Drinfeld modules over rings of characteristic zero

This module provides the classes sage.rings.function_fields.drinfeld_module.charzero_drinfeld_module.DrinfeldModule_charzero and sage.rings.function_fields.drinfeld_module.charzero_drinfeld_module.DrinfeldModule_rational, which both inherit sage.rings.function_fields.drinfeld_module.drinfeld_module.DrinfeldModule.

AUTHORS:

  • David Ayotte (2023-09)

  • Xavier Caruso (2024-12) - computation of class polynomials

class sage.rings.function_field.drinfeld_modules.charzero_drinfeld_module.DrinfeldModule_charzero(gen, category)[source]

Bases: DrinfeldModule

This class implements Drinfeld \(\mathbb{F}_q[T]\)-modules defined over fields of \(\mathbb{F}_q[T]\)-characteristic zero.

Recall that the \(\mathbb{F}_q[T]\)-characteristic is defined as the kernel of the underlying structure morphism. For general definitions and help on Drinfeld modules, see class sage.rings.function_fields.drinfeld_module.drinfeld_module.DrinfeldModule.

Construction:

The user does not ever need to directly call DrinfeldModule_charzero — the metaclass DrinfeldModule is responsible for instantiating the right class depending on the input:

sage: A = GF(3)['T']
sage: K.<T> = Frac(A)
sage: phi = DrinfeldModule(A, [T, 1])
sage: phi
Drinfeld module defined by T |--> t + T

>>> from sage.all import *
>>> A = GF(Integer(3))['T']
>>> K = Frac(A, names=('T',)); (T,) = K._first_ngens(1)
>>> phi = DrinfeldModule(A, [T, Integer(1)])
>>> phi
Drinfeld module defined by T |--> t + T


sage: isinstance(phi, DrinfeldModule)
True
sage: from sage.rings.function_field.drinfeld_modules.charzero_drinfeld_module import DrinfeldModule_charzero
sage: isinstance(phi, DrinfeldModule_charzero)
True

>>> from sage.all import *
>>> isinstance(phi, DrinfeldModule)
True
>>> from sage.rings.function_field.drinfeld_modules.charzero_drinfeld_module import DrinfeldModule_charzero
>>> isinstance(phi, DrinfeldModule_charzero)
True

Logarithm and exponential

It is possible to calculate the logarithm and the exponential of any Drinfeld modules of characteristic zero:

sage: A = GF(2)['T']
sage: K.<T> = Frac(A)
sage: phi = DrinfeldModule(A, [T, 1])
sage: phi.exponential()
z + ((1/(T^2+T))*z^2) + ((1/(T^8+T^6+T^5+T^3))*z^4) + O(z^8)
sage: phi.logarithm()
z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^8)

>>> from sage.all import *
>>> A = GF(Integer(2))['T']
>>> K = Frac(A, names=('T',)); (T,) = K._first_ngens(1)
>>> phi = DrinfeldModule(A, [T, Integer(1)])
>>> phi.exponential()
z + ((1/(T^2+T))*z^2) + ((1/(T^8+T^6+T^5+T^3))*z^4) + O(z^8)
>>> phi.logarithm()
z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^8)

Goss polynomials

Goss polynomials are a sequence of polynomials related with the analytic theory of Drinfeld module. They provide a function field analogue of certain classical trigonometric functions:

sage: A = GF(2)['T']
sage: K.<T> = Frac(A)
sage: phi = DrinfeldModule(A, [T, 1])
sage: phi.goss_polynomial(1)
X
sage: phi.goss_polynomial(2)
X^2
sage: phi.goss_polynomial(3)
X^3 + (1/(T^2 + T))*X^2

>>> from sage.all import *
>>> A = GF(Integer(2))['T']
>>> K = Frac(A, names=('T',)); (T,) = K._first_ngens(1)
>>> phi = DrinfeldModule(A, [T, Integer(1)])
>>> phi.goss_polynomial(Integer(1))
X
>>> phi.goss_polynomial(Integer(2))
X^2
>>> phi.goss_polynomial(Integer(3))
X^3 + (1/(T^2 + T))*X^2

