Graded rings of Drinfeld modular forms¶

This module defines a class named DrinfeldModularForms. Currently, the implementation only supports the full modular group \(\mathrm{GL}_r(A)\) where \(A = \mathbb{F}_q[T]\).

The implementation is based on the following identification:

\[M^r(\mathrm{GL}_r(A)) = \mathbb{C}_{\infty}[g_1, \ldots, g_{r-1}, g_{r}].\]

where \(g_i\) is the \(i\)-th coefficient form of weight \(q^{i} - 1\).

AUTHORS:

David Ayotte (2022): initial version

class sage.modular.drinfeld_modform.ring.DrinfeldModularForms(base_ring, rank, group, has_type, names)[source]¶

Bases: Parent, UniqueRepresentation

Base class for the graded ring of Drinfeld modular forms.

If \(K = \mathrm{Frac}(A)\) where \(A = \mathbb{F}_q[T]\), then the ring of Drinfeld modular forms over \(K\) of rank \(r\) and type zero for \(\mathrm{GL}_r(A)\) is

\[M^{r, 0}(\mathrm{GL}_r(A)) = K[g_1, \ldots, g_{r-1}, g_{r}].\]

where \(g_i\) the \(i\)-th coefficient form of weight \(q^{i} - 1\) at \(T\).

Similarly, the ring of Drinfeld modular forms over \(K\) of rank \(r\) and arbitrary type is

\[M^{r}(\mathrm{GL}_r(A)) = K[g_1, \ldots, g_{r-1}, h_{r}].\]

where \(h_r\) is a form of weight \((q^r - 1)/(q - 1)\) and type \(1\).

We will see the elements of this ring as formal objects given by algebraic combination of the generator of the ring. See the class DrinfeldModularFormsElement for more details about their implementation.

INPUT:

base_ring – the fraction field of a univariate polynomial ring over \(\mathbb{F}_q\)
rank integer (default: None); the rank of the ring. If the rank is None, then the names of the generators must be specified.
group – (not implemented, default: None) the group of the ring. The current implementation only supports the full modular group \(\mathrm{GL}_r(A)\).
has_type – boolean (default: False); if set to True, returns the graded ring of arbitrary type
names – string, tuple or list (default: None); a single character, a tuple or list of character, or comma separated string of character representing the names of the generators. If this parameter is set to None and the rank is specified, then the default names for the generators will be:
- g1, g2, ..., gr for the type zero forms
- g1, g2, ..., hr for the arbitrary type forms.
If this parameter is a single character, for example f, and a rank is specified, then the names will be of the form f1, f2, ..., fr. Finally, if this parameter is a list, a tupe or a string of comma separated characters, then each character will corresponds to a generator. Note that in this case, it not necessary to specify the rank.

EXAMPLES:

Sage

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

Python

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

Sage Live

q = 3
A = GF(q)['T']
K.<T> = Frac(A)
M = DrinfeldModularForms(K, 3)
M

Use the gens() method to obtain the generators of the ring:

Sage

sage: M.gens()
[g1, g2, g3]
sage: M.inject_variables()  # assign the variable g1, g2, g3
Defining g1, g2, g3
sage: T*g1*g2 + g3
g3 + T*g1*g2

Python

>>> from sage.all import *
>>> M.gens()
[g1, g2, g3]
>>> M.inject_variables()  # assign the variable g1, g2, g3
Defining g1, g2, g3
>>> T*g1*g2 + g3
g3 + T*g1*g2

Sage Live

M.gens()
M.inject_variables()  # assign the variable g1, g2, g3
T*g1*g2 + g3

When creating the ring, one can name the generators in various ways:

Sage

sage: M.<F, G, H> = DrinfeldModularForms(K)
sage: M.gens()
[F, G, H]
sage: M = DrinfeldModularForms(K, 5, names='f')  # must specify the rank
sage: M.gens()
[f1, f2, f3, f4, f5]
sage: M = DrinfeldModularForms(K, names='u, v, w, x')
sage: M.gens()
[u, v, w, x]
sage: M = DrinfeldModularForms(K, names=['F', 'G', 'H'])
sage: M.gens()
[F, G, H]

