gh-35728: New feature annotations `# optional - sage.schemes sage.modular sage.libs.flint` etc.
    
<!-- Please provide a concise, informative and self-explanatory title.
-->
<!-- Don't put issue numbers in the title. Put it in the Description
below. -->
<!-- For example, instead of "Fixes #12345", use "Add a new method to
multiply two integers" -->

### 📚 Description

<!-- Describe your changes here in detail. -->
<!-- Why is this change required? What problem does it solve? -->
<!-- If this PR resolves an open issue, please link to it here. For
example "Fixes #12345". -->
- Part of: #29705
- Cherry-picked from: #35095

<!-- If your change requires a documentation PR, please link it
appropriately. -->

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. It should be `[x]` not `[x
]`. -->

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation accordingly.

### ⌛ Dependencies

<!-- List all open PRs that this PR logically depends on
- #12345: short description why this is a dependency
- #34567: ...
-->

<!-- If you're unsure about any of these, don't hesitate to ask. We're
here to help! -->
    
URL: #35728
Reported by: Matthias Köppe
Reviewer(s): David Coudert, Matthias Köppe

Author: Release Manager

src/sage/arith/functions.pyx

@@ -72,25 +72,25 @@ def lcm(a, b=None):
 
     Make sure we try `\QQ` and not merely `\ZZ` (:trac:`13014`)::
 
-        sage: bool(lcm(2/5, 3/7) == lcm(SR(2/5), SR(3/7)))
+        sage: bool(lcm(2/5, 3/7) == lcm(SR(2/5), SR(3/7)))                              # optional - sage.symbolic
         True
 
     Make sure that the lcm of Expressions stays symbolic::
 
         sage: parent(lcm(2, 4))
         Integer Ring
-        sage: parent(lcm(SR(2), 4))
+        sage: parent(lcm(SR(2), 4))                                                     # optional - sage.symbolic
         Symbolic Ring
-        sage: parent(lcm(2, SR(4)))
+        sage: parent(lcm(2, SR(4)))                                                     # optional - sage.symbolic
         Symbolic Ring
-        sage: parent(lcm(SR(2), SR(4)))
+        sage: parent(lcm(SR(2), SR(4)))                                                 # optional - sage.symbolic
         Symbolic Ring
 
     Verify that objects without lcm methods but which can't be
     coerced to `\ZZ` or `\QQ` raise an error::
 
-        sage: F.<x,y> = FreeMonoid(2)
-        sage: lcm(x,y)
+        sage: F.<x,y> = FreeMonoid(2)                                                   # optional - sage.groups
+        sage: lcm(x,y)                                                                  # optional - sage.groups
         Traceback (most recent call last):
         ...
         TypeError: unable to find lcm of x and y

src/sage/arith/misc.py

@@ -1,3 +1,4 @@
+# sage.doctest: optional - primecountpy
 """
 Utility classes for multi-modular algorithms
 """

src/sage/arith/multi_modular.pyx

@@ -432,16 +432,16 @@ cdef class PrecomposedAction(Action):
     We demonstrate that an example discussed on :trac:`14711` did not become a
     problem::
 
-        sage: E = ModularSymbols(11).2
-        sage: s = E.modular_symbol_rep()
-        sage: del E,s
-        sage: import gc
-        sage: _ = gc.collect()
-        sage: E = ModularSymbols(11).2
-        sage: v = E.manin_symbol_rep()
-        sage: c,x = v[0]
-        sage: y = x.modular_symbol_rep()
-        sage: coercion_model.get_action(QQ, parent(y), op=operator.mul)
+        sage: E = ModularSymbols(11).2                                                  # optional - sage.modular
+        sage: s = E.modular_symbol_rep()                                                # optional - sage.modular
+        sage: del E,s                                                                   # optional - sage.modular
+        sage: import gc                                                                 # optional - sage.modular
+        sage: _ = gc.collect()                                                          # optional - sage.modular
+        sage: E = ModularSymbols(11).2                                                  # optional - sage.modular
+        sage: v = E.manin_symbol_rep()                                                  # optional - sage.modular
+        sage: c,x = v[0]                                                                # optional - sage.modular
+        sage: y = x.modular_symbol_rep()                                                # optional - sage.modular
+        sage: coercion_model.get_action(QQ, parent(y), op=operator.mul)                 # optional - sage.modular
         Left scalar multiplication by Rational Field
          on Abelian Group of all Formal Finite Sums over Rational Field
          with precomposition on right by Coercion map:
@@ -483,12 +483,12 @@ cdef class PrecomposedAction(Action):
 
         Check that this action can be pickled (:trac:`29031`)::
 