Base fields of \(\mathbb{F}_q[T]\)-characteristic zero

The base fields need not only be fraction fields of polynomials ring. In the following example, we construct a Drinfeld module over \(\mathbb{F}_q((1/T))\), the completion of the rational function field at the place \(1/T\):

sage: A.<T> = GF(2)[]
sage: L.<s> = LaurentSeriesRing(GF(2))  # s = 1/T
sage: phi = DrinfeldModule(A, [1/s, s + s^2 + s^5 + O(s^6), 1+1/s])
sage: phi(T)
(s^-1 + 1)*t^2 + (s + s^2 + s^5 + O(s^6))*t + s^-1

>>> from sage.all import *
>>> A = GF(Integer(2))['T']; (T,) = A._first_ngens(1)
>>> L = LaurentSeriesRing(GF(Integer(2)), names=('s',)); (s,) = L._first_ngens(1)# s = 1/T
>>> phi = DrinfeldModule(A, [Integer(1)/s, s + s**Integer(2) + s**Integer(5) + O(s**Integer(6)), Integer(1)+Integer(1)/s])
>>> phi(T)
(s^-1 + 1)*t^2 + (s + s^2 + s^5 + O(s^6))*t + s^-1

One can also construct Drinfeld modules over SageMath’s global function fields:

sage: A.<T> = GF(5)[]
sage: K.<z> = FunctionField(GF(5))  # z = T
sage: phi = DrinfeldModule(A, [z, 1, z^2])
sage: phi(T)
z^2*t^2 + t + z

>>> from sage.all import *
>>> A = GF(Integer(5))['T']; (T,) = A._first_ngens(1)
>>> K = FunctionField(GF(Integer(5)), names=('z',)); (z,) = K._first_ngens(1)# z = T
>>> phi = DrinfeldModule(A, [z, Integer(1), z**Integer(2)])
>>> phi(T)
z^2*t^2 + t + z
exponential(prec=+Infinity, name='z')[source]

Return the exponential of this Drinfeld module.

Note that the exponential is only defined when the \(\mathbb{F}_q[T]\)-characteristic is zero.

INPUT:

  • prec – an integer or Infinity (default: Infinity); the precision at which the series is returned; if Infinity, a lazy power series in returned, else, a classical power series is returned.

  • name – string (default: 'z'); the name of the generator of the lazy power series ring

EXAMPLES:

sage: A = GF(2)['T']
sage: K.<T> = Frac(A)
sage: phi = DrinfeldModule(A, [T, 1])
sage: q = A.base_ring().cardinality()

>>> from sage.all import *
>>> A = GF(Integer(2))['T']
>>> K = Frac(A, names=('T',)); (T,) = K._first_ngens(1)
>>> phi = DrinfeldModule(A, [T, Integer(1)])
>>> q = A.base_ring().cardinality()

When prec is Infinity (which is the default), the exponential is returned as a lazy power series, meaning that any of its coefficients can be computed on demands:

sage: exp = phi.exponential(); exp
z + ((1/(T^2+T))*z^2) + ((1/(T^8+T^6+T^5+T^3))*z^4) + O(z^8)
sage: exp[2^4]
1/(T^64 + T^56 + T^52 + ... + T^27 + T^23 + T^15)
sage: exp[2^5]
1/(T^160 + T^144 + T^136 + ... + T^55 + T^47 + T^31)

>>> from sage.all import *
>>> exp = phi.exponential(); exp
z + ((1/(T^2+T))*z^2) + ((1/(T^8+T^6+T^5+T^3))*z^4) + O(z^8)
>>> exp[Integer(2)**Integer(4)]
1/(T^64 + T^56 + T^52 + ... + T^27 + T^23 + T^15)
>>> exp[Integer(2)**Integer(5)]
1/(T^160 + T^144 + T^136 + ... + T^55 + T^47 + T^31)

On the contrary, when prec is a finite number, all the required coefficients are computed at once:

sage: phi.exponential(prec=10)
z + (1/(T^2 + T))*z^2 + (1/(T^8 + T^6 + T^5 + T^3))*z^4 + (1/(T^24 + T^20 + T^18 + T^17 + T^14 + T^13 + T^11 + T^7))*z^8 + O(z^10)

