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Limit the use of RingExtensions in Drinfeld modules #40421

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Limit the use of RingExtensions in Drinfeld modules #40421

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a1b901e
remove RingExtensions in Ore polynomials
Jul 16, 2025
58c4b14
fix lint
Jul 16, 2025
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60 changes: 26 additions & 34 deletions src/sage/categories/drinfeld_modules.py
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Original file line number Diff line number Diff line change
Expand Up @@ -70,7 +70,7 @@ class DrinfeldModules(Category_over_base_ring):
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
Category of Drinfeld modules over Finite Field in z of size 11^4

The output tells the user that the category is only defined by its
base.
Expand All @@ -88,7 +88,7 @@ class DrinfeldModules(Category_over_base_ring):
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
To: Finite Field in z of size 11^4
Defn: T |--> z^3 + 7*z^2 + 6*z + 10

The so-called constant coefficient --- which is the same for all
Expand Down Expand Up @@ -123,7 +123,7 @@ class DrinfeldModules(Category_over_base_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
Ore Polynomial Ring in t over Finite Field in z of size 11^4 twisted by z |--> z^11
sage: C.ore_polring() is phi.ore_polring()
True

Expand Down Expand Up @@ -165,20 +165,24 @@ class DrinfeldModules(Category_over_base_ring):
sage: K.<z> = Fq.extension(4)
sage: from sage.categories.drinfeld_modules import DrinfeldModules
sage: base = Hom(A, K)(0)
sage: C = DrinfeldModules(base)
sage: C = DrinfeldModules(base) # known bug (blankline)
<BLANKLINE>
Traceback (most recent call last):
...
TypeError: base field must be a ring extension

::
Note that `C.base_morphism()` has codomain `K` while
the defining morphism of `C.base()` has codomain `K` viewed
as an `A`-field. Thus, they differ::

sage: C.base().defining_morphism() == C.base_morphism()
True
False

::

sage: base = Hom(A, A)(1)
sage: C = DrinfeldModules(base)
sage: C = DrinfeldModules(base) # known bug (blankline)
<BLANKLINE>
Traceback (most recent call last):
...
TypeError: base field must be a ring extension
Expand All @@ -203,7 +207,7 @@ class DrinfeldModules(Category_over_base_ring):
TypeError: function ring base must be a finite field
"""

def __init__(self, base_field, name='t'):
def __init__(self, base_morphism, name='t'):
r"""
Initialize ``self``.

Expand All @@ -223,34 +227,23 @@ def __init__(self, base_field, name='t'):
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: ore_polring.<t> = OrePolynomialRing(phi.base(), phi.base().frobenius_endomorphism())
sage: ore_polring.<t> = OrePolynomialRing(K, K.frobenius_endomorphism())
sage: C._ore_polring is ore_polring
True
sage: i = phi.base().coerce_map_from(K)
sage: base_morphism = Hom(A, K)(p_root)
sage: C.base() == K.over(base_morphism)
True
sage: C._base_morphism == i * base_morphism
True
sage: C._function_ring is A
True
sage: C._constant_coefficient == base_morphism(T)
sage: C._constant_coefficient == C._base_morphism(T)
True
sage: C._characteristic(C._constant_coefficient)
0
"""
# Check input is a ring extension
if not isinstance(base_field, RingExtension_generic):
raise TypeError('base field must be a ring extension')
base_morphism = base_field.defining_morphism()
self._base_morphism = base_morphism
function_ring = self._function_ring = base_morphism.domain()
base_field = self._base_field = base_morphism.codomain()
# Check input is a field
if not base_field.is_field():
raise TypeError('input must be a field')
self._base_field = base_field
self._function_ring = base_morphism.domain()
# Check domain of base morphism is Fq[T]
function_ring = self._function_ring
if not isinstance(function_ring, PolynomialRing_generic):
raise NotImplementedError('function ring must be a polynomial '
'ring')
Expand All @@ -262,7 +255,7 @@ def __init__(self, base_field, name='t'):
Fq = function_ring_base
A = function_ring
T = A.gen()
K = base_field # A ring extension
K = base_field
# Build K{t}
d = log(Fq.cardinality(), Fq.characteristic())
tau = K.frobenius_endomorphism(d)
Expand All @@ -284,7 +277,7 @@ def __init__(self, base_field, name='t'):
i = A.coerce_map_from(Fq)
Fq_to_K = self._base_morphism * i
self._base_over_constants_field = base_field.over(Fq_to_K)
super().__init__(base=base_field)
super().__init__(base=base_field.over(base_morphism))

def _latex_(self):
r"""
Expand Down Expand Up @@ -321,7 +314,7 @@ def _repr_(self):
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
Category of Drinfeld modules over Finite Field in z of size 11^4
"""
return f'Category of Drinfeld modules over {self._base_field}'