Python

>>> from sage.all import *
>>> M = DrinfeldModularForms(K, names=('F', 'G', 'H',)); (F, G, H,) = M._first_ngens(3)
>>> M.gens()
[F, G, H]
>>> M = DrinfeldModularForms(K, Integer(5), names='f')  # must specify the rank
>>> M.gens()
[f1, f2, f3, f4, f5]
>>> M = DrinfeldModularForms(K, names='u, v, w, x')
>>> M.gens()
[u, v, w, x]
>>> M = DrinfeldModularForms(K, names=['F', 'G', 'H'])
>>> M.gens()
[F, G, H]

Sage Live

M.<F, G, H> = DrinfeldModularForms(K)
M.gens()
M = DrinfeldModularForms(K, 5, names='f')  # must specify the rank
M.gens()
M = DrinfeldModularForms(K, names='u, v, w, x')
M.gens()
M = DrinfeldModularForms(K, names=['F', 'G', 'H'])
M.gens()

Set the keyword parameter has_type to True in order to create the ring of Drinfeld modular forms of arbitrary type:

Sage

sage: M = DrinfeldModularForms(K, 4, has_type=True)
sage: M.gens()
[g1, g2, g3, h4]
sage: h4 = M.3
sage: h4.type()
1

Python

>>> from sage.all import *
>>> M = DrinfeldModularForms(K, Integer(4), has_type=True)
>>> M.gens()
[g1, g2, g3, h4]
>>> h4 = M.gen(3)
>>> h4.type()
1

Sage Live

M = DrinfeldModularForms(K, 4, has_type=True)
M.gens()
h4 = M.3
h4.type()

To obtain a generating set of the subspace of forms of a fixed weight, use the methode basis_of_weight():

Sage

sage: M = DrinfeldModularForms(K, 2)
sage: M.basis_of_weight(q^3 - 1)
[g1*g2^3, g1^5*g2^2, g1^9*g2, g1^13]

Python

>>> from sage.all import *
>>> M = DrinfeldModularForms(K, Integer(2))
>>> M.basis_of_weight(q**Integer(3) - Integer(1))
[g1*g2^3, g1^5*g2^2, g1^9*g2, g1^13]

Sage Live

M = DrinfeldModularForms(K, 2)
M.basis_of_weight(q^3 - 1)

In order to compute the coefficient forms, use the methods coefficient_form() and coefficient_forms():

Sage

sage: M = DrinfeldModularForms(K, 3)
sage: M.coefficient_form(1)
g1
sage: M.coefficient_form(2)
g2
sage: M.coefficient_form(3)
g3
sage: M.coefficient_forms(T)
[g1, g2, g3]
sage: M.coefficient_forms(T^2)
[(T^3 + T)*g1,
 g1^4 + (T^9 + T)*g2,
 g1^9*g2 + g1*g2^3 + (T^27 + T)*g3,
 g1^27*g3 + g1*g3^3 + g2^10,
 g2^27*g3 + g2*g3^9,
 g3^28]

Python

>>> from sage.all import *
>>> M = DrinfeldModularForms(K, Integer(3))
>>> M.coefficient_form(Integer(1))
g1
>>> M.coefficient_form(Integer(2))
g2
>>> M.coefficient_form(Integer(3))
g3
>>> M.coefficient_forms(T)
[g1, g2, g3]
>>> M.coefficient_forms(T**Integer(2))
[(T^3 + T)*g1,
 g1^4 + (T^9 + T)*g2,
 g1^9*g2 + g1*g2^3 + (T^27 + T)*g3,
 g1^27*g3 + g1*g3^3 + g2^10,
 g2^27*g3 + g2*g3^9,
 g3^28]

Sage Live

M = DrinfeldModularForms(K, 3)
M.coefficient_form(1)
M.coefficient_form(2)
M.coefficient_form(3)
M.coefficient_forms(T)
M.coefficient_forms(T^2)

REFERENCE:

For a quick introduction to Drinfeld modular forms, see the tutorial. For more extensive references, see [Gek1988] and [BRP2018].