>>> from sage.all import *
>>> phi.exponential(prec=Integer(10))
z + (1/(T^2 + T))*z^2 + (1/(T^8 + T^6 + T^5 + T^3))*z^4 + (1/(T^24 + T^20 + T^18 + T^17 + T^14 + T^13 + T^11 + T^7))*z^8 + O(z^10)

Example in higher rank:

sage: A = GF(5)['T']
sage: K.<T> = Frac(A)
sage: phi = DrinfeldModule(A, [T, T^2, T + T^2 + T^4, 1])
sage: exp = phi.exponential(); exp
z + ((T/(T^4+4))*z^5) + O(z^8)

>>> from sage.all import *
>>> A = GF(Integer(5))['T']
>>> K = Frac(A, names=('T',)); (T,) = K._first_ngens(1)
>>> phi = DrinfeldModule(A, [T, T**Integer(2), T + T**Integer(2) + T**Integer(4), Integer(1)])
>>> exp = phi.exponential(); exp
z + ((T/(T^4+4))*z^5) + O(z^8)

The exponential is the compositional inverse of the logarithm (see logarithm()):

sage: log = phi.logarithm(); log
z + ((4*T/(T^4+4))*z^5) + O(z^8)
sage: exp.compose(log)
z + O(z^8)
sage: log.compose(exp)
z + O(z^8)

>>> from sage.all import *
>>> log = phi.logarithm(); log
z + ((4*T/(T^4+4))*z^5) + O(z^8)
>>> exp.compose(log)
z + O(z^8)
>>> log.compose(exp)
z + O(z^8)

REFERENCE:

See section 4.6 of [Gos1998] for the definition of the exponential.

goss_polynomial(n, var='X')[source]

Return the \(n\)-th Goss polynomial of the Drinfeld module.

Note that Goss polynomials are only defined for Drinfeld modules of characteristic zero.

INPUT:

  • n – integer; the index of the Goss polynomial

  • var– string (default: 'X'); the name of polynomial variable

OUTPUT: a univariate polynomial in var over the base \(A\)-field

EXAMPLES:

sage: A = GF(3)['T']
sage: K.<T> = Frac(A)
sage: phi = DrinfeldModule(A, [T, 1])  # Carlitz module
sage: phi.goss_polynomial(1)
X
sage: phi.goss_polynomial(2)
X^2
sage: phi.goss_polynomial(4)
X^4 + (1/(T^3 + 2*T))*X^2
sage: phi.goss_polynomial(5)
X^5 + (2/(T^3 + 2*T))*X^3
sage: phi.goss_polynomial(10)
X^10 + (1/(T^3 + 2*T))*X^8 + (1/(T^6 + T^4 + T^2))*X^6 + (1/(T^9 + 2*T^3))*X^4 + (1/(T^18 + 2*T^12 + 2*T^10 + T^4))*X^2

>>> from sage.all import *
>>> A = GF(Integer(3))['T']
>>> K = Frac(A, names=('T',)); (T,) = K._first_ngens(1)
>>> phi = DrinfeldModule(A, [T, Integer(1)])  # Carlitz module
>>> phi.goss_polynomial(Integer(1))
X
>>> phi.goss_polynomial(Integer(2))
X^2
>>> phi.goss_polynomial(Integer(4))
X^4 + (1/(T^3 + 2*T))*X^2
>>> phi.goss_polynomial(Integer(5))
X^5 + (2/(T^3 + 2*T))*X^3
>>> phi.goss_polynomial(Integer(10))
X^10 + (1/(T^3 + 2*T))*X^8 + (1/(T^6 + T^4 + T^2))*X^6 + (1/(T^9 + 2*T^3))*X^4 + (1/(T^18 + 2*T^12 + 2*T^10 + T^4))*X^2

REFERENCE:

Section 3 of [Gek1988] provides an exposition of Goss polynomials.

logarithm(prec=+Infinity, name='z')[source]

Return the logarithm of the given Drinfeld module.

By definition, the logarithm is the compositional inverse of the exponential (see exponential()). Note that the logarithm is only defined when the \(\mathbb{F}_q[T]\)-characteristic is zero.

