Drinfeld modules over a base

This module provides the class sage.category.drinfeld_modules.DrinfeldModules.

AUTHORS:

• Antoine Leudière (2022-04)

• Xavier Caruso (2022-06)

class sage.categories.drinfeld_modules.DrinfeldModules(base_field, name='t')

Bases: Category_over_base_ring

This class implements the category of Drinfeld \(\mathbb{F}_q[T]\)-modules on a given base field.

Let \(\mathbb{F}_q[T]\) be a polynomial ring with coefficients in a finite field \(\mathbb{F}_q\) and let \(K\) be a field. Fix a ring morphism \(\gamma: \mathbb{F}_q[T] \to K\); we say that \(K\) is an \(\mathbb{F}_q[T]`*-field*. Let `K\{\tau\}\) be the ring of Ore polynomials with coefficients in \(K\), whose multiplication is given by the rule \(\tau \lambda = \lambda^q \tau\) for any \(\lambda \in K\).

The extension \(K\)/\(\mathbb{F}_q[T]\) (represented as an instance of the class sage.rings.ring_extension.RingExtension) is the base field of the category; its defining morphism \(\gamma\) is called the base morphism.

The monic polynomial that generates the kernel of \(\gamma\) is called the \(\mathbb{F}_q[T]\)-characteristic, or function-field characteristic, of the base field. We say that \(\mathbb{F}_q[T]\) is the function ring of the category; \(K\{\tau\}\) is the Ore polynomial ring. The constant coefficient of the category is the image of \(T\) under the base morphism.

Construction

Generally, Drinfeld modules objects are created before their category, and the category is retrieved as an attribute of the Drinfeld module:

sage: Fq = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(4)
sage: p_root = z^3 + 7*z^2 + 6*z + 10
sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
sage: C = phi.category()
sage: C
Category of Drinfeld modules over Finite Field in z of size 11^4 over its base

The output tells the user that the category is only defined by its base.

Properties of the category

The base field is retrieved using the method base().

sage: C.base() Finite Field in z of size 11^4 over its base

Equivalently, one can use base_morphism() to retrieve the base morphism:

sage: C.base_morphism()
Ring morphism:
  From: Univariate Polynomial Ring in T over Finite Field of size 11
  To:   Finite Field in z of size 11^4 over its base
  Defn: T |--> z^3 + 7*z^2 + 6*z + 10

The so-called constant coefficient — which is the same for all Drinfeld modules in the category — is simply the image of \(T\) by the base morphism:

sage: C.constant_coefficient()
z^3 + 7*z^2 + 6*z + 10
sage: C.base_morphism()(T) == C.constant_coefficient()
True

Similarly, the function ring-characteristic of the category is either \(0\) or the unique monic polynomial in \(\mathbb{F}_q[T]\) that generates the kernel of the base:

sage: C.characteristic()
T^2 + 7*T + 2
sage: C.base_morphism()(C.characteristic())
0

The base field, base morphism, function ring and Ore polynomial ring are the same for the category and its objects:

sage: C.base() is phi.base()
True
sage: C.base_morphism() is phi.base_morphism()
True

sage: C.function_ring()
Univariate Polynomial Ring in T over Finite Field of size 11
sage: C.function_ring() is phi.function_ring()
True

sage: C.ore_polring()
Ore Polynomial Ring in t over Finite Field in z of size 11^4 over its base twisted by Frob
sage: C.ore_polring() is phi.ore_polring()
True

Creating Drinfeld module objects from the category

Calling object() with an Ore polynomial creates a Drinfeld module object in the category whose generator is the input:

sage: psi = C.object([p_root, 1])
sage: psi
Drinfeld module defined by T |--> t + z^3 + 7*z^2 + 6*z + 10
sage: psi.category() is C
True

Of course, the constant coefficient of the input must be the same as the category:

sage: C.object([z, 1])
Traceback (most recent call last):
...
ValueError: constant coefficient must equal that of the category

It is also possible to create a random object in the category. The input is the desired rank:

sage: rho = C.random_object(2)
sage: rho  # random
Drinfeld module defined by T |--> (7*z^3 + 7*z^2 + 10*z + 2)*t^2 + (9*z^3 + 5*z^2 + 2*z + 7)*t + z^3 + 7*z^2 + 6*z + 10
sage: rho.rank() == 2
True
sage: rho.category() is C
True

Endsets()

Return the category of endsets.

