5 + z^3 + z + 1
sage: psi = phi.velu(ore_pol)
sage: morphism = Hom(phi, psi)(ore_pol)
sage: morphism
Drinfeld Module morphism:
  From: Drinfeld module defined by T |--> (z^2 + z)*t^2 + (z^2 + z)*t + z
  To:   Drinfeld module defined by T |--> (z^5 + z^2 + z + 1)*t^2 + (z^4 + z + 1)*t + z
  Defn: t + z^5 + z^3 + z + 1

The given Ore polynomial must indeed define a morphism:

sage: morphism = Hom(phi, psi)(1)
Traceback (most recent call last):
...
ValueError: Ore polynomial does not define a morphism

One can get basic data on the morphism:

sage: morphism.domain()
Drinfeld module defined by T |--> (z^2 + z)*t^2 + (z^2 + z)*t + z
sage: morphism.domain() is phi
True

sage: morphism.codomain()
Drinfeld module defined by T |--> (z^5 + z^2 + z + 1)*t^2 + (z^4 + z + 1)*t + z
sage: morphism.codomain() is psi
True

sage: morphism.ore_polynomial()
t + z^5 + z^3 + z + 1
sage: morphism.ore_polynomial() is ore_pol
True

One can check various properties:

sage: morphism.is_zero()
False
sage: morphism.is_isogeny()
True
sage: morphism.is_endomorphism()
False
sage: morphism.is_isomorphism()
False

characteristic_polynomial(var='X')

Return the characteristic polynomial of this endomorphism.

INPUT:

• var – a string (default: X), the name of the variable of the characteristic polynomial

EXAMPLES:

sage: Fq = GF(5)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(3)
sage: phi = DrinfeldModule(A, [z, 0, 1, z])

sage: f = phi.frobenius_endomorphism()
sage: f.characteristic_polynomial()
X^3 + (T + 1)*X^2 + (2*T + 3)*X + 2*T^3 + T + 1

We verify, on an example, that the caracteristic polynomial of a morphism corresponding to \(\phi_a\) is \((X-a)^r\) where \(r\) is the rank:

sage: g = phi.hom(T^2 + 1)
sage: chi = g.characteristic_polynomial()
sage: chi.factor()
(X + 4*T^2 + 4)^3

An example with another variable name:

sage: f.characteristic_polynomial(var='Y')
Y^3 + (T + 1)*Y^2 + (2*T + 3)*Y + 2*T^3 + T + 1

charpoly(var='X')

Return the characteristic polynomial of this endomorphism.

INPUT:

• var – a string (default: X), the name of the variable of the characteristic polynomial

EXAMPLES:

sage: Fq = GF(5)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(3)
sage: phi = DrinfeldModule(A, [z, 0, 1, z])

sage: f = phi.frobenius_endomorphism()
sage: chi = f.charpoly()
sage: chi
X^3 + (T + 1)*X^2 + (2*T + 3)*X + 2*T^3 + T + 1

We check that the characteristic polynomial annihilates the morphism (Cayley-Hamilton’s theorem):

sage: chi(f)
Endomorphism of Drinfeld module defined by T |--> z*t^3 + t^2 + z
  Defn: 0

We verify, on an example, that the caracteristic polynomial of the morphism corresponding to \(\phi_a\) is \((X-a)^r\) where \(r\) is the rank:

sage: g = phi.hom(T^2 + 1)
sage: g.charpoly().factor()
(X + 4*T^2 + 4)^3

An example with another variable name:

sage: f.charpoly(var='Y')
Y^3 + (T + 1)*Y^2 + (2*T + 3)*Y + 2*T^3 + T + 1

dual_isogeny()

Return a dual isogeny to this morphism.

By definition, a dual isogeny of \(f : \phi \to \psi\) is an isogeny \(g : \psi \to \phi\) such that the composite \(g \circ f\) is the multiplication by a generator of the norm of \(f\).