INPUT:

  • prec – an integer or Infinity (default: Infinity); the precision at which the series is returned; if Infinity, a lazy power series in returned

  • name – string (default: 'z'); the name of the generator of the lazy power series ring

EXAMPLES:

sage: A = GF(2)['T']
sage: K.<T> = Frac(A)
sage: phi = DrinfeldModule(A, [T, 1])

>>> from sage.all import *
>>> A = GF(Integer(2))['T']
>>> K = Frac(A, names=('T',)); (T,) = K._first_ngens(1)
>>> phi = DrinfeldModule(A, [T, Integer(1)])

When prec is Infinity (which is the default), the logarithm is returned as a lazy power series, meaning that any of its coefficients can be computed on demands:

sage: log = phi.logarithm(); log
z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^8)
sage: log[2^4]
1/(T^30 + T^29 + T^27 + ... + T^7 + T^5 + T^4)
sage: log[2^5]
1/(T^62 + T^61 + T^59 + ... + T^8 + T^6 + T^5)

>>> from sage.all import *
>>> log = phi.logarithm(); log
z + ((1/(T^2+T))*z^2) + ((1/(T^6+T^5+T^3+T^2))*z^4) + O(z^8)
>>> log[Integer(2)**Integer(4)]
1/(T^30 + T^29 + T^27 + ... + T^7 + T^5 + T^4)
>>> log[Integer(2)**Integer(5)]
1/(T^62 + T^61 + T^59 + ... + T^8 + T^6 + T^5)

If prec is a finite number, all the required coefficients are computed at once:

sage: phi.logarithm(prec=10)
z + (1/(T^2 + T))*z^2 + (1/(T^6 + T^5 + T^3 + T^2))*z^4 + (1/(T^14 + T^13 + T^11 + T^10 + T^7 + T^6 + T^4 + T^3))*z^8 + O(z^10)

>>> from sage.all import *
>>> phi.logarithm(prec=Integer(10))
z + (1/(T^2 + T))*z^2 + (1/(T^6 + T^5 + T^3 + T^2))*z^4 + (1/(T^14 + T^13 + T^11 + T^10 + T^7 + T^6 + T^4 + T^3))*z^8 + O(z^10)

Example in higher rank:

sage: A = GF(5)['T']
sage: K.<T> = Frac(A)
sage: phi = DrinfeldModule(A, [T, T^2, T + T^2 + T^4, 1])
sage: phi.logarithm()
z + ((4*T/(T^4+4))*z^5) + O(z^8)

>>> from sage.all import *
>>> A = GF(Integer(5))['T']
>>> K = Frac(A, names=('T',)); (T,) = K._first_ngens(1)
>>> phi = DrinfeldModule(A, [T, T**Integer(2), T + T**Integer(2) + T**Integer(4), Integer(1)])
>>> phi.logarithm()
z + ((4*T/(T^4+4))*z^5) + O(z^8)
class sage.rings.function_field.drinfeld_modules.charzero_drinfeld_module.DrinfeldModule_rational(gen, category)[source]

Bases: DrinfeldModule_charzero

A class for Drinfeld modules defined over the fraction field of the underlying function field.

class_polynomial()[source]

Return the class polynomial, that is the Fitting ideal of the class module, of this Drinfeld module.

We refer to [Tae2012] for the definition and basic properties of the class module.

EXAMPLES:

We check that the class module of the Carlitz module is trivial:

sage: q = 5
sage: Fq = GF(q)
sage: A = Fq['T']
sage: K.<T> = Frac(A)
sage: C = DrinfeldModule(A, [T, 1]); C
Drinfeld module defined by T |--> t + T
sage: C.class_polynomial()
1

>>> from sage.all import *
>>> q = Integer(5)
>>> Fq = GF(q)
>>> A = Fq['T']
>>> K = Frac(A, names=('T',)); (T,) = K._first_ngens(1)
>>> C = DrinfeldModule(A, [T, Integer(1)]); C
Drinfeld module defined by T |--> t + T
>>> C.class_polynomial()
1

When the coefficients of the Drinfeld module have small enough degrees, the class module is always trivial:

sage: gs = [T] + [A.random_element(degree = q^i)
....:             for i in range(1, 5)]
sage: phi = DrinfeldModule(A, gs)
sage: phi.class_polynomial()
1

