Elements of Drinfeld modular forms rings¶

This module defines the elements of the class DrinfeldModularForms.

AUTHORS:

David Ayotte (2022): initial version

class sage.modular.drinfeld_modform.element.DrinfeldModularFormsElement(parent, polynomial)[source]¶

Bases: ModuleElement

Element class of rings of Drinfeld modular forms.

Recall that a graded Drinfeld form is a sum of Drinfeld modular forms having potentially different weights:

\[F = f_{k_1} + f_{k_2} + \cdots + f_{k_n}\]

where \(f_{k_i}\) is a Drinfeld modular form of weight \(k_i\). We also say that \(f_{k_i}\) is an homogeneous component of weight \(k_i\). If \(n=1\), then we say that \(F\) is homogeneous of weight \(k_1\).

EXAMPLES: use the inject_variable method of the parent to quickly assign variables names to the generators:

Sage

sage: A = GF(3)['T']
sage: K.<T> = Frac(A)
sage: M = DrinfeldModularForms(K, 2)
sage: M.inject_variables()
Defining g1, g2
sage: g1 in M
True
sage: g2.parent()
Ring of Drinfeld modular forms of rank 2 over Fraction Field of Univariate Polynomial Ring in T over Finite Field of size 3

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']
>>> K = Frac(A, names=('T',)); (T,) = K._first_ngens(1)
>>> M = DrinfeldModularForms(K, Integer(2))
>>> M.inject_variables()
Defining g1, g2
>>> g1 in M
True
>>> g2.parent()
Ring of Drinfeld modular forms of rank 2 over Fraction Field of Univariate Polynomial Ring in T over Finite Field of size 3

Sage Live

A = GF(3)['T']
K.<T> = Frac(A)
M = DrinfeldModularForms(K, 2)
M.inject_variables()
g1 in M
g2.parent()

Next, via algebraic combination of the generator, we may create any element of the ring:

Sage

sage: F = g1*g2 + g2
sage: F
g1*g2 + g2
sage: F.is_homogeneous()
False
sage: F.homogeneous_components()
{8: g2, 10: g1*g2}

Python

>>> from sage.all import *
>>> F = g1*g2 + g2
>>> F
g1*g2 + g2
>>> F.is_homogeneous()
False
>>> F.homogeneous_components()
{8: g2, 10: g1*g2}

Sage Live

F = g1*g2 + g2
F
F.is_homogeneous()
F.homogeneous_components()

If the created form is homogeneous, we can ask for its weight in which case it will be a Drinfeld modular form:

Sage

sage: H = g1^4*g2^9 + T*g1^8*g2^8 + (T^2 - 1)*g1^28*g2^3
sage: H.is_homogeneous()
True
sage: H.weight()
80

Python

>>> from sage.all import *
>>> H = g1**Integer(4)*g2**Integer(9) + T*g1**Integer(8)*g2**Integer(8) + (T**Integer(2) - Integer(1))*g1**Integer(28)*g2**Integer(3)
>>> H.is_homogeneous()
True
>>> H.weight()
80

Sage Live

H = g1^4*g2^9 + T*g1^8*g2^8 + (T^2 - 1)*g1^28*g2^3
H.is_homogeneous()
H.weight()

You can also construct an element by simply passing a multivariate polynomial to the parent:

Sage

sage: f1, f2 = polygens(K, 2, 'f1, f2')
sage: M(f1)
g1
sage: M(f2)
g2
sage: M(T*f1 + f2^3 + T^2 + 1)
g2^3 + T*g1 + (T^2 + 1)

Python

>>> from sage.all import *
>>> f1, f2 = polygens(K, Integer(2), 'f1, f2')
>>> M(f1)
g1
>>> M(f2)
g2
>>> M(T*f1 + f2**Integer(3) + T**Integer(2) + Integer(1))
g2^3 + T*g1 + (T^2 + 1)

Sage Live

f1, f2 = polygens(K, 2, 'f1, f2')
M(f1)
M(f2)
M(T*f1 + f2^3 + T^2 + 1)

Note

This class should not be directly instanciated, instead create an instance of the parent DrinfeldModularForms and access its elements using the relevant methods.

homogeneous_components()[source]¶

Return the homogeneous components of this graded Drinfeld form.