EXAMPLES:

sage: Fq = GF(5)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(3)
sage: phi = DrinfeldModule(A, [z, 0, 1, z])
sage: t = phi.ore_variable()
sage: f = phi.hom(t + 1)
sage: f
Drinfeld Module morphism:
  From: Drinfeld module defined by T |--> z*t^3 + t^2 + z
  To:   Drinfeld module defined by T |--> (2*z^2 + 4*z + 4)*t^3 + (3*z^2 + 2*z + 2)*t^2 + (2*z^2 + 3*z + 4)*t + z
  Defn: t + 1
sage: g = f.dual_isogeny()
sage: g
Drinfeld Module morphism:
  From: Drinfeld module defined by T |--> (2*z^2 + 4*z + 4)*t^3 + (3*z^2 + 2*z + 2)*t^2 + (2*z^2 + 3*z + 4)*t + z
  To:   Drinfeld module defined by T |--> z*t^3 + t^2 + z
  Defn: z*t^2 + (4*z + 1)*t + z + 4

We check that \(f \circ g\) (resp. \(g \circ f\)) is the multiplication by the norm of \(f\):

sage: a = f.norm().gen(); a
T + 4
sage: g * f == phi.hom(a)
True

sage: psi = f.codomain()
sage: f * g == psi.hom(a)
True

inverse()

Return the inverse of this morphism.

Only morphisms defined by constant nonzero Ore polynomials are invertible.

EXAMPLES:

sage: Fq = GF(5)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(3)
sage: phi = DrinfeldModule(A, [z, 1, z, z^2])
sage: f = phi.hom(2); f
Endomorphism of Drinfeld module defined by T |--> z^2*t^3 + z*t^2 + t + z
  Defn: 2
sage: f.inverse()
Endomorphism of Drinfeld module defined by T |--> z^2*t^3 + z*t^2 + t + z
  Defn: 3

Inversion of general isomorphisms between different Drinfeld modules also works:

sage: g = phi.hom(z); g
Drinfeld Module morphism:
  From: Drinfeld module defined by T |--> z^2*t^3 + z*t^2 + t + z
  To:   Drinfeld module defined by T |--> z^2*t^3 + (z^2 + 2*z + 3)*t^2 + (z^2 + 3*z)*t + z
  Defn: z
sage: g.inverse()
Drinfeld Module morphism:
  From: Drinfeld module defined by T |--> z^2*t^3 + (z^2 + 2*z + 3)*t^2 + (z^2 + 3*z)*t + z
  To:   Drinfeld module defined by T |--> z^2*t^3 + z*t^2 + t + z
  Defn: 3*z^2 + 4

When the morphism is not invertible, an error is raised:

sage: F = phi.frobenius_endomorphism()
sage: F.inverse()
Traceback (most recent call last):
...
ZeroDivisionError: this morphism is not invertible

is_identity()

Return True whether the morphism is the identity morphism.

EXAMPLES:

sage: Fq = GF(2)
sage: A.<T> = Fq[]
sage: K.<z6> = Fq.extension(6)
sage: phi = DrinfeldModule(A, [z6, 1, 1])
sage: morphism = End(phi)(1)
sage: morphism.is_identity()
True

sage: psi = DrinfeldModule(A, [z6, z6^4 + z6^2 + 1, 1])
sage: t = phi.ore_polring().gen()
sage: morphism = Hom(phi, psi)(t + z6^5 + z6^2 + 1)
sage: morphism.is_identity()
False

is_isogeny()

Return True whether the morphism is an isogeny.

EXAMPLES:

sage: Fq = GF(2)
sage: A.<T> = Fq[]
sage: K.<z6> = Fq.extension(6)
sage: phi = DrinfeldModule(A, [z6, 1, 1])
sage: psi = DrinfeldModule(A, [z6, z6^4 + z6^2 + 1, 1])
sage: t = phi.ore_polring().gen()
sage: morphism = Hom(phi, psi)(t + z6^5 + z6^2 + 1)
sage: morphism.is_isogeny()
True

sage: zero_morphism = End(phi)(0)
sage: zero_morphism.is_isogeny()
False

sage: identity_morphism = End(phi)(1)
sage: identity_morphism.is_isogeny()
True

sage: frobenius_endomorphism = phi.frobenius_endomorphism()
sage: frobenius_endomorphism.is_isogeny()
True

is_isomorphism()

Return True whether the morphism is an isomorphism.