EXAMPLES:

sage: Fq = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(4)
sage: p_root = z^3 + 7*z^2 + 6*z + 10
sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
sage: C = phi.category()

sage: from sage.categories.homsets import Homsets
sage: C.Endsets() is Homsets().Endsets()
True

Homsets()

Return the category of homsets.

EXAMPLES:

sage: Fq = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(4)
sage: p_root = z^3 + 7*z^2 + 6*z + 10
sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
sage: C = phi.category()

sage: from sage.categories.homsets import Homsets
sage: C.Homsets() is Homsets()
True

class ParentMethods

Bases: object

base()

Return the base field of this Drinfeld module, viewed as an algebra over the function ring.

This is an instance of the class sage.rings.ring_extension.RingExtension.

EXAMPLES:

sage: Fq = GF(25)
sage: A.<T> = Fq[]
sage: K.<z12> = Fq.extension(6)
sage: p_root = 2*z12^11 + 2*z12^10 + z12^9 + 3*z12^8 + z12^7 + 2*z12^5 + 2*z12^4 + 3*z12^3 + z12^2 + 2*z12
sage: phi = DrinfeldModule(A, [p_root, z12^3, z12^5])
sage: phi.base()
Finite Field in z12 of size 5^12 over its base

The base can be infinite:

sage: sigma = DrinfeldModule(A, [Frac(A).gen(), 1])
sage: sigma.base()
Fraction Field of Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2 over its base

base_morphism()

Return the base morphism of this Drinfeld module.

EXAMPLES:

sage: Fq = GF(25)
sage: A.<T> = Fq[]
sage: K.<z12> = Fq.extension(6)
sage: p_root = 2*z12^11 + 2*z12^10 + z12^9 + 3*z12^8 + z12^7 + 2*z12^5 + 2*z12^4 + 3*z12^3 + z12^2 + 2*z12
sage: phi = DrinfeldModule(A, [p_root, z12^3, z12^5])
sage: phi.base_morphism()
Ring morphism:
  From: Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2
  To:   Finite Field in z12 of size 5^12 over its base
  Defn: T |--> 2*z12^11 + 2*z12^10 + z12^9 + 3*z12^8 + z12^7 + 2*z12^5 + 2*z12^4 + 3*z12^3 + z12^2 + 2*z12

The base field can be infinite:

sage: sigma = DrinfeldModule(A, [Frac(A).gen(), 1])
sage: sigma.base_morphism()
Ring morphism:
  From: Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2
  To:   Fraction Field of Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2 over its base
  Defn: T |--> T

base_over_constants_field()

Return the base field, seen as an extension over the constants field \(\mathbb{F}_q\).

This is an instance of the class sage.rings.ring_extension.RingExtension.

EXAMPLES:

sage: Fq = GF(25)
sage: A.<T> = Fq[]
sage: K.<z12> = Fq.extension(6)
sage: p_root = 2*z12^11 + 2*z12^10 + z12^9 + 3*z12^8 + z12^7 + 2*z12^5 + 2*z12^4 + 3*z12^3 + z12^2 + 2*z12
sage: phi = DrinfeldModule(A, [p_root, z12^3, z12^5])
sage: phi.base_over_constants_field()
Field in z12 with defining polynomial x^6 + (4*z2 + 3)*x^5 + x^4 + (3*z2 + 1)*x^3 + x^2 + (4*z2 + 1)*x + z2 over its base

characteristic()

Return the function ring-characteristic.

EXAMPLES:

sage: Fq = GF(25)
sage: A.<T> = Fq[]
sage: K.<z12> = Fq.extension(6)
sage: p_root = 2*z12^11 + 2*z12^10 + z12^9 + 3*z12^8 + z12^7 + 2*z12^5 + 2*z12^4 + 3*z12^3 + z12^2 + 2*z12
sage: phi = DrinfeldModule(A, [p_root, z12^3, z12^5])
sage: phi.characteristic()
T^2 + (4*z2 + 2)*T + 2
sage: phi.base_morphism()(phi.characteristic())
0

sage: B.<Y> = Fq[]
sage: L = Frac(B)
sage: psi = DrinfeldModule(A, [L(1), 0, 0, L(1)])
sage: psi.characteristic()
Traceback (most recent call last):
...
NotImplementedError: function ring characteristic not implemented in this case

constant_coefficient()

Return the constant coefficient of the generator of this Drinfeld module.