EXAMPLES:

Sage

sage: A = GF(3)['T']; K = Frac(A)
sage: M = DrinfeldModularForms(K, 2)
sage: M.inject_variables()
Defining g1, g2
sage: F = g1 + g1^2 + g1*g2^2 + g2^4
sage: D = F.homogeneous_components(); D
{2: g1, 4: g1^2, 18: g1*g2^2, 32: g2^4}
sage: D[32]
g2^4

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']; K = Frac(A)
>>> M = DrinfeldModularForms(K, Integer(2))
>>> M.inject_variables()
Defining g1, g2
>>> F = g1 + g1**Integer(2) + g1*g2**Integer(2) + g2**Integer(4)
>>> D = F.homogeneous_components(); D
{2: g1, 4: g1^2, 18: g1*g2^2, 32: g2^4}
>>> D[Integer(32)]
g2^4

Sage Live

A = GF(3)['T']; K = Frac(A)
M = DrinfeldModularForms(K, 2)
M.inject_variables()
F = g1 + g1^2 + g1*g2^2 + g2^4
D = F.homogeneous_components(); D
D[32]

is_homogeneous()[source]¶

Return whether the graded form is homogeneous in the weight.

We recall that elements of Drinfeld modular forms ring are not necessarily modular forms as they may have mixed weight components.

EXAMPLES:

Sage

sage: A = GF(3)['T']; K = Frac(A)
sage: M = DrinfeldModularForms(K, 2)
sage: M.inject_variables()
Defining g1, g2
sage: f = g1^5*g2^2  # homogeneous polynomial
sage: f.is_homogeneous()
True
sage: g = g1 + g2  # mixed weight components
sage: g.is_homogeneous()
False

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']; K = Frac(A)
>>> M = DrinfeldModularForms(K, Integer(2))
>>> M.inject_variables()
Defining g1, g2
>>> f = g1**Integer(5)*g2**Integer(2)  # homogeneous polynomial
>>> f.is_homogeneous()
True
>>> g = g1 + g2  # mixed weight components
>>> g.is_homogeneous()
False

Sage Live

A = GF(3)['T']; K = Frac(A)
M = DrinfeldModularForms(K, 2)
M.inject_variables()
f = g1^5*g2^2  # homogeneous polynomial
f.is_homogeneous()
g = g1 + g2  # mixed weight components
g.is_homogeneous()

is_one()[source]¶

Return True whether this graded Drinfeld form is the multiplicative identity.

EXAMPLES:

Sage

sage: A = GF(3)['T']; K = Frac(A)
sage: M = DrinfeldModularForms(K, 2)
sage: u = M.one()
sage: u.is_one()
True
sage: (M.0).is_one()
False

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']; K = Frac(A)
>>> M = DrinfeldModularForms(K, Integer(2))
>>> u = M.one()
>>> u.is_one()
True
>>> (M.gen(0)).is_one()
False

Sage Live

A = GF(3)['T']; K = Frac(A)
M = DrinfeldModularForms(K, 2)
u = M.one()
u.is_one()
(M.0).is_one()

is_zero()[source]¶

Return True whether this graded Drinfeld form is the additive identity.

EXAMPLES:

Sage

sage: A = GF(3)['T']; K = Frac(A)
sage: M = DrinfeldModularForms(K, 2)
sage: z = M.zero()
sage: z.is_zero()
True
sage: f = M.0
sage: f.is_zero()
False
sage: (f - f).is_zero()
True
sage: (0 * M.0).is_zero()
True

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']; K = Frac(A)
>>> M = DrinfeldModularForms(K, Integer(2))
>>> z = M.zero()
>>> z.is_zero()
True
>>> f = M.gen(0)
>>> f.is_zero()
False
>>> (f - f).is_zero()
True
>>> (Integer(0) * M.gen(0)).is_zero()
True

Sage Live

A = GF(3)['T']; K = Frac(A)
M = DrinfeldModularForms(K, 2)
z = M.zero()
z.is_zero()
f = M.0
f.is_zero()
(f - f).is_zero()
(0 * M.0).is_zero()

polynomial()[source]¶

Return this graded Drinfeld forms as a multivariate polynomial over the generators of the ring.

OUTPUT: a multivariate polynomial over the base ring

EXAMPLES:

Sage

sage: A = GF(3)['T']; K = Frac(A)
sage: M = DrinfeldModularForms(K, 2)
sage: M.inject_variables()
Defining g1, g2
sage: P1 = g1.polynomial();
sage: P2 = g2.polynomial();
sage: P2^2 + P1^2 + P1
g2^2 + g1^2 + g1
sage: P1.parent()
Multivariate Polynomial Ring in g1, g2 over Fraction Field of Univariate Polynomial Ring in T over Finite Field of size 3

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']; K = Frac(A)
>>> M = DrinfeldModularForms(K, Integer(2))
>>> M.inject_variables()
Defining g1, g2
>>> P1 = g1.polynomial();
>>> P2 = g2.polynomial();
>>> P2**Integer(2) + P1**Integer(2) + P1
g2^2 + g1^2 + g1
>>> P1.parent()
Multivariate Polynomial Ring in g1, g2 over Fraction Field of Univariate Polynomial Ring in T over Finite Field of size 3