>>> from sage.all import *
>>> gs = [T] + [A.random_element(degree = q**i)
...             for i in range(Integer(1), Integer(5))]
>>> phi = DrinfeldModule(A, gs)
>>> phi.class_polynomial()
1

Here is an example with a nontrivial class module:

sage: phi = DrinfeldModule(A, [T, 2*T^14 + 2*T^4])
sage: phi.class_polynomial()
T + 3

>>> from sage.all import *
>>> phi = DrinfeldModule(A, [T, Integer(2)*T**Integer(14) + Integer(2)*T**Integer(4)])
>>> phi.class_polynomial()
T + 3
coefficient_in_function_ring(n)[source]

Return the \(n\)-th coefficient of this Drinfeld module as an element of the underlying function ring.

INPUT:

  • n – an integer

EXAMPLES:

sage: q = 5
sage: Fq = GF(q)
sage: A = Fq['T']
sage: R = Fq['U']
sage: K.<U> = Frac(R)
sage: phi = DrinfeldModule(A, [U, 0, U^2, U^3])
sage: phi.coefficient_in_function_ring(2)
T^2

>>> from sage.all import *
>>> q = Integer(5)
>>> Fq = GF(q)
>>> A = Fq['T']
>>> R = Fq['U']
>>> K = Frac(R, names=('U',)); (U,) = K._first_ngens(1)
>>> phi = DrinfeldModule(A, [U, Integer(0), U**Integer(2), U**Integer(3)])
>>> phi.coefficient_in_function_ring(Integer(2))
T^2

Compare with the method meth:\(coefficient\):

sage: phi.coefficient(2)
U^2

>>> from sage.all import *
>>> phi.coefficient(Integer(2))
U^2

If the required coefficient is not a polynomials, an error is raised:

sage: psi = DrinfeldModule(A, [U, 1/U])
sage: psi.coefficient_in_function_ring(0)
T
sage: psi.coefficient_in_function_ring(1)
Traceback (most recent call last):
...
ValueError: coefficient is not polynomial

>>> from sage.all import *
>>> psi = DrinfeldModule(A, [U, Integer(1)/U])
>>> psi.coefficient_in_function_ring(Integer(0))
T
>>> psi.coefficient_in_function_ring(Integer(1))
Traceback (most recent call last):
...
ValueError: coefficient is not polynomial
coefficients_in_function_ring(sparse=True)[source]

Return the coefficients of this Drinfeld module as elements of the underlying function ring.

INPUT:

  • sparse – a boolean (default: True); if True, only return the nonzero coefficients; otherwise, return all of them.

EXAMPLES:

sage: q = 5
sage: Fq = GF(q)
sage: A = Fq['T']
sage: R = Fq['U']
sage: K.<U> = Frac(R)
sage: phi = DrinfeldModule(A, [U, 0, U^2, U^3])
sage: phi.coefficients_in_function_ring()
[T, T^2, T^3]
sage: phi.coefficients_in_function_ring(sparse=False)
[T, 0, T^2, T^3]

>>> from sage.all import *
>>> q = Integer(5)
>>> Fq = GF(q)
>>> A = Fq['T']
>>> R = Fq['U']
>>> K = Frac(R, names=('U',)); (U,) = K._first_ngens(1)
>>> phi = DrinfeldModule(A, [U, Integer(0), U**Integer(2), U**Integer(3)])
>>> phi.coefficients_in_function_ring()
[T, T^2, T^3]
>>> phi.coefficients_in_function_ring(sparse=False)
[T, 0, T^2, T^3]

Compare with the method meth:\(coefficients\):

sage: phi.coefficients()
[U, U^2, U^3]

>>> from sage.all import *
>>> phi.coefficients()
[U, U^2, U^3]

If the coefficients are not polynomials, an error is raised:

sage: psi = DrinfeldModule(A, [U, 1/U])
sage: psi.coefficients_in_function_ring()
Traceback (most recent call last):
...
ValueError: coefficients are not polynomials

>>> from sage.all import *
>>> psi = DrinfeldModule(A, [U, Integer(1)/U])
>>> psi.coefficients_in_function_ring()
Traceback (most recent call last):
...
ValueError: coefficients are not polynomials