OUTPUT: an element in the base field

EXAMPLES:

sage: Fq = GF(25)
sage: A.<T> = Fq[]
sage: K.<z12> = Fq.extension(6)
sage: p_root = 2*z12^11 + 2*z12^10 + z12^9 + 3*z12^8 + z12^7 + 2*z12^5 + 2*z12^4 + 3*z12^3 + z12^2 + 2*z12
sage: phi = DrinfeldModule(A, [p_root, z12^3, z12^5])
sage: phi.constant_coefficient() == p_root
True

Let \(\mathbb{F}_q[T]\) be the function ring, and let \(\gamma\) be the base of the Drinfeld module. The constant coefficient is \(\gamma(T)\):

sage: C = phi.category()
sage: base = C.base()
sage: base(T) == phi.constant_coefficient()
True

Naturally, two Drinfeld modules in the same category have the same constant coefficient:

sage: t = phi.ore_polring().gen()
sage: psi = C.object(phi.constant_coefficient() + t^3)
sage: psi
Drinfeld module defined by T |--> t^3 + 2*z12^11 + 2*z12^10 + z12^9 + 3*z12^8 + z12^7 + 2*z12^5 + 2*z12^4 + 3*z12^3 + z12^2 + 2*z12

Reciprocally, it is impossible to create two Drinfeld modules in this category if they do not share the same constant coefficient:

sage: rho = C.object(phi.constant_coefficient() + 1 + t^3)
Traceback (most recent call last):
...
ValueError: constant coefficient must equal that of the category

function_ring()

Return the function ring of this Drinfeld module.

EXAMPLES:

sage: Fq = GF(25)
sage: A.<T> = Fq[]
sage: K.<z12> = Fq.extension(6)
sage: p_root = 2*z12^11 + 2*z12^10 + z12^9 + 3*z12^8 + z12^7 + 2*z12^5 + 2*z12^4 + 3*z12^3 + z12^2 + 2*z12
sage: phi = DrinfeldModule(A, [p_root, z12^3, z12^5])
sage: phi.function_ring()
Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2
sage: phi.function_ring() is A
True

ore_polring()

Return the Ore polynomial ring of this Drinfeld module.

EXAMPLES:

sage: Fq = GF(25)
sage: A.<T> = Fq[]
sage: K.<z12> = Fq.extension(6)
sage: p_root = 2*z12^11 + 2*z12^10 + z12^9 + 3*z12^8 + z12^7 + 2*z12^5 + 2*z12^4 + 3*z12^3 + z12^2 + 2*z12
sage: phi = DrinfeldModule(A, [p_root, z12^3, z12^5])
sage: S = phi.ore_polring()
sage: S
Ore Polynomial Ring in t over Finite Field in z12 of size 5^12 over its base twisted by Frob^2

The Ore polynomial ring can also be retrieved from the category of the Drinfeld module:

sage: S is phi.category().ore_polring()
True

The generator of the Drinfeld module is in the Ore polynomial ring:

sage: phi(T) in S
True

ore_variable()

Return the variable of the Ore polynomial ring of this Drinfeld module.

EXAMPLES:

sage: Fq = GF(25)
sage: A.<T> = Fq[]
sage: K.<z12> = Fq.extension(6)
sage: p_root = 2*z12^11 + 2*z12^10 + z12^9 + 3*z12^8 + z12^7 + 2*z12^5 + 2*z12^4 + 3*z12^3 + z12^2 + 2*z12
sage: phi = DrinfeldModule(A, [p_root, z12^3, z12^5])

sage: phi.ore_polring()
Ore Polynomial Ring in t over Finite Field in z12 of size 5^12 over its base twisted by Frob^2
sage: phi.ore_variable()
t

base_morphism()

Return the base morphism of the category.