Element[source]¶: alias of DrinfeldModularFormsElement

basis(k)[source]¶

Return a list of Drinfeld modular forms which forms a basis for the subspace of weight \(k\).

Note that if \(k\not\equiv 0\) modulo \(q-1\), then the subspace is 0.

An alias of this method is basis.

INPUT:

k – integer

EXAMPLES:

Sage

sage: q = 3; A = GF(q)['T']; K = Frac(A);
sage: M = DrinfeldModularForms(K, 2)
sage: M.basis_of_weight(q - 1)
[g1]
sage: M.basis_of_weight(q^2 - 1)
[g2, g1^4]
sage: M.basis_of_weight(q^3 - 1)
[g1*g2^3, g1^5*g2^2, g1^9*g2, g1^13]
sage: M.basis_of_weight(19*(q-1))
[g1^3*g2^4, g1^7*g2^3, g1^11*g2^2, g1^15*g2, g1^19]

Python

>>> from sage.all import *
>>> q = Integer(3); A = GF(q)['T']; K = Frac(A);
>>> M = DrinfeldModularForms(K, Integer(2))
>>> M.basis_of_weight(q - Integer(1))
[g1]
>>> M.basis_of_weight(q**Integer(2) - Integer(1))
[g2, g1^4]
>>> M.basis_of_weight(q**Integer(3) - Integer(1))
[g1*g2^3, g1^5*g2^2, g1^9*g2, g1^13]
>>> M.basis_of_weight(Integer(19)*(q-Integer(1)))
[g1^3*g2^4, g1^7*g2^3, g1^11*g2^2, g1^15*g2, g1^19]

Sage Live

q = 3; A = GF(q)['T']; K = Frac(A);
M = DrinfeldModularForms(K, 2)
M.basis_of_weight(q - 1)
M.basis_of_weight(q^2 - 1)
M.basis_of_weight(q^3 - 1)
M.basis_of_weight(19*(q-1))

basis_of_weight(k)[source]¶

Return a list of Drinfeld modular forms which forms a basis for the subspace of weight \(k\).

Note that if \(k\not\equiv 0\) modulo \(q-1\), then the subspace is 0.

An alias of this method is basis.

INPUT:

k – integer

EXAMPLES:

Sage

sage: q = 3; A = GF(q)['T']; K = Frac(A);
sage: M = DrinfeldModularForms(K, 2)
sage: M.basis_of_weight(q - 1)
[g1]
sage: M.basis_of_weight(q^2 - 1)
[g2, g1^4]
sage: M.basis_of_weight(q^3 - 1)
[g1*g2^3, g1^5*g2^2, g1^9*g2, g1^13]
sage: M.basis_of_weight(19*(q-1))
[g1^3*g2^4, g1^7*g2^3, g1^11*g2^2, g1^15*g2, g1^19]

Python

>>> from sage.all import *
>>> q = Integer(3); A = GF(q)['T']; K = Frac(A);
>>> M = DrinfeldModularForms(K, Integer(2))
>>> M.basis_of_weight(q - Integer(1))
[g1]
>>> M.basis_of_weight(q**Integer(2) - Integer(1))
[g2, g1^4]
>>> M.basis_of_weight(q**Integer(3) - Integer(1))
[g1*g2^3, g1^5*g2^2, g1^9*g2, g1^13]
>>> M.basis_of_weight(Integer(19)*(q-Integer(1)))
[g1^3*g2^4, g1^7*g2^3, g1^11*g2^2, g1^15*g2, g1^19]

