Implement relative Frobenius for Drinfeld modules

src/sage/categories/drinfeld_modules.py

@@ -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
@@ -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
 
@@ -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

@@ -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

@@ -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
 
     ::
 
@@ -107,7 +107,7 @@ class DrinfeldModule_charzero(DrinfeldModule):
         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):
@@ -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'>
     """
@@ -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