Fix documentation for Drinfeld modules

src/sage/categories/drinfeld_modules.py

@@ -36,23 +36,23 @@
 class DrinfeldModules(Category_over_base_ring):
     r"""
     This class implements the category of Drinfeld
-    `\mathbb{F}_q[T]`-modules on a given base field.
+    `\GF{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
+    Let `\GF{q}[T]` be a polynomial ring with coefficients in a
+    finite field `\GF{q}` and let `K` be a field. Fix a ring
+    morphism `\gamma: \GF{q}[T] \to K`; we say that `K` is an
+    `\GF{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 extension `K/\GF{q}[T]` (represented as an instance of
     the class :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 `\GF{q}[T]`-*characteristic*, or *function-field
+    characteristic*, of the base field. We say that `\GF{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.
@@ -77,7 +77,7 @@ class DrinfeldModules(Category_over_base_ring):
 
     .. RUBRIC:: Properties of the category
 
-    The base field is retrieved using the method :meth:`base`.
+    The base field is retrieved using the method :meth:`base`::
 
         sage: C.base()
         Finite Field in z of size 11^4 over its base
@@ -101,7 +101,7 @@ class DrinfeldModules(Category_over_base_ring):
         True
 
     Similarly, the function ring-characteristic of the category is
-    either `0` or the unique monic polynomial in `\mathbb{F}_q[T]` that
+    either `0` or the unique monic polynomial in `\GF{q}[T]` that
     generates the kernel of the base::
 
         sage: C.characteristic()
@@ -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 twisted by z |--> z^11
+        Ore Polynomial Ring in τ over Finite Field in z of size 11^4 twisted by z |--> z^11
         sage: C.ore_polring() is phi.ore_polring()
         True
 
@@ -135,7 +135,7 @@ class DrinfeldModules(Category_over_base_ring):
 
         sage: psi = C.object([p_root, 1])
         sage: psi
-        Drinfeld module defined by T |--> t + z^3 + 7*z^2 + 6*z + 10
+        Drinfeld module defined by T |--> τ + z^3 + 7*z^2 + 6*z + 10
         sage: psi.category() is C
         True
 
@@ -207,7 +207,7 @@ class DrinfeldModules(Category_over_base_ring):
         TypeError: function ring base must be a finite field
     """
 
-    def __init__(self, base_morphism, name='t'):
+    def __init__(self, base_morphism, name='τ'):
         r"""
         Initialize ``self``.
 
@@ -216,7 +216,7 @@ def __init__(self, base_morphism, name='t'):
         - ``base_field`` -- the base field, which is a ring extension
           over a base
 
-        - ``name`` -- (default: ``'t'``) the name of the Ore polynomial
+        - ``name`` -- (default: ``'τ'``) the name of the Ore polynomial
           variable
 
         TESTS::
@@ -227,7 +227,7 @@ def __init__(self, base_morphism, 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(K, K.frobenius_endomorphism())
+            sage: ore_polring.<τ> = OrePolynomialRing(K, K.frobenius_endomorphism())
             sage: C._ore_polring is ore_polring
             True
             sage: C._function_ring is A
@@ -364,7 +364,7 @@ def A_field(self):
         This is an instance of the class
         :class:`sage.rings.ring_extension.RingExtension`.
 
-        NOTE::
+        .. NOTE::
 
             This method has the same behavior as :meth:`base`.
 
@@ -406,7 +406,7 @@ def base_morphism(self):
     def base_over_constants_field(self):
         r"""
         Return the base field, seen as an extension over the constants
-        field `\mathbb{F}_q`.
+        field `\GF{q}`.
 
         EXAMPLES::
 
@@ -507,7 +507,7 @@ def object(self, gen):
 
             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
+            Drinfeld module defined by T |--> τ^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
@@ -534,7 +534,7 @@ 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 twisted by z |--> z^11
+            Ore Polynomial Ring in τ over Finite Field in z of size 11^4 twisted by z |--> z^11
         """
         return self._ore_polring
 
@@ -600,7 +600,7 @@ def A_field(self):
             This is an instance of the class
             :class:`sage.rings.ring_extension.RingExtension`.
 