Sage Live

q = 3; A = GF(q)['T']; K = Frac(A);
M = DrinfeldModularForms(K, 2)
M.basis_of_weight(q - 1)
M.basis_of_weight(q^2 - 1)
M.basis_of_weight(q^3 - 1)
M.basis_of_weight(19*(q-1))

coefficient_form(i, a=None)[source]¶

Return the \(i\)-th coefficient form of the universal Drinfeld module over \(\Omega^r(\mathbb{C}_{\infty})\):

\[\phi_{w, a} = a + g_{1, a}\tau + \cdots + g_{r d_a, a}\tau^{r d_a}\]

where \(d_a := \mathrm{deg}(a)\).

INPUT:

i – integer between 1 and \(r d_a\)
a – (default: None) an element in the ring of regular functions. If \(a\) is None, then the method returns the \(i\)-th coefficient form of \(\phi_{w, T}\).

EXAMPLES:

Sage

sage: q = 3
sage: A = GF(q)['T']
sage: K.<T> = Frac(A)
sage: M = DrinfeldModularForms(K, 3)
sage: M.coefficient_form(1)
g1
sage: M.coefficient_form(2)
g2
sage: M.coefficient_form(3)
g3
sage: M.coefficient_form(3, T^2)
g1^9*g2 + g1*g2^3 + (T^27 + T)*g3

Python

>>> from sage.all import *
>>> q = Integer(3)
>>> A = GF(q)['T']
>>> K = Frac(A, names=('T',)); (T,) = K._first_ngens(1)
>>> M = DrinfeldModularForms(K, Integer(3))
>>> M.coefficient_form(Integer(1))
g1
>>> M.coefficient_form(Integer(2))
g2
>>> M.coefficient_form(Integer(3))
g3
>>> M.coefficient_form(Integer(3), T**Integer(2))
g1^9*g2 + g1*g2^3 + (T^27 + T)*g3

Sage Live

q = 3
A = GF(q)['T']
K.<T> = Frac(A)
M = DrinfeldModularForms(K, 3)
M.coefficient_form(1)
M.coefficient_form(2)
M.coefficient_form(3)
M.coefficient_form(3, T^2)

Sage

sage: M = DrinfeldModularForms(K, 2, has_type=True)
sage: M.coefficient_form(1)
g1
sage: M.coefficient_form(2)
h2^2
sage: M.coefficient_form(2, T^3 + T^2 + T)
(T^9 + T^3 + T + 1)*g1^4 + (T^18 + T^10 + T^9 + T^2 + T + 1)*h2^2

Python

>>> from sage.all import *
>>> M = DrinfeldModularForms(K, Integer(2), has_type=True)
>>> M.coefficient_form(Integer(1))
g1
>>> M.coefficient_form(Integer(2))
h2^2
>>> M.coefficient_form(Integer(2), T**Integer(3) + T**Integer(2) + T)
(T^9 + T^3 + T + 1)*g1^4 + (T^18 + T^10 + T^9 + T^2 + T + 1)*h2^2

Sage Live

M = DrinfeldModularForms(K, 2, has_type=True)
M.coefficient_form(1)
M.coefficient_form(2)
M.coefficient_form(2, T^3 + T^2 + T)

coefficient_forms(a=None)[source]¶

Return the list of all coefficients of the universal Drinfeld module at \(a\).

See also coefficient_form() for definitions.

INPUT:

a – (default: None) an element in the ring of regular functions. If \(a\) is None, then the method returns the coefficients forms at \(a = T\).

OUTPUT: list of Drinfeld modular forms. The \(i\)-th element of that list corresponds to the \((i+1)\)-th coefficient form at \(a\).