Expand Down Expand Up @@ -378,7 +371,7 @@ def base_morphism(self):
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
To: Finite Field in z of size 11^4
Defn: T |--> z^3 + 7*z^2 + 6*z + 10

sage: C.constant_coefficient() == C.base_morphism()(T)
Expand Down Expand Up @@ -517,7 +510,7 @@ def ore_polring(self):
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
Ore Polynomial Ring in t over Finite Field in z of size 11^4 twisted by z |--> z^11
"""
return self._ore_polring

Expand Down Expand Up @@ -615,17 +608,16 @@ def base_morphism(self):
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
To: Finite Field in z12 of size 5^12
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:
Coercion map:
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
To: Fraction Field of Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2
"""
return self.category().base_morphism()

Expand Down Expand Up @@ -753,7 +745,7 @@ def ore_polring(self):
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
Ore Polynomial Ring in t over Finite Field in z12 of size 5^12 twisted by z12 |--> z12^(5^2)

The Ore polynomial ring can also be retrieved from the category
of the Drinfeld module::
Expand Down Expand Up @@ -782,7 +774,7 @@ def ore_variable(self):
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
Ore Polynomial Ring in t over Finite Field in z12 of size 5^12 twisted by z12 |--> z12^(5^2)
sage: phi.ore_variable()
t
"""
Expand Down
2 changes: 0 additions & 2 deletions src/sage/rings/function_field/drinfeld_modules/charzero_drinfeld_module.py
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Expand Up @@ -504,7 +504,6 @@ def coefficient_in_function_ring(self, n):
"""
A = self.function_ring()
g = self.coefficient(n)
g = g.backend(force=True)
if g.denominator().is_one():
return A(g.numerator().list())
else:
Expand Down Expand Up @@ -550,7 +549,6 @@ def coefficients_in_function_ring(self, sparse=True):
A = self.function_ring()
gs = []
for g in self.coefficients(sparse):
g = g.backend(force=True)
if g.denominator().is_one():
gs.append(A(g.numerator().list()))
else:
Expand Down
44 changes: 18 additions & 26 deletions src/sage/rings/function_field/drinfeld_modules/drinfeld_module.py
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Expand Up @@ -194,7 +194,7 @@ class DrinfeldModule(Parent, UniqueRepresentation):
:class:`sage.categories.drinfeld_modules.DrinfeldModules`)::

sage: phi.category()
Category of Drinfeld modules over Finite Field in z of size 3^12 over its base
Category of Drinfeld modules over Finite Field in z of size 3^12
sage: phi.category() is psi.category()
False
sage: phi.category() is rho.category()
Expand All @@ -219,7 +219,7 @@ class DrinfeldModule(Parent, UniqueRepresentation):
sage: phi.base_morphism()
Ring morphism:
From: Univariate Polynomial Ring in T over Finite Field in z2 of size 3^2
To: Finite Field in z of size 3^12 over its base
To: Finite Field in z of size 3^12
Defn: T |--> z

Note that the base field is *not* the field `K`. Rather, it is a
Expand All @@ -237,14 +237,13 @@ class DrinfeldModule(Parent, UniqueRepresentation):
sage: phi.base_morphism()
Ring morphism:
From: Univariate Polynomial Ring in T over Finite Field in z2 of size 3^2
To: Finite Field in z of size 3^12 over its base
To: Finite Field in z of size 3^12
Defn: T |--> z

::

sage: phi.ore_polring() # K{t}
Ore Polynomial Ring in t over Finite Field in z of size 3^12 over its base
twisted by Frob^2
Ore Polynomial Ring in t over Finite Field in z of size 3^12 twisted by z |--> z^(3^2)

::

