Hom spaces between Drinfeld modules

src/doc/en/reference/references/index.rst

@@ -5240,6 +5240,9 @@ REFERENCES:
 .. [Mur1983] \G. E. Murphy. *The idempotents of the symmetric group
              and Nakayama's conjecture*. J. Algebra **81** (1983). 258-265.
 
+.. [Mus2023] Yossef Musleh. *Algorithms for Drinfeld Modules*. 
+             PhD thesis, University of Waterloo, 2023.
+
 .. [Muth2019] Robert Muth. *Super RSK correspondence with symmetry*.
               Electron. J. Combin. **26** (2019), no. 2, Paper 2.27, 29 pp.
               https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i2p27,
@@ -6914,6 +6917,9 @@ REFERENCES:
                *A new keystream generator MUGI*; in
                FSE, (2002), pp. 179-194.
 
+.. [Wes2022] \B. Wesolowski. 2022. *Computing isogenies between finite Drinfeld modules*.
+             Cryptology ePrint Archive, Paper 2022/438. https://eprint.iacr.org/2022/438
+
 .. [Whi1932] \H. Whitney, *Congruent graphs and the connectivity of graphs*,
              American Journal of Mathematics,
              pages 150--168, 1932,

src/sage/categories/drinfeld_modules.py

@@ -1,3 +1,4 @@
+# -*- coding: utf-8 -*-
 # sage_setup: distribution = sagemath-categories
 # sage.doctest: needs sage.rings.finite_rings
 r"""
@@ -123,7 +124,7 @@ 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 +136,7 @@ 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 +208,7 @@ 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 +217,7 @@ 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 +228,7 @@ 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
@@ -438,7 +439,7 @@ def characteristic(self):
 
         ::
 
-            sage: psi = DrinfeldModule(A, [Frac(A).gen(), 1])
+            sage: psi = DrinfeldModule(A, [T, 1])
             sage: C = psi.category()
             sage: C.characteristic()
             0
@@ -507,7 +508,7 @@ 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 +535,7 @@ 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
 
@@ -639,7 +640,7 @@ def base(self):
 
             The base can be infinite::
 
-                sage: sigma = DrinfeldModule(A, [Frac(A).gen(), 1])
+                sage: sigma = DrinfeldModule(A, [T, 1])
                 sage: sigma.base()
                 Fraction Field of Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2 over its base
             """
@@ -664,7 +665,7 @@ def base_morphism(self):
 
             The base field can be infinite::
 
-                sage: sigma = DrinfeldModule(A, [Frac(A).gen(), 1])
+                sage: sigma = DrinfeldModule(A, [T, 1])
                 sage: sigma.base_morphism()
                 Coercion map:
                   From: Univariate Polynomial Ring in T over Finite Field in z2 of size 5^2
@@ -711,8 +712,7 @@ def characteristic(self):
             ::
 
                 sage: B.<Y> = Fq[]
-                sage: L = Frac(B)
-                sage: psi = DrinfeldModule(A, [L(1), 0, 0, L(1)])
+                sage: psi = DrinfeldModule(A, [B(1), 0, 0, 1])
                 sage: psi.characteristic()
                 Traceback (most recent call last):
                 ...
@@ -770,7 +770,7 @@ 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
@@ -796,7 +796,7 @@ 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::
@@ -825,8 +825,8 @@ 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

@@ -56,11 +56,10 @@ class DrinfeldModule_charzero(DrinfeldModule):
     responsible for instantiating the right class depending on the
     input::
 
-        sage: A = GF(3)['T']
-        sage: K.<T> = Frac(A)
+        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
 
     ::
 
@@ -75,8 +74,7 @@ class DrinfeldModule_charzero(DrinfeldModule):
     It is possible to calculate the logarithm and the exponential of
     any Drinfeld modules of characteristic zero::
 
-        sage: A = GF(2)['T']
-        sage: K.<T> = Frac(A)
+        sage: A.<T> = GF(2)[]
         sage: phi = DrinfeldModule(A, [T, 1])
         sage: phi.exponential()
         z + ((1/(T^2+T))*z^2) + ((1/(T^8+T^6+T^5+T^3))*z^4) + O(z^8)
@@ -89,8 +87,7 @@ class DrinfeldModule_charzero(DrinfeldModule):
     analytic theory of Drinfeld module. They provide a function field
     analogue of certain classical trigonometric functions::
 
-        sage: A = GF(2)['T']
-        sage: K.<T> = Frac(A)
+        sage: A.<T> = GF(2)[]
         sage: phi = DrinfeldModule(A, [T, 1])
         sage: phi.goss_polynomial(1)
         X
@@ -110,7 +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::
@@ -119,7 +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):
@@ -133,8 +130,7 @@ def _compute_coefficient_exp(self, k):
 
         TESTS::
 
-            sage: A = GF(2)['T']
-            sage: K.<T> = Frac(A)
+            sage: A.<T> = GF(2)[]
             sage: phi = DrinfeldModule(A, [T, 1])
             sage: q = A.base_ring().cardinality()
             sage: phi._compute_coefficient_exp(0)
@@ -175,8 +171,7 @@ def exponential(self, prec=Infinity, name='z'):
 