EXAMPLES:

Sage

sage: q = 3
sage: A = GF(q)['T']
sage: K.<T> = Frac(A)
sage: M = DrinfeldModularForms(K, 2)
sage: M.coefficient_forms()
[g1, g2]
sage: M.coefficient_forms(T^2)
[(T^3 + T)*g1, g1^4 + (T^9 + T)*g2, g1^9*g2 + g1*g2^3, g2^10]
sage: M.coefficient_forms(T^3)
[(T^6 + T^4 + T^2)*g1,
 (T^9 + T^3 + T)*g1^4 + (T^18 + T^10 + T^2)*g2,
 g1^13 + (T^27 + T^9 + T)*g1^9*g2 + (T^27 + T^3 + T)*g1*g2^3,
 g1^36*g2 + g1^28*g2^3 + g1^4*g2^9 + (T^81 + T^9 + T)*g2^10,
 g1^81*g2^10 + g1^9*g2^28 + g1*g2^30,
 g2^91]

Python

>>> from sage.all import *
>>> q = Integer(3)
>>> A = GF(q)['T']
>>> K = Frac(A, names=('T',)); (T,) = K._first_ngens(1)
>>> M = DrinfeldModularForms(K, Integer(2))
>>> M.coefficient_forms()
[g1, g2]
>>> M.coefficient_forms(T**Integer(2))
[(T^3 + T)*g1, g1^4 + (T^9 + T)*g2, g1^9*g2 + g1*g2^3, g2^10]
>>> M.coefficient_forms(T**Integer(3))
[(T^6 + T^4 + T^2)*g1,
 (T^9 + T^3 + T)*g1^4 + (T^18 + T^10 + T^2)*g2,
 g1^13 + (T^27 + T^9 + T)*g1^9*g2 + (T^27 + T^3 + T)*g1*g2^3,
 g1^36*g2 + g1^28*g2^3 + g1^4*g2^9 + (T^81 + T^9 + T)*g2^10,
 g1^81*g2^10 + g1^9*g2^28 + g1*g2^30,
 g2^91]

Sage Live

q = 3
A = GF(q)['T']
K.<T> = Frac(A)
M = DrinfeldModularForms(K, 2)
M.coefficient_forms()
M.coefficient_forms(T^2)
M.coefficient_forms(T^3)

gen(n)[source]¶

Return the \(n\)-th generator of this ring.

EXAMPLES:

Sage

sage: A = GF(3)['T']; K = Frac(A); T = K.gen()
sage: M = DrinfeldModularForms(K, 2)
sage: M.gen(0)
g1
sage: M.1  # equivalent to M.gen(1)
g2

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']; K = Frac(A); T = K.gen()
>>> M = DrinfeldModularForms(K, Integer(2))
>>> M.gen(Integer(0))
g1
>>> M.gen(1)  # equivalent to M.gen(1)
g2

Sage Live

A = GF(3)['T']; K = Frac(A); T = K.gen()
M = DrinfeldModularForms(K, 2)
M.gen(0)
M.1  # equivalent to M.gen(1)

Note

Recall that the ring of Drinfeld modular forms is generated by the \(r\) coefficient forms of the universal Drinfeld module at \(T\), \(g_1, g_2, \ldots, g_r\), see coefficient_forms(). We highlight however that we make a shift in the indexing so that the \(i\)-th generator corresponds to the \(i+1\)-th coefficient form for \(0\leq i \leq r-1\).

gens()[source]¶

Return a list of generators of this ring.

EXAMPLES:

Sage

sage: A = GF(3)['T']; K = Frac(A); T = K.gen()
sage: M = DrinfeldModularForms(K, 5)
sage: M.gens()
[g1, g2, g3, g4, g5]

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']; K = Frac(A); T = K.gen()
>>> M = DrinfeldModularForms(K, Integer(5))
>>> M.gens()
[g1, g2, g3, g4, g5]

Sage Live

A = GF(3)['T']; K = Frac(A); T = K.gen()
M = DrinfeldModularForms(K, 5)
M.gens()

ngens()[source]¶

Return the number of generators of this ring.

Note that the number of generators is equal to the rank.