-            sage: E = ModularSymbols(11).2
-            sage: v = E.manin_symbol_rep()
-            sage: c,x = v[0]
-            sage: y = x.modular_symbol_rep()
-            sage: act = coercion_model.get_action(QQ, parent(y), op=operator.mul)
-            sage: loads(dumps(act)) is not None
+            sage: E = ModularSymbols(11).2                                              # optional - sage.modular
+            sage: v = E.manin_symbol_rep()                                              # optional - sage.modular
+            sage: c,x = v[0]                                                            # optional - sage.modular
+            sage: y = x.modular_symbol_rep()                                            # optional - sage.modular
+            sage: act = coercion_model.get_action(QQ, parent(y), op=operator.mul)       # optional - sage.modular
+            sage: loads(dumps(act)) is not None                                         # optional - sage.modular
             True
         """
         return (type(self), (self._action, self.G_precomposition, self.S_precomposition))

src/sage/categories/action.pyx

@@ -103,7 +103,7 @@ class ParentMethods:
 
         def _Hom_(self, Y, category):
             r"""
-            Returns the homset from ``self`` to ``Y`` in the category ``category``
+            Return the homset from ``self`` to ``Y`` in the category ``category``
 
             INPUT:
 
@@ -121,15 +121,15 @@ def _Hom_(self, Y, category):
 
             EXAMPLES::
 
-                sage: M = ModularForms(Gamma0(7), 4)
-                sage: H = M._Hom_(M, category = HeckeModules(QQ)); H
+                sage: M = ModularForms(Gamma0(7), 4)                                    # optional - sage.modular
+                sage: H = M._Hom_(M, category=HeckeModules(QQ)); H                      # optional - sage.modular
                 Set of Morphisms
                  from Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(7) of weight 4 over Rational Field
                    to Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(7) of weight 4 over Rational Field
                    in Category of Hecke modules over Rational Field
-                sage: H.__class__
+                sage: H.__class__                                                       # optional - sage.modular
                 <class 'sage.modular.hecke.homspace.HeckeModuleHomspace_with_category'>
-                sage: TestSuite(H).run(skip=["_test_elements", "_test_an_element", "_test_elements_eq",
+                sage: TestSuite(H).run(skip=["_test_elements", "_test_an_element", "_test_elements_eq",     # optional - sage.modular
                 ....:                        "_test_elements_eq_reflexive", "_test_elements_eq_transitive",
                 ....:                        "_test_elements_eq_symmetric", "_test_elements_neq", "_test_some_elements",
                 ....:                        "_test_zero", "_test_additive_associativity",
@@ -142,7 +142,7 @@ def _Hom_(self, Y, category):
 
             TESTS::
 
-                sage: H = M._Hom_(M, category = HeckeModules(GF(5))); H
+                sage: H = M._Hom_(M, category=HeckeModules(GF(5))); H                   # optional - sage.modular sage.rings.finite_rings
                 Traceback (most recent call last):
                 ...
                 TypeError: Category of Hecke modules over Finite Field of size 5

src/sage/categories/hecke_modules.py

@@ -592,15 +592,15 @@ class Homset(Set_generic):
 
     Conversely, homsets of non-unique parents are non-unique::
 
-        sage: P11 = ProductProjectiveSpaces(QQ, [1, 1])
-        sage: H = End(P11)
-        sage: loads(dumps(P11)) is ProductProjectiveSpaces(QQ, [1, 1])
+        sage: P11 = ProductProjectiveSpaces(QQ, [1, 1])                                 # optional - sage.schemes
+        sage: H = End(P11)                                                              # optional - sage.schemes
+        sage: loads(dumps(P11)) is ProductProjectiveSpaces(QQ, [1, 1])                  # optional - sage.schemes
         False
-        sage: loads(dumps(P11)) == ProductProjectiveSpaces(QQ, [1, 1])
+        sage: loads(dumps(P11)) == ProductProjectiveSpaces(QQ, [1, 1])                  # optional - sage.schemes
         True
-        sage: loads(dumps(H)) is H
+        sage: loads(dumps(H)) is H                                                      # optional - sage.schemes
         False
-        sage: loads(dumps(H)) == H
+        sage: loads(dumps(H)) == H                                                      # optional - sage.schemes
         True
     """
     def __init__(self, X, Y, category=None, base=None, check=True):

src/sage/categories/homset.py

@@ -165,7 +165,7 @@ def _call_(self, x):
         If ``x`` itself is not a module with basis, but there is a
         canonical one associated to it, the latter is returned::
 