         EXAMPLES::
 
-            sage: A = GF(2)['T']
-            sage: K.<T> = Frac(A)
+            sage: A.<T> = GF(2)[]
             sage: phi = DrinfeldModule(A, [T, 1])
             sage: q = A.base_ring().cardinality()
 
@@ -199,8 +194,7 @@ def exponential(self, prec=Infinity, name='z'):
 
         Example in higher rank::
 
-            sage: A = GF(5)['T']
-            sage: K.<T> = Frac(A)
+            sage: A.<T> = GF(5)[]
             sage: phi = DrinfeldModule(A, [T, T^2, T + T^2 + T^4, 1])
             sage: exp = phi.exponential(); exp
             z + ((T/(T^4+4))*z^5) + O(z^8)
@@ -217,8 +211,7 @@ def exponential(self, prec=Infinity, name='z'):
 
         TESTS::
 
-            sage: A = GF(2)['T']
-            sage: K.<T> = Frac(A)
+            sage: A.<T> = GF(2)[]
             sage: phi = DrinfeldModule(A, [T, 1])
             sage: exp = phi.exponential()
             sage: exp[2] == 1/(T**q - T)  # expected value
@@ -259,8 +252,7 @@ def _compute_coefficient_log(self, k):
 
         TESTS::
 
-            sage: A = GF(2)['T']
-            sage: K.<T> = Frac(A)
+            sage: A.<T> = GF(2)[]
             sage: phi = DrinfeldModule(A, [T, 1])
             sage: q = A.base_ring().cardinality()
             sage: phi._compute_coefficient_log(0)
@@ -305,8 +297,7 @@ def logarithm(self, prec=Infinity, name='z'):
 
         EXAMPLES::
 
-            sage: A = GF(2)['T']
-            sage: K.<T> = Frac(A)
+            sage: A.<T> = GF(2)[]
             sage: phi = DrinfeldModule(A, [T, 1])
 
         When ``prec`` is ``Infinity`` (which is the default),
@@ -328,16 +319,14 @@ def logarithm(self, prec=Infinity, name='z'):
 
         Example in higher rank::
 
-            sage: A = GF(5)['T']
-            sage: K.<T> = Frac(A)
+            sage: A.<T> = GF(5)[]
             sage: phi = DrinfeldModule(A, [T, T^2, T + T^2 + T^4, 1])
             sage: phi.logarithm()
             z + ((4*T/(T^4+4))*z^5) + O(z^8)
 
         TESTS::
 
-            sage: A = GF(2)['T']
-            sage: K.<T> = Frac(A)
+            sage: A.<T> = GF(2)[]
             sage: phi = DrinfeldModule(A, [T, 1])
             sage: q = 2
             sage: log[2] == -1/((T**q - T))  # expected value
@@ -374,8 +363,7 @@ def _compute_goss_polynomial(self, n, q, poly_ring, X):
 
         TESTS::
 
-            sage: A = GF(2^2)['T']
-            sage: K.<T> = Frac(A)
+            sage: A.<T> = GF(2^2)[]
             sage: phi = DrinfeldModule(A, [T, T+1, T^2, 1])
             sage: poly_ring = phi.base()['X']
             sage: X = poly_ring.gen()
@@ -423,8 +411,7 @@ def goss_polynomial(self, n, var='X'):
 
         EXAMPLES::
 
-            sage: A = GF(3)['T']
-            sage: K.<T> = Frac(A)
+            sage: A.<T> = GF(3)[]
             sage: phi = DrinfeldModule(A, [T, 1])  # Carlitz module
             sage: phi.goss_polynomial(1)
             X
@@ -459,10 +446,9 @@ class DrinfeldModule_rational(DrinfeldModule_charzero):
 
         sage: q = 9
         sage: Fq = GF(q)
-        sage: A = Fq['T']
-        sage: K.<T> = Frac(A)
+        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'>
     """
@@ -479,9 +465,8 @@ def coefficient_in_function_ring(self, n):
 
             sage: q = 5
             sage: Fq = GF(q)
-            sage: A = Fq['T']
-            sage: R = Fq['U']
-            sage: K.<U> = Frac(R)
+            sage: A.<T> = Fq[]
+            sage: R.<U> = Fq[]
             sage: phi = DrinfeldModule(A, [U, 0, U^2, U^3])
             sage: phi.coefficient_in_function_ring(2)
             T^2
@@ -524,9 +509,8 @@ def coefficients_in_function_ring(self, sparse=True):
 
             sage: q = 5
             sage: Fq = GF(q)
-            sage: A = Fq['T']
-            sage: R = Fq['U']
-            sage: K.<U> = Frac(R)
+            sage: A.<T> = Fq[]
+            sage: R.<U> = Fq[]
             sage: phi = DrinfeldModule(A, [U, 0, U^2, U^3])
             sage: phi.coefficients_in_function_ring()
             [T, T^2, T^3]
@@ -570,10 +554,9 @@ def class_polynomial(self):
 
             sage: q = 5
             sage: Fq = GF(q)
-            sage: A = Fq['T']
-            sage: K.<T> = Frac(A)
+            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