Sage Live

A = GF(3)['T']; K = Frac(A)
M = DrinfeldModularForms(K, 2)
M.inject_variables()
P1 = g1.polynomial();
P2 = g2.polynomial();
P2^2 + P1^2 + P1
P1.parent()

The degree of each variables corresponds to the weight of the generator:

Sage

sage: P1.degree()
2
sage: P2.degree()
8

Python

>>> from sage.all import *
>>> P1.degree()
2
>>> P2.degree()
8

Sage Live

P1.degree()
P2.degree()

rank()[source]¶

Return the rank of this graded Drinfeld form.

Note that the rank is independent of the chosen form and depends only on the parent.

EXAMPLES:

Sage

sage: A = GF(3)['T']; K = Frac(A)
sage: M2 = DrinfeldModularForms(K, 2)
sage: (M2.0).rank()
2
sage: M5 = DrinfeldModularForms(K, 5)
sage: (M5.0 + M5.3).rank()
5

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']; K = Frac(A)
>>> M2 = DrinfeldModularForms(K, Integer(2))
>>> (M2.gen(0)).rank()
2
>>> M5 = DrinfeldModularForms(K, Integer(5))
>>> (M5.gen(0) + M5.gen(3)).rank()
5

Sage Live

A = GF(3)['T']; K = Frac(A)
M2 = DrinfeldModularForms(K, 2)
(M2.0).rank()
M5 = DrinfeldModularForms(K, 5)
(M5.0 + M5.3).rank()

type()[source]¶

Return the type of this graded Drinfeld form.

Recall that the type is the integer \(0 \leq m \leq q-1\) such that

\[f(\gamma(w)) = \mathrm{det}(\gamma)^m j(\gamma, w)^k f(w).\]

EXAMPLES:

Sage

sage: A = GF(11)['T']; K = Frac(A)
sage: M = DrinfeldModularForms(K, 2, has_type=True)
sage: M.inject_variables()
Defining g1, h2
sage: F = g1*h2^9
sage: F.type()
9
sage: (h2^11).type()
1
sage: g1.type()
0

Python

>>> from sage.all import *
>>> A = GF(Integer(11))['T']; K = Frac(A)
>>> M = DrinfeldModularForms(K, Integer(2), has_type=True)
>>> M.inject_variables()
Defining g1, h2
>>> F = g1*h2**Integer(9)
>>> F.type()
9
>>> (h2**Integer(11)).type()
1
>>> g1.type()
0

Sage Live

A = GF(11)['T']; K = Frac(A)
M = DrinfeldModularForms(K, 2, has_type=True)
M.inject_variables()
F = g1*h2^9
F.type()
(h2^11).type()
g1.type()

The type only makes sense when the form is homogeneous:

Sage

sage: F = g1^4 + h2
sage: F.type()
Traceback (most recent call last):
...
ValueError: the graded form is not homogeneous

Python

>>> from sage.all import *
>>> F = g1**Integer(4) + h2
>>> F.type()
Traceback (most recent call last):
...
ValueError: the graded form is not homogeneous

Sage Live

F = g1^4 + h2
F.type()

weight()[source]¶

Return the weight of this graded Drinfeld modular form.

EXAMPLES:

Sage

sage: A = GF(3)['T']; K = Frac(A)
sage: M = DrinfeldModularForms(K, 2)
sage: M.inject_variables()
Defining g1, g2
sage: g1.weight()
2
sage: g2.weight()
8
sage: f = g1^5*g2^2
sage: f.weight()
26

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']; K = Frac(A)
>>> M = DrinfeldModularForms(K, Integer(2))
>>> M.inject_variables()
Defining g1, g2
>>> g1.weight()
2
>>> g2.weight()
8
>>> f = g1**Integer(5)*g2**Integer(2)
>>> f.weight()
26

Sage Live

A = GF(3)['T']; K = Frac(A)
M = DrinfeldModularForms(K, 2)
M.inject_variables()
g1.weight()
g2.weight()
f = g1^5*g2^2
f.weight()

If the form is not homogeneous, then the method returns an error:

Sage

sage: f = g1 + g2
sage: f.weight()
Traceback (most recent call last):
...
ValueError: the graded form is not homogeneous

Python

>>> from sage.all import *
>>> f = g1 + g2
>>> f.weight()
Traceback (most recent call last):
...
ValueError: the graded form is not homogeneous

Sage Live

f = g1 + g2
f.weight()