is_ordinary/is_supersingular + more doctests

Author: xcaruso

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

@@ -1341,27 +1341,6 @@ def is_isomorphic(self, other, absolutely=False):
                 raise NotImplementedError(f"cannot solve the equation u^{e} == {ue}")
             return True
 
-    def is_supersingular(self):
-        r"""
-        Return ``True`` if the Drinfeld module is supersingular.
-
-        A Drinfeld module is supersingular if and only if its
-        height equals its rank.
-
-        EXAMPLES::
-
-            sage: Fq = GF(343)
-            sage: A.<T> = Fq[]
-            sage: K.<z6> = Fq.extension(2)
-            sage: phi = DrinfeldModule(A, [1, 0, z6])
-            sage: phi.is_supersingular()
-            True
-            sage: phi(phi.characteristic()) # Purely inseparable
-            z6*t^2
-
-        """
-        return self.height() == self.rank()
-
     def is_finite(self) -> bool:
         r"""
         Return ``True`` if this Drinfeld module is finite,

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

@@ -487,13 +487,43 @@ def invert(self, ore_pol):
             pass
         raise ValueError('input must be in the image of the Drinfeld module')
 
+    def is_supersingular(self):
+        r"""
+        Return ``True`` if this Drinfeld module is supersingular.
+
+        A Drinfeld module is supersingular if and only if its
+        height equals its rank.
+
+        EXAMPLES::
+
+            sage: Fq = GF(343)
+            sage: A.<T> = Fq[]
+            sage: K.<z6> = Fq.extension(2)
+            sage: phi = DrinfeldModule(A, [1, 0, z6])
+            sage: phi.is_supersingular()
+            True
+            sage: phi(phi.characteristic())   # Purely inseparable
+            z6*t^2
+
+        In rank two, a Drinfeld module is either ordinary or
+        supersinguler. In higher ranks, it could be neither of
+        the two::
+
+            sage: psi = DrinfeldModule(A, [1, 0, z6, z6])
+            sage: psi.is_ordinary()
+            False
+            sage: psi.is_supersingular()
+            False
+
+        """
+        return self.height() == self.rank()
+
     def is_ordinary(self):
         r"""
-        Return ``True`` if the Drinfeld module is ordinary; raise a
-        NotImplementedError if the rank is not two.
+        Return ``True`` if this Drinfeld module is ordinary.
 
-        A rank two Drinfeld module is *ordinary* if and only if it is
-        not supersingular; see :meth:`is_supersingular`.
+        A Drinfeld module is supersingular if and only if its
+        height is one.
 
         EXAMPLES::
 
@@ -503,9 +533,12 @@ def is_ordinary(self):
             sage: phi = DrinfeldModule(A, [1, 0, z6])
             sage: phi.is_ordinary()
             False
-            sage: phi_p = phi(phi.characteristic())
-            sage: phi_p  # Purely inseparable
-            z6*t^2
+
+        ::
+
+            sage: phi = DrinfeldModule(A, [1, z6, 0, z6])
+            sage: phi.is_ordinary()
+            True
+
         """
-        self._check_rank_two()
-        return not self.is_supersingular()
+        return self.height() == 1

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

@@ -33,8 +33,39 @@
 class DrinfeldModuleMorphismAction(Action):
     r"""
     Action of the function ring on the homset of a Drinfeld module.
-    """
 
+    EXAMPLES::
+
+        sage: Fq = GF(5)
+        sage: A.<T> = Fq[]
+        sage: K.<z> = Fq.extension(3)
+        sage: phi = DrinfeldModule(A, [z, 1, z])
+        sage: psi = DrinfeldModule(A, [z, z^2 + 4*z + 3, 2*z^2 + 4*z + 4])
+        sage: H = Hom(phi, psi)
+        sage: t = phi.ore_variable()
+        sage: f = H(t + 2)
+
+    Left action::
+
+        sage: (T + 1) * f
+        Drinfeld Module morphism:
+          From: Drinfeld module defined by T |--> z*t^2 + t + z
+          To:   Drinfeld module defined by T |--> (2*z^2 + 4*z + 4)*t^2 + (z^2 + 4*z + 3)*t + z
+          Defn: (2*z^2 + 4*z + 4)*t^3 + (2*z + 1)*t^2 + (2*z^2 + 4*z + 2)*t + 2*z + 2
+
+    Right action currently does not work (it is a known bug, due to an
+    incompatibility between multiplication of morphisms and the coercion
+    system)::
+
+        sage: f * (T + 1)
+        Traceback (most recent call last):
+        ...
+        TypeError: right (=T + 1) must be a map to multiply it by Drinfeld Module morphism:
+          From: Drinfeld module defined by T |--> z*t^2 + t + z
+          To:   Drinfeld module defined by T |--> (2*z^2 + 4*z + 4)*t^2 + (z^2 + 4*z + 3)*t + z
+          Defn: t + 2
+
+    """
     def __init__(self, A, H, is_left, op):
         r"""
         Initialize this action.
@@ -349,8 +380,6 @@ def _element_constructor_(self, *args, **kwds):
         Return the Drinfeld module morphism defined by the given Ore
         polynomial.
 
-        INPUT: an Ore polynomial
-
         EXAMPLES::
 
             sage: Fq = GF(27)
@@ -365,6 +394,13 @@ def _element_constructor_(self, *args, **kwds):
             sage: identity_morphism
             Identity morphism of Drinfeld module defined by T |--> 2*t^2 + z6*t + z6
 
+        ::
+
+            sage: scalar_multiplication = E(T)
+            sage: scalar_multiplication
+            Endomorphism of Drinfeld module defined by T |--> 2*t^2 + z6*t + z6
+              Defn: 2*t^2 + z6*t + z6
+
         ::
 
             sage: frobenius_endomorphism = E(t^6)

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

@@ -135,17 +135,24 @@ def __classcall_private__(cls, parent, x):
 
         - ``parent`` -- the Drinfeld module homset
 
-        - ``x`` -- the Ore polynomial defining the morphism or a
-          DrinfeldModuleMorphism
+        - ``x`` -- a morphism of Drinfeld modules or an element
+          (either an Ore polynomial or an element in the base
+          ring) defining it
 
         TESTS::
 
             sage: Fq = GF(2)
             sage: A.<T> = Fq[]
             sage: K.<z6> = Fq.extension(6)
             sage: phi = DrinfeldModule(A, [z6, 1, 1])
-            sage: psi = DrinfeldModule(A, [z6, z6^4 + z6^2 + 1, 1])
+            sage: End(phi)(T + 1)
+            Endomorphism of Drinfeld module defined by T |--> t^2 + t + z6
+              Defn: t^2 + t + z6 + 1
+
+        ::
+
             sage: t = phi.ore_polring().gen()
+            sage: psi = DrinfeldModule(A, [z6, z6^4 + z6^2 + 1, 1])
             sage: morphism = Hom(phi, psi)(t + z6^5 + z6^2 + 1)
             sage: morphism is Hom(phi, psi)(morphism)
             True