gh-40851: add some -> tuple annotations
    
just add a few annotations that gens must be `tuple`

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #40851
Reported by: Frédéric Chapoton
Reviewer(s): Vincent Macri

Author: Release Manager

src/sage/algebras/lie_algebras/lie_algebra.py

@@ -953,7 +953,7 @@ def _an_element_(self):
         return self.lie_algebra_generators()[self._indices.an_element()]
 
 # Do we want this to return lie_algebra_generators()? Perhaps in the category?
-#    def gens(self):
+#    def gens(self) -> tuple:
 #        """
 #        Return a tuple whose entries are the generators for this
 #        object, in some order.

src/sage/algebras/lie_conformal_algebras/finitely_freely_generated_lca.py

@@ -6,21 +6,19 @@
 
 - Reimundo Heluani (2019-08-09): Initial implementation.
 """
-
-#******************************************************************************
+# ****************************************************************************
 #       Copyright (C) 2019 Reimundo Heluani <[email protected]>
 #
 # This program is free software: you can redistribute it and/or modify
 # it under the terms of the GNU General Public License as published by
 # the Free Software Foundation, either version 2 of the License, or
 # (at your option) any later version.
-#                  http://www.gnu.org/licenses/
-#*****************************************************************************
-
-from sage.misc.cachefunc import cached_method
+#                  https://www.gnu.org/licenses/
+# ***************************************************************************
 from sage.categories.lie_conformal_algebras import LieConformalAlgebras
 from .freely_generated_lie_conformal_algebra import \
-                                            FreelyGeneratedLieConformalAlgebra
+    FreelyGeneratedLieConformalAlgebra
+from sage.misc.cachefunc import cached_method
 
 
 class FinitelyFreelyGeneratedLCA(FreelyGeneratedLieConformalAlgebra):
@@ -33,7 +31,7 @@ class FinitelyFreelyGeneratedLCA(FreelyGeneratedLieConformalAlgebra):
     """
     def __init__(self, R, index_set=None, central_elements=None, category=None,
                  element_class=None, prefix=None, names=None, latex_names=None,
-                 **kwds):
+                 **kwds) -> None:
         """
         Initialize ``self``.
 
@@ -53,7 +51,8 @@ def __init__(self, R, index_set=None, central_elements=None, category=None,
             raise TypeError("index_set must be a finite set")
 
         super().__init__(R,
-                         index_set=index_set, central_elements=central_elements,
+                         index_set=index_set,
+                         central_elements=central_elements,
                          category=category, element_class=element_class,
                          prefix=prefix, **kwds)
         self._ngens = len(self._generators)
@@ -88,7 +87,7 @@ def _an_element_(self):
         """
         return self.sum(self.gens())
 
-    def ngens(self):
+    def ngens(self) -> int:
         """
         The number of generators of this Lie conformal algebra.
 
@@ -102,7 +101,7 @@ def ngens(self):
         return self._ngens
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         """
         The generators for this Lie conformal algebra.
 
@@ -126,13 +125,14 @@ def gens(self):
         return self.lie_conformal_algebra_generators()
 
     @cached_method
-    def central_elements(self):
+    def central_elements(self) -> tuple:
         """
         The central elements of this Lie conformal algebra.
 
         EXAMPLES::
 
-            sage: R = lie_conformal_algebras.NeveuSchwarz(QQ); R.central_elements()
+            sage: R = lie_conformal_algebras.NeveuSchwarz(QQ)
+            sage: R.central_elements()
             (C,)
         """
         return tuple(FreelyGeneratedLieConformalAlgebra.central_elements(self))

src/sage/algebras/weyl_algebra.py

@@ -1530,10 +1530,11 @@ def polynomial_gens(self):
             True
         """
         from sage.sets.non_negative_integers import NonNegativeIntegers
-        return Family(NonNegativeIntegers(), lambda x: self.gen(x), name=self.variable_names()[0])
+        return Family(NonNegativeIntegers(), lambda x: self.gen(x),
+                      name=self.variable_names()[0])
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return the algebra generators of ``self``.
 

src/sage/modular/abvar/homology.py

@@ -138,10 +138,11 @@ def __richcmp__(self, other, op):
             return richcmp((self.abelian_variety(), self.base_ring()),
                            (other.abelian_variety(), other.base_ring()), op)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
-        Return string representation of ``self``. This must be defined in the
-        derived class.
+        Return string representation of ``self``.
+
+        This must be defined in the derived class.
 