-            NOTE::
+            .. NOTE::
 
                 This method has the same behavior as :meth:`base`.
 
@@ -623,7 +623,7 @@ def base(self):
             This is an instance of the class
             :class:`sage.rings.ring_extension.RingExtension`.
 
-            NOTE::
+            .. NOTE::
 
                 This method has the same behavior as :meth:`A_field`.
 
@@ -675,7 +675,7 @@ def base_morphism(self):
         def base_over_constants_field(self):
             r"""
             Return the base field, seen as an extension over the constants
-            field `\mathbb{F}_q`.
+            field `\GF{q}`.
 
             This is an instance of the class
             :class:`sage.rings.ring_extension.RingExtension`.
@@ -754,7 +754,7 @@ def constant_coefficient(self):
                 sage: phi.constant_coefficient() == p_root
                 True
 
-            Let `\mathbb{F}_q[T]` be the function ring, and let `\gamma` be
+            Let `\GF{q}[T]` be the function ring, and let `\gamma` be
             the base of the Drinfeld module. The constant coefficient is
             `\gamma(T)`::
 
@@ -769,7 +769,7 @@ def constant_coefficient(self):
                 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
+                Drinfeld module defined by T |--> τ^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
@@ -795,7 +795,7 @@ 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 twisted by z12 |--> z12^(5^2)
+                Ore Polynomial Ring in τ 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::
@@ -824,8 +824,8 @@ 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 twisted by z12 |--> z12^(5^2)
+                Ore Polynomial Ring in τ over Finite Field in z12 of size 5^12 twisted by z12 |--> z12^(5^2)
                 sage: phi.ore_variable()
-                t
+                τ
             """
             return self.category().ore_polring().gen()

src/sage/rings/function_field/drinfeld_modules/action.py

@@ -28,9 +28,9 @@
 class DrinfeldModuleAction(Action):
     r"""
     This class implements the module action induced by a Drinfeld
-    `\mathbb{F}_q[T]`-module.
+    `\GF{q}[T]`-module.
 
-    Let `\phi` be a Drinfeld `\mathbb{F}_q[T]`-module over a field `K`
+    Let `\phi` be a Drinfeld `\GF{q}[T]`-module over a field `K`
     and let `L/K` be a field extension. Let `x \in L` and let `a` be a
     function ring element; the action is defined as `(a, x) \mapsto
     \phi_a(x)`.
@@ -42,12 +42,12 @@ class DrinfeldModuleAction(Action):
     .. NOTE::
 
         The user should never explicitly instantiate the class
-        `DrinfeldModuleAction`.
+        :class:`DrinfeldModuleAction`.
 
     .. WARNING::
 
-        This class may be replaced later on. See issues #34833 and
-        #34834.
+        This class may be replaced later on. See issues
+        :issue:`34833` and :issue:`34834`.
 
     INPUT: the Drinfeld module
 
@@ -60,7 +60,7 @@ class DrinfeldModuleAction(Action):
         sage: action = phi.action()
         sage: action
         Action on Finite Field in z of size 11^2 over its base
-         induced by Drinfeld module defined by T |--> t^3 + z
+         induced by Drinfeld module defined by T |--> τ^3 + z
 
     The action on elements is computed as follows::
 
@@ -154,7 +154,7 @@ def _latex_(self):
             sage: phi = DrinfeldModule(A, [z, 0, 0, 1])
             sage: action = phi.action()
             sage: latex(action)
-            \text{Action{ }on{ }}\Bold{F}_{11^{2}}\text{{ }induced{ }by{ }}\phi: T \mapsto t^{3} + z
+            \text{Action{ }on{ }}\Bold{F}_{11^{2}}\text{{ }induced{ }by{ }}\phi: T \mapsto τ^{3} + z
         """
         return f'\\text{{Action{{ }}on{{ }}}}' \
                f'{latex(self._base)}\\text{{{{ }}' \
@@ -174,7 +174,7 @@ def _repr_(self):
             sage: phi = DrinfeldModule(A, [z, 0, 0, 1])
             sage: action = phi.action()
             sage: action
-            Action on Finite Field in z of size 11^2 over its base induced by Drinfeld module defined by T |--> t^3 + z
+            Action on Finite Field in z of size 11^2 over its base induced by Drinfeld module defined by T |--> τ^3 + z
         """
         return f'Action on {self._base} induced by ' \
                f'{self._drinfeld_module}'