-            sage: CQ(AbelianVariety(Gamma0(37)))  # indirect doctest                    # optional - sage.modules
+            sage: CQ(AbelianVariety(Gamma0(37)))  # indirect doctest                    # optional - sage.modular sage.modules
             Vector space of dimension 4 over Rational Field
         """
         try:

src/sage/categories/modules_with_basis.py

@@ -63,9 +63,11 @@ def __init__(self):
         """
         # sage.combinat will be a namespace package.
         # Testing whether sage.combinat itself can be imported is meaningless.
+        # Some modules providing basic combinatorics are already included in sagemath-categories.
         # Hence, we test a Python module within the package.
         JoinFeature.__init__(self, 'sage.combinat',
-                             [PythonModule('sage.combinat.combination')])
+                             [PythonModule('sage.combinat.tableau')],
+                             spkg='sagemath_combinat')
 
 
 class sage__geometry__polyhedron(PythonModule):
@@ -87,7 +89,8 @@ def __init__(self):
             sage: isinstance(sage__geometry__polyhedron(), sage__geometry__polyhedron)
             True
         """
-        PythonModule.__init__(self, 'sage.geometry.polyhedron')
+        PythonModule.__init__(self, 'sage.geometry.polyhedron',
+                              spkg='sagemath_polyhedra')
 
 
 class sage__graphs(JoinFeature):
@@ -109,7 +112,31 @@ def __init__(self):
             True
         """
         JoinFeature.__init__(self, 'sage.graphs',
-                             [PythonModule('sage.graphs.graph')])
+                             [PythonModule('sage.graphs.graph')],
+                             spkg='sagemath_graphs')
+
+
+class sage__modular(JoinFeature):
+    r"""
+    A :class:`~sage.features.Feature` describing the presence of :mod:`sage.modular`.
+
+    EXAMPLES::
+
+        sage: from sage.features.sagemath import sage__modular
+        sage: sage__modular().is_present()                                              # optional - sage.modular
+        FeatureTestResult('sage.modular', True)
+    """
+    def __init__(self):
+        r"""
+        TESTS::
+
+            sage: from sage.features.sagemath import sage__modular
+            sage: isinstance(sage__modular(), sage__modular)
+            True
+        """
+        JoinFeature.__init__(self, 'sage.modular',
+                             [PythonModule('sage.modular.modform.eisenstein_submodule')],
+                             spkg='sagemath_schemes')
 
 
 class sage__groups(JoinFeature):
@@ -134,6 +161,55 @@ def __init__(self):
                              [PythonModule('sage.groups.perm_gps.permgroup')])
 
 
+class sage__libs__flint(JoinFeature):
+    r"""
+    A :class:`sage.features.Feature` describing the presence of :mod:`sage.libs.flint`
+    and other modules depending on FLINT and arb.
+
+    EXAMPLES::
+
+        sage: from sage.features.sagemath import sage__libs__flint
+        sage: sage__libs__flint().is_present()                                          # optional - sage.libs.flint
+        FeatureTestResult('sage.libs.flint', True)
+    """
+    def __init__(self):
+        r"""
+        TESTS::
+
+            sage: from sage.features.sagemath import sage__libs__flint
+            sage: isinstance(sage__libs__flint(), sage__libs__flint)
+            True
+        """
+        JoinFeature.__init__(self, 'sage.libs.flint',
+                             [PythonModule('sage.libs.flint.flint'),
+                              PythonModule('sage.libs.arb.arith')],
+                             spkg='sagemath_flint')
+
+
+class sage__libs__ntl(JoinFeature):
+    r"""
+    A :class:`sage.features.Feature` describing the presence of :mod:`sage.libs.ntl`
+    and other modules depending on NTL and arb.
+
+    EXAMPLES::
+
+        sage: from sage.features.sagemath import sage__libs__ntl
+        sage: sage__libs__ntl().is_present()                                            # optional - sage.libs.ntl
+        FeatureTestResult('sage.libs.ntl', True)
+    """
+    def __init__(self):
+        r"""
+        TESTS::
+
+            sage: from sage.features.sagemath import sage__libs__ntl
+            sage: isinstance(sage__libs__ntl(), sage__libs__ntl)
+            True
+        """
+        JoinFeature.__init__(self, 'sage.libs.ntl',
+                             [PythonModule('sage.libs.ntl.convert')],
+                             spkg='sagemath_ntl')
+
+
 class sage__libs__pari(JoinFeature):
     r"""
     A :class:`sage.features.Feature` describing the presence of :mod:`sage.libs.pari`.
@@ -153,7 +229,8 @@ def __init__(self):
             True
         """
         JoinFeature.__init__(self, 'sage.libs.pari',
-                             [PythonModule('sage.libs.pari.convert_sage')])
+                             [PythonModule('sage.libs.pari.convert_sage')],
+                             spkg='sagemath_pari')
 