EXAMPLES:

Sage

sage: A = GF(3)['T']; K = Frac(A); T = K.gen()
sage: M = DrinfeldModularForms(K, 5)
sage: M.ngens()
5

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']; K = Frac(A); T = K.gen()
>>> M = DrinfeldModularForms(K, Integer(5))
>>> M.ngens()
5

Sage Live

A = GF(3)['T']; K = Frac(A); T = K.gen()
M = DrinfeldModularForms(K, 5)
M.ngens()

one()[source]¶

Return the multiplicative unit.

EXAMPLES:

Sage

sage: A = GF(3)['T']; K = Frac(A); T = K.gen()
sage: M = DrinfeldModularForms(K, 2)
sage: M.one()
1
sage: M.one() * M.0
g1
sage: M.one().is_one()
True

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']; K = Frac(A); T = K.gen()
>>> M = DrinfeldModularForms(K, Integer(2))
>>> M.one()
1
>>> M.one() * M.gen(0)
g1
>>> M.one().is_one()
True

Sage Live

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

polynomial_ring()[source]¶

Return the multivariate polynomial ring over the base ring where each variable corresponds to a generator of this Drinfeld modular forms ring.

EXAMPLES:

Sage

sage: q = 3; A = GF(q)['T']; K = Frac(A);
sage: M = DrinfeldModularForms(K, 2)
sage: P = M.polynomial_ring()
sage: P
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 *
>>> q = Integer(3); A = GF(q)['T']; K = Frac(A);
>>> M = DrinfeldModularForms(K, Integer(2))
>>> P = M.polynomial_ring()
>>> P
Multivariate Polynomial Ring in g1, g2 over Fraction Field of Univariate Polynomial Ring in T over Finite Field of size 3

Sage Live

q = 3; A = GF(q)['T']; K = Frac(A);
M = DrinfeldModularForms(K, 2)
P = M.polynomial_ring()
P

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

Sage

sage: P.inject_variables()
Defining g1, g2
sage: g1.degree()
2
sage: g2.degree()
8

Python

>>> from sage.all import *
>>> P.inject_variables()
Defining g1, g2
>>> g1.degree()
2
>>> g2.degree()
8

Sage Live

P.inject_variables()
g1.degree()
g2.degree()

rank()[source]¶

Return the rank of this ring of Drinfeld modular forms.

EXAMPLES:

Sage

sage: A = GF(3)['T']; K = Frac(A);
sage: DrinfeldModularForms(K, 2).rank()
2
sage: DrinfeldModularForms(K, 3).rank()
3
sage: DrinfeldModularForms(K, 4).rank()
4

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']; K = Frac(A);
>>> DrinfeldModularForms(K, Integer(2)).rank()
2
>>> DrinfeldModularForms(K, Integer(3)).rank()
3
>>> DrinfeldModularForms(K, Integer(4)).rank()
4

Sage Live

A = GF(3)['T']; K = Frac(A);
DrinfeldModularForms(K, 2).rank()
DrinfeldModularForms(K, 3).rank()
DrinfeldModularForms(K, 4).rank()

zero()[source]¶

Return the additive identity.

EXAMPLES:

Sage

sage: A = GF(3)['T']; K = Frac(A); T = K.gen()
sage: M = DrinfeldModularForms(K, 2)
sage: M.zero()
0
sage: M.zero() + M.1
g2
sage: M.zero() * M.1
0
sage: M.zero().is_zero()
True

Python

>>> from sage.all import *
>>> A = GF(Integer(3))['T']; K = Frac(A); T = K.gen()
>>> M = DrinfeldModularForms(K, Integer(2))
>>> M.zero()
0
>>> M.zero() + M.gen(1)
g2
>>> M.zero() * M.gen(1)
0
>>> M.zero().is_zero()
True

Sage Live

A = GF(3)['T']; K = Frac(A); T = K.gen()
M = DrinfeldModularForms(K, 2)
M.zero()
M.zero() + M.1
M.zero() * M.1
M.zero().is_zero()