Expand All @@ -268,9 +267,8 @@ class DrinfeldModule(Parent, UniqueRepresentation):
sage: phi.morphism() # The Drinfeld module as a morphism
Ring morphism:
From: Univariate Polynomial Ring in T over Finite Field in z2 of size 3^2
To: Ore Polynomial Ring in t
over Finite Field in z of size 3^12 over its base
twisted by Frob^2
To: Ore Polynomial Ring in t over Finite Field in z of size 3^12
twisted by z |--> z^(3^2)
Defn: T |--> t^2 + t + z

One can compute the rank and height::
Expand Down Expand Up @@ -578,42 +576,37 @@ def __classcall_private__(cls, function_ring, gen, name='t'):
if isinstance(gen, OrePolynomial):
ore_polring = gen.parent()
# Base ring without morphism structure:
base_field_noext = ore_polring.base()
base_field = ore_polring.base()
name = ore_polring.variable_name()
# `gen` is a list of coefficients (function_ring = Fq[T]):
elif isinstance(gen, (list, tuple)):
ore_polring = None
# Base ring without morphism structure:
base_field_noext = Sequence(gen).universe()
base_field = Sequence(gen).universe()
else:
raise TypeError('generator must be list of coefficients or Ore '
'polynomial')
# Constant coefficient must be nonzero:
if gen[0].is_zero():
raise ValueError('constant coefficient must be nonzero')
# The coefficients are in a base field that has coercion from Fq:
if not (hasattr(base_field_noext, 'has_coerce_map_from') and
base_field_noext.has_coerce_map_from(function_ring.base_ring())):
if not (hasattr(base_field, 'has_coerce_map_from') and
base_field.has_coerce_map_from(function_ring.base_ring())):
raise ValueError('function ring base must coerce into base field')

# Build the category
T = function_ring.gen()
if isinstance(base_field_noext, RingExtension_generic):
base_field = base_field_noext
elif base_field_noext.has_coerce_map_from(function_ring) \
and T == gen[0]:
base_morphism = base_field_noext.coerce_map_from(function_ring)
base_field = base_field_noext.over(base_morphism)
if base_field.has_coerce_map_from(function_ring) and T == gen[0]:
base_morphism = base_field.coerce_map_from(function_ring)
else:
base_morphism = Hom(function_ring, base_field_noext)(gen[0])
base_field = base_field_noext.over(base_morphism)
base_morphism = Hom(function_ring, base_field)(gen[0])

# This test is also done in the category. We put it here also
# to have a friendlier error message
if not base_field.is_field():
raise ValueError('generator coefficients must live in a field')

category = DrinfeldModules(base_field, name=name)
category = DrinfeldModules(base_morphism, name=name)

# Check gen as Ore polynomial
ore_polring = category.ore_polring() # Sanity cast
Expand All @@ -622,12 +615,11 @@ def __classcall_private__(cls, function_ring, gen, name='t'):
raise ValueError('generator must have positive degree')

# Instantiate the appropriate class:
backend = base_field.backend(force=True)
if backend.is_finite():
if base_field.is_finite():
from sage.rings.function_field.drinfeld_modules.finite_drinfeld_module import DrinfeldModule_finite
return DrinfeldModule_finite(gen, category)
if isinstance(backend, FractionField_generic):
ring = backend.ring()
if isinstance(base_field, FractionField_generic):
ring = base_field.ring()
if (isinstance(ring, PolynomialRing_generic)
and ring.base_ring() is function_ring_base
and base_morphism(T) == ring.gen()):
Expand Down Expand Up @@ -1792,7 +1784,7 @@ def morphism(self):
Ring morphism:
From: Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2
To: Ore Polynomial Ring in t over Finite Field in z12 of size 5^12
over its base twisted by Frob^2
twisted by z12 |--> z12^(5^2)
Defn: T |--> z12^5*t^2 + z12^3*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
sage: from sage.rings.morphism import RingHomomorphism
Expand Down
2 changes: 1 addition & 1 deletion src/sage/rings/function_field/drinfeld_modules/morphism.py
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Expand Up @@ -593,7 +593,7 @@ def _motive_matrix(self):
# The next rows:
# each row is obtained from the previous one by
# applying the semi-linear transformation f |-> t*f
inv = K(phiT[r]).inverse()
inv = ~phiT[r]
B = inv * phiT
T = KT.gen()
for i in range(1, r):
Expand Down
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