EXAMPLES:

sage: Fq = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(4)
sage: p_root = z^3 + 7*z^2 + 6*z + 10
sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
sage: C = phi.category()
sage: C.base_morphism()
Ring morphism:
  From: Univariate Polynomial Ring in T over Finite Field of size 11
  To:   Finite Field in z of size 11^4 over its base
  Defn: T |--> z^3 + 7*z^2 + 6*z + 10

sage: C.constant_coefficient() == C.base_morphism()(T)
True

base_over_constants_field()

Return the base field, seen as an extension over the constants field \(\mathbb{F}_q\).

EXAMPLES:

sage: Fq = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(4)
sage: p_root = z^3 + 7*z^2 + 6*z + 10
sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
sage: C = phi.category()
sage: C.base_over_constants_field()
Field in z with defining polynomial x^4 + 8*x^2 + 10*x + 2 over its base

characteristic()

Return the function ring-characteristic.

EXAMPLES:

sage: Fq = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(4)
sage: p_root = z^3 + 7*z^2 + 6*z + 10
sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
sage: C = phi.category()
sage: C.characteristic()
T^2 + 7*T + 2

sage: psi = DrinfeldModule(A, [Frac(A).gen(), 1])
sage: C = psi.category()
sage: C.characteristic()
0

constant_coefficient()

Return the constant coefficient of the category.

EXAMPLES:

sage: Fq = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(4)
sage: p_root = z^3 + 7*z^2 + 6*z + 10
sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
sage: C = phi.category()
sage: C.constant_coefficient()
z^3 + 7*z^2 + 6*z + 10
sage: C.constant_coefficient() == C.base()(T)
True

function_ring()

Return the function ring of the category.

EXAMPLES:

sage: Fq = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(4)
sage: p_root = z^3 + 7*z^2 + 6*z + 10
sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
sage: C = phi.category()
sage: C.function_ring()
Univariate Polynomial Ring in T over Finite Field of size 11
sage: C.function_ring() is A
True

object(gen)

Return a Drinfeld module object in the category whose generator is the input.

INPUT:

• gen – the generator of the Drinfeld module, given as an Ore polynomial or a list of coefficients

EXAMPLES:

sage: Fq = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(4)
sage: p_root = z^3 + 7*z^2 + 6*z + 10
sage: psi = DrinfeldModule(A, [p_root, 1])
sage: C = psi.category()

sage: phi = C.object([p_root, 0, 1])
sage: phi
Drinfeld module defined by T |--> t^2 + z^3 + 7*z^2 + 6*z + 10
sage: t = phi.ore_polring().gen()
sage: C.object(t^2 + z^3 + 7*z^2 + 6*z + 10) is phi
True

ore_polring()

Return the Ore polynomial ring of the category

EXAMPLES:

sage: Fq = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(4)
sage: p_root = z^3 + 7*z^2 + 6*z + 10
sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
sage: C = phi.category()
sage: C.ore_polring()
Ore Polynomial Ring in t over Finite Field in z of size 11^4 over its base twisted by Frob

random_object(rank)

Return a random Drinfeld module in the category with given rank.

INPUT:

• rank – an integer, the rank of the Drinfeld module

EXAMPLES:

sage: Fq = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(4)
sage: p_root = z^3 + 7*z^2 + 6*z + 10
sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
sage: C = phi.category()

sage: psi = C.random_object(3)  # random
Drinfeld module defined by T |--> (6*z^3 + 4*z^2 + 10*z + 9)*t^3 + (4*z^3 + 8*z^2 + 8*z)*t^2 + (10*z^3 + 3*z^2 + 6*z)*t + z^3 + 7*z^2 + 6*z + 10
sage: psi.rank() == 3
True

super_categories()

EXAMPLES:

sage: Fq = GF(11)
sage: A.<T> = Fq[]
sage: K.<z> = Fq.extension(4)
sage: p_root = z^3 + 7*z^2 + 6*z + 10
sage: phi = DrinfeldModule(A, [p_root, 0, 0, 1])
sage: C = phi.category()
sage: C.super_categories()
[Category of objects]