gh-35884: `sage.{modular,schemes}`: Modularization fixes for imports; update `sage -fiximports`, add relint pattern 
    
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### 📚 Description

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- Part of: #29705
- Cherry-picked from: #35095

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### ⌛ Dependencies

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- Depends on #35974 (to soothe the linter)

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URL: #35884
Reported by: Matthias Köppe
Reviewer(s): Kwankyu Lee, Matthias Köppe

Author: Release Manager

src/.relint.yml

@@ -51,7 +51,18 @@
   # Keep in sync with SAGE_ROOT/src/sage/misc/replace_dot_all.py
   pattern: 'from\s+sage(|[.](arith|categories|combinat|crypto|databases|data_structures|dynamics|ext|game_theory|games|geometry|graphs|groups|interfaces|manifolds|matrix|matroids|misc|modules|monoids|numerical|probability|quadratic_forms|quivers|rings|sat|schemes|sets|stats|symbolic|tensor)[a-z0-9_.]*|[.]libs)[.]all\s+import'
   # imports from .all are allowed in all.py; also allow in some modules that need sage.all
-  filePattern: '(.*/|)(?!(all|benchmark|dev_tools|parsing|sage_eval))[^/.]*[.](py|pyx|pxi)$'
+  filePattern: '(.*/|)(?!(all|benchmark|dev_tools|parsing|sage_eval|explain_pickle|.*_test))[^/.]*[.](py|pyx|pxi)$'
+
+- name: 'namespace_pkg_all_import_2: Module-level import of .all of a namespace package'
+  hint: |
+    Sage library code should not import sage.PAC.KAGE.all when sage.PAC.KAGE is an implicit
+    namespace package. Type import_statements("SOME_IDENTIFIER") to find a more specific import,
+    and rewrite the import statement as "from sage.PAC.KAGE.MODULE import ..."
+    or "lazy_import('sage.PAC.KAGE.MODULE', '...')".
+  # Keep in sync with above; but for now we ignore sage.{arith,categories}
+  pattern: '^import\s+sage(|[.](combinat|crypto|databases|data_structures|dynamics|ext|game_theory|games|geometry|graphs|groups|interfaces|manifolds|matrix|matroids|misc|modules|monoids|numerical|probability|quadratic_forms|quivers|rings|sat|schemes|sets|stats|symbolic|tensor)[a-z0-9_.]*|[.]libs)[.]all'
+  # imports from .all are allowed in all.py; also allow in some modules that need sage.all
+  filePattern: '(.*/|)(?!(all|benchmark|dev_tools|parsing|sage_eval|explain_pickle|.*_test))[^/.]*[.](py|pyx|pxi)$'
 
 # Magic doctest comments
 

src/sage/lfunctions/lcalc.py

@@ -30,9 +30,14 @@
 import os
 
 from sage.structure.sage_object import SageObject
+from sage.misc.lazy_import import lazy_import
 from sage.misc.pager import pager
-import sage.rings.all
-import sage.schemes.elliptic_curves.ell_generic
+from sage.rings.integer_ring import ZZ
+from sage.rings.rational_field import QQ
+
+lazy_import('sage.rings.complex_mpfr', 'ComplexField')
+lazy_import('sage.rings.real_mpfr', 'RealField')
+lazy_import('sage.schemes.elliptic_curves.ell_generic', 'EllipticCurve_generic', as_='EllipticCurve')
 
 prec = 32
 
@@ -75,9 +80,8 @@ def _compute_L(self, L):
             if L == 'tau':
                 return '--tau'
             return L
-        import sage.schemes.all
-        if sage.schemes.elliptic_curves.ell_generic.is_EllipticCurve(L):
-            if L.base_ring() == sage.rings.all.RationalField():
+        if isinstance(L, EllipticCurve):
+            if L.base_ring() == QQ:
                 L = L.minimal_model()
                 return '-e --a1 %s --a2 %s --a3 %s --a4 %s --a6 %s' % tuple(L.a_invariants())
         raise TypeError("$L$-function of %s not known" % L)
@@ -127,7 +131,7 @@ def zeros(self, n, L=''):
             [0.000000000, 5.00317001, 6.87039122]
         """
         L = self._compute_L(L)
-        RR = sage.rings.all.RealField(prec)
+        RR = RealField(prec)
         X = self('-z %s %s' % (int(n), L))
         return [RR(z) for z in X.split()]
 
@@ -162,7 +166,7 @@ def zeros_in_interval(self, x, y, stepsize, L=''):
             [(14.1347251, 0.184672916), (21.0220396, -0.0677893290), (25.0108576, -0.0555872781)]
         """
         L = self._compute_L(L)
-        RR = sage.rings.all.RealField(prec)
+        RR = RealField(prec)
         X = self('--zeros-interval -x %s -y %s --stepsize=%s %s' % (
             float(x), float(y), float(stepsize), L))
         return [tuple([RR(z) for z in t.split()]) for t in X.split('\n')]
@@ -193,7 +197,7 @@ def value(self, s, L=''):
             2.69261988568132 - 0.0203860296025982*I
         """
         L = self._compute_L(L)
-        CC = sage.rings.all.ComplexField(prec)
+        CC = ComplexField(prec)
         s = CC(s)
         x, y = self('-v -x %s -y %s %s' % (s.real(), s.imag(), L)).split()
         return CC((float(x), float(y)))
@@ -275,7 +279,7 @@ def values_along_line(self, s0, s1, number_samples, L=''):
 
         """
         L = self._compute_L(L)
-        CC = sage.rings.all.ComplexField(prec)
+        CC = ComplexField(prec)
         s0 = CC(s0)
         s1 = CC(s1)
         v = self('--value-line-segment -x %s -y %s -X %s -Y %s --number-samples %s %s' % (
@@ -343,8 +347,7 @@ def twist_values(self, s, dmin, dmax, L=''):
             0.373691713 + 0.0*I
         """
         L = self._compute_L(L)
-        CC = sage.rings.all.ComplexField(prec)
-        Z = sage.rings.all.Integer
+        CC = ComplexField(prec)
         s = CC(s)
         typ = '--twist-quadratic'
         dmin = int(dmin)
@@ -358,7 +361,7 @@ def twist_values(self, s, dmin, dmax, L=''):
             return w
         for a in v.split('\n'):
             d, x, y = a.split()
-            w.append((Z(d), CC(x, y)))
+            w.append((ZZ(d), CC(x, y)))
         return w
 
     def twist_zeros(self, n, dmin, dmax, L=''):
@@ -393,8 +396,7 @@ def twist_zeros(self, n, dmin, dmax, L=''):
             {-3: [8.03973716, 11.2492062, 15.7046192], 5: [6.64845335, 9.83144443, 11.9588456]}
         """
         L = self._compute_L(L)
-        RR = sage.rings.all.RealField(prec)
-        Z = sage.rings.all.Integer
+        RR = RealField(prec)
         typ = '--twist-quadratic'
         n = int(n)
         v = self('-z %s %s --start %s --finish %s %s' % (
@@ -405,7 +407,7 @@ def twist_zeros(self, n, dmin, dmax, L=''):
         for a in v.split('\n'):
             d, x = a.split()
             x = RR(x)
-            d = Z(d)
+            d = ZZ(d)
             if d in w:
                 w[d].append(x)
             else:
@@ -439,10 +441,9 @@ def analytic_rank(self, L=''):
             1
         """
         L = self._compute_L(L