EXAMPLES:

sage: Fq = GF(2)
sage: A.<T> = Fq[]
sage: K.<z6> = Fq.extension(6)
sage: phi = DrinfeldModule(A, [z6, 1, 1])
sage: psi = DrinfeldModule(A, [z6, z6^4 + z6^2 + 1, 1])
sage: t = phi.ore_polring().gen()
sage: morphism = Hom(phi, psi)(t + z6^5 + z6^2 + 1)
sage: morphism.is_isomorphism()
False

sage: zero_morphism = End(phi)(0)
sage: zero_morphism.is_isomorphism()
False

sage: identity_morphism = End(phi)(1)
sage: identity_morphism.is_isomorphism()
True

sage: frobenius_endomorphism = phi.frobenius_endomorphism()
sage: frobenius_endomorphism.is_isomorphism()
False

is_zero()

Return True whether the morphism is the zero morphism.

EXAMPLES:

sage: Fq = GF(2)
sage: A.<T> = Fq[]
sage: K.<z6> = Fq.extension(6)
sage: phi = DrinfeldModule(A, [z6, 1, 1])
sage: psi = DrinfeldModule(A, [z6, z6^4 + z6^2 + 1, 1])
sage: t = phi.ore_polring().gen()
sage: morphism = Hom(phi, psi)(t + z6^5 + z6^2 + 1)
sage: morphism.is_zero()
False

sage: zero_morphism = End(phi)(0)
sage: zero_morphism.is_zero()
True

norm(ideal=True)

Return the norm of this isogeny.

INPUT:

• ideal – a boolean (default: True); if True, return the norm as an ideal in the function ring of the Drinfeld modules; if False, return the norm as an element in this function ring (only relevant for endomorphisms)

EXAMPLES:

sage: Fq = GF(5)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(3)
sage: phi = DrinfeldModule(A, [z, 0, 1, z])
sage: t = phi.ore_variable()
sage: f = phi.hom(t + 1)
sage: f.norm()
Principal ideal (T + 4) of Univariate Polynomial Ring in T over Finite Field of size 5

The norm of the Frobenius endomorphism is equal to the characteristic:

sage: F = phi.frobenius_endomorphism()
sage: F.norm()
Principal ideal (T^3 + 3*T + 3) of Univariate Polynomial Ring in T over Finite Field of size 5
sage: phi.characteristic()
T^3 + 3*T + 3

For \(a\) in the underlying function ring, the norm of the endomorphism given by \(\phi_a\) is \(a^r\) where \(r\) is the rank:

sage: g = phi.hom(T)
sage: g.norm()
Principal ideal (T^3) of Univariate Polynomial Ring in T over Finite Field of size 5

sage: h = phi.hom(T+1)
sage: h.norm()
Principal ideal (T^3 + 3*T^2 + 3*T + 1) of Univariate Polynomial Ring in T over Finite Field of size 5

For endomorphisms, the norm is not an ideal of \(A\) but it makes sense as an actual element of \(A\). We can get this element by passing in the argument ideal=False:

sage: phi.hom(2*T).norm(ideal=False)
3*T^3

sage: f.norm(ideal=False)
Traceback (most recent call last):
...
ValueError: norm is defined as an actual element only for endomorphisms

ore_polynomial()

Return the Ore polynomial that defines the morphism.

EXAMPLES:

sage: Fq = GF(2)
sage: A.<T> = Fq[]
sage: K.<z6> = Fq.extension(6)
sage: phi = DrinfeldModule(A, [z6, 1, 1])
sage: psi = DrinfeldModule(A, [z6, z6^4 + z6^2 + 1, 1])
sage: t = phi.ore_polring().gen()
sage: morphism = Hom(phi, psi)(t + z6^5 + z6^2 + 1)
sage: ore_pol = morphism.ore_polynomial()
sage: ore_pol
t + z6^5 + z6^2 + 1

sage: ore_pol * phi(T) == psi(T) * ore_pol
True