         EXAMPLES::
 
@@ -154,7 +155,7 @@ def _repr_(self):
         """
         raise NotImplementedError("please override this in the derived class")
 
-    def gens(self):
+    def gens(self) -> tuple:
         """
         Return generators of ``self``.
 

src/sage/modules/fg_pid/fgp_module.py

@@ -1198,21 +1198,20 @@ def gens_vector(self, x, reduce=False):
 
         We create a derived class and overwrite :meth:`gens`::
 
-             sage: from sage.modules.fg_pid.fgp_module import FGP_Module_class
-             sage: W = ZZ^3
-             sage: V = W.span(matrix.diagonal([1/6, 1/3, 1/12]))
-             sage: class FGP_with_gens(FGP_Module_class):
-             ....:     def __init__(self, V, W, gens):
-             ....:         FGP_Module_class.__init__(self, V, W)
-             ....:         self._gens = tuple([self(g) for g in gens])
-             ....:     def gens(self):
-             ....:         return self._gens
-             sage: gens = [(1/2, 0, 0), (0, 0, 1/4), (1/3, 0, 0), (0, 1/3, 0), (0, 0, 2/3)]
-             sage: gens = [V(g) for g in gens]
-             sage: D = FGP_with_gens(V, W, gens)
-             sage: D.gens()
-             ((0, 3, 0), (0, 0, 3), (0, 4, 0), (1, 2, 0), (0, 0, 8))
-
+            sage: from sage.modules.fg_pid.fgp_module import FGP_Module_class
+            sage: W = ZZ^3
+            sage: V = W.span(matrix.diagonal([1/6, 1/3, 1/12]))
+            sage: class FGP_with_gens(FGP_Module_class):
+            ....:     def __init__(self, V, W, gens):
+            ....:         FGP_Module_class.__init__(self, V, W)
+            ....:         self._gens = tuple([self(g) for g in gens])
+            ....:     def gens(self) -> tuple:
+            ....:         return self._gens
+            sage: gens = [(1/2, 0, 0), (0, 0, 1/4), (1/3, 0, 0), (0, 1/3, 0), (0, 0, 2/3)]
+            sage: gens = [V(g) for g in gens]
+            sage: D = FGP_with_gens(V, W, gens)
+            sage: D.gens()
+            ((0, 3, 0), (0, 0, 3), (0, 4, 0), (1, 2, 0), (0, 0, 8))
 
         We create some element of ``D``::
 
@@ -1620,7 +1619,7 @@ def _hom_general(self, im_gens, check=True):
         EXAMPLES::
 
             sage: class SillyModule(sage.modules.fg_pid.fgp_module.FGP_Module_class):
-            ....:     def gens(self):
+            ....:     def gens(self) -> tuple:
             ....:         return tuple(flatten([[x,x] for x in self.smith_form_gens()]))
             sage: A = SillyModule(ZZ**1, span([[3]], ZZ))
             sage: A.gen(0)
@@ -1662,7 +1661,7 @@ def _hom_from_smith(self, im_smith_gens, check=True):
         EXAMPLES::
 
             sage: class SillyModule(sage.modules.fg_pid.fgp_module.FGP_Module_class):
-            ....:     def gens(self):
+            ....:     def gens(self) -> tuple:
             ....:         return tuple(flatten([[x,x] for x in self.smith_form_gens()]))
             sage: A = SillyModule(ZZ**1, span([[3]], ZZ))
             sage: A.gen(0)

src/sage/schemes/elliptic_curves/ell_finite_field.py

@@ -705,7 +705,7 @@ def cardinality_pari(self):
         return Integer(self.__pari__().ellcard())
 
     @cached_method
-    def gens(self):
+    def gens(self) -> tuple:
         r"""
         Return points which generate the abelian group of points on
         this elliptic curve.
@@ -800,7 +800,7 @@ def gens(self):
             self._order = ZZ(card)
         pts = tuple(self.point(list(P)) for P in pts)
         if len(pts) >= 1:
-            pts[0]._order = ZZ(ords[0]) # PARI documentation: "P is of order d_1"
+            pts[0]._order = ZZ(ords[0])  # PARI documentation: "P is of order d_1"
         return pts
 
     def __iter__(self):