 
 class sage__modules(JoinFeature):
@@ -175,7 +252,8 @@ def __init__(self):
             True
         """
         JoinFeature.__init__(self, 'sage.modules',
-                             [PythonModule('sage.modules.free_module')])
+                             [PythonModule('sage.modules.free_module')],
+                             spkg='sagemath_modules')
 
 
 class sage__plot(JoinFeature):
@@ -197,7 +275,8 @@ def __init__(self):
             True
         """
         JoinFeature.__init__(self, 'sage.plot',
-                             [PythonModule('sage.plot.plot')])
+                             [PythonModule('sage.plot.plot')],
+                             spkg='sagemath_symbolics')
 
 
 class sage__rings__finite_rings(JoinFeature):
@@ -290,6 +369,29 @@ def __init__(self):
                              [PythonModule('sage.rings.padics.factory')])
 
 
+class sage__rings__polynomial__pbori(JoinFeature):
+    r"""
+    A :class:`sage.features.Feature` describing the presence of :mod:`sage.rings.polynomial.pbori`.
+
+    EXAMPLES::
+
+        sage: from sage.features.sagemath import sage__rings__polynomial__pbori
+        sage: sage__rings__polynomial__pbori().is_present()                             # optional - sage.rings.polynomial.pbori
+        FeatureTestResult('sage.rings.polynomial.pbori', True)
+    """
+    def __init__(self):
+        r"""
+        TESTS::
+
+            sage: from sage.features.sagemath import sage__rings__polynomial__pbori
+            sage: isinstance(sage__rings__polynomial__pbori(), sage__rings__polynomial__pbori)
+            True
+        """
+        JoinFeature.__init__(self, 'sage.rings.polynomial.pbori',
+                             [PythonModule('sage.rings.polynomial.pbori.pbori')],
+                             spkg='sagemath_brial')
+
+
 class sage__rings__real_double(PythonModule):
     r"""
     A :class:`~sage.features.Feature` describing the presence of :mod:`sage.rings.real_double`.
@@ -329,7 +431,31 @@ def __init__(self):
             sage: isinstance(sage__rings__real_mpfr(), sage__rings__real_mpfr)
             True
         """
-        PythonModule.__init__(self, 'sage.rings.real_mpfr')
+        PythonModule.__init__(self, 'sage.rings.real_mpfr',
+                              spkg='sagemath_modules')
+
+
+class sage__schemes(JoinFeature):
+    r"""
+    A :class:`~sage.features.Feature` describing the presence of :mod:`sage.schemes`.
+
+    EXAMPLES::
+
+        sage: from sage.features.sagemath import sage__schemes
+        sage: sage__schemes().is_present()                                              # optional - sage.schemes
+        FeatureTestResult('sage.schemes', True)
+    """
+    def __init__(self):
+        r"""
+        TESTS::
+
+            sage: from sage.features.sagemath import sage__schemes
+            sage: isinstance(sage__schemes(), sage__schemes)
+            True
+        """
+        JoinFeature.__init__(self, 'sage.schemes',
+                             [PythonModule('sage.schemes.elliptic_curves.ell_generic')],
+                             spkg="sagemath_schemes")
 
 
 class sage__symbolic(JoinFeature):
@@ -382,13 +508,18 @@ def all_features():
             sage__geometry__polyhedron(),
             sage__graphs(),
             sage__groups(),
+            sage__libs__flint(),
+            sage__libs__ntl(),
             sage__libs__pari(),
+            sage__modular(),
             sage__modules(),
             sage__plot(),
             sage__rings__finite_rings(),
             sage__rings__function_field(),
             sage__rings__number_field(),
             sage__rings__padics(),
+            sage__rings__polynomial__pbori(),
             sage__rings__real_double(),
             sage__rings__real_mpfr(),
+            sage__schemes(),
             sage__symbolic()]

src/sage/features/sagemath.py

@@ -3707,8 +3707,9 @@ class disk_cached_function:
             sage: dir = tmp_dir()
             sage: @disk_cached_function(dir)
             ....: def foo(x): return ModularSymbols(x)
-            sage: foo(389)
-            Modular Symbols space of dimension 65 for Gamma_0(389) of weight 2 with sign 0 over Rational Field
+            sage: foo(389)                                                              # optional - sage.modular
+            Modular Symbols space of dimension 65 for Gamma_0(389) of weight 2
+             with sign 0 over Rational Field
         """
         return DiskCachedFunction(f, self._dir, memory_cache=self._memory_cache, key=self._key)