src/sage/rings/function_field/drinfeld_modules/charzero_drinfeld_module.py

@@ -41,10 +41,10 @@
 
 class DrinfeldModule_charzero(DrinfeldModule):
     r"""
-    This class implements Drinfeld `\mathbb{F}_q[T]`-modules defined
-    over fields of `\mathbb{F}_q[T]`-characteristic zero.
+    This class implements Drinfeld `\GF{q}[T]`-modules defined
+    over fields of `\GF{q}[T]`-characteristic zero.
 
-    Recall that the `\mathbb{F}_q[T]`-*characteristic* is defined as the
+    Recall that the `\GF{q}[T]`-*characteristic* is defined as the
     kernel of the underlying structure morphism. For general definitions
     and help on Drinfeld modules, see class
     :class:`sage.rings.function_fields.drinfeld_module.drinfeld_module.DrinfeldModule`.
@@ -59,7 +59,7 @@ class DrinfeldModule_charzero(DrinfeldModule):
         sage: A.<T> = GF(3)[]
         sage: phi = DrinfeldModule(A, [T, 1])
         sage: phi
-        Drinfeld module defined by T |--> t + T
+        Drinfeld module defined by T |--> τ + T
 
     ::
 
@@ -96,18 +96,18 @@ class DrinfeldModule_charzero(DrinfeldModule):
         sage: phi.goss_polynomial(3)
         X^3 + (1/(T^2 + T))*X^2
 
-    .. RUBRIC:: Base fields of `\mathbb{F}_q[T]`-characteristic zero
+    .. RUBRIC:: Base fields of `\GF{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
+    `\GF{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
+        (s^-1 + 1)*τ^2 + (s + s^2 + s^5 + O(s^6))*τ + s^-1
 
     One can also construct Drinfeld modules over SageMath's global
     function fields::
@@ -116,7 +116,7 @@ class DrinfeldModule_charzero(DrinfeldModule):
         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
+        z^2*τ^2 + τ + z
     """
     @cached_method
     def _compute_coefficient_exp(self, k):
@@ -157,7 +157,7 @@ def exponential(self, prec=Infinity, name='z'):
         Return the exponential of this Drinfeld module.
 
         Note that the exponential is only defined when the
-        `\mathbb{F}_q[T]`-characteristic is zero.
+        `\GF{q}[T]`-characteristic is zero.
 
         INPUT:
 
@@ -283,7 +283,7 @@ def logarithm(self, prec=Infinity, name='z'):
 
         By definition, the logarithm is the compositional inverse of the
         exponential (see :meth:`exponential`). Note that the logarithm
-        is only defined when the `\mathbb{F}_q[T]`-characteristic is
+        is only defined when the `\GF{q}[T]`-characteristic is
         zero.
 
         INPUT:
@@ -448,7 +448,7 @@ class DrinfeldModule_rational(DrinfeldModule_charzero):
         sage: Fq = GF(q)
         sage: A.<T> = Fq[]
         sage: C = DrinfeldModule(A, [T, 1]); C
-        Drinfeld module defined by T |--> t + T
+        Drinfeld module defined by T |--> τ + T
         sage: type(C)
         <class 'sage.rings.function_field.drinfeld_modules.charzero_drinfeld_module.DrinfeldModule_rational_with_category'>
     """
@@ -471,7 +471,7 @@ def coefficient_in_function_ring(self, n):
             sage: phi.coefficient_in_function_ring(2)
             T^2
 
-        Compare with the method meth:`coefficient`::
+        Compare with the method :meth:`coefficient`::
 
             sage: phi.coefficient(2)
             U^2
@@ -556,7 +556,7 @@ def class_polynomial(self):
             sage: Fq = GF(q)
             sage: A.<T> = Fq[]
             sage: C = DrinfeldModule(A, [T, 1]); C
-            Drinfeld module defined by T |--> t + T
+            Drinfeld module defined by T |--> τ + T
             sage: C.class_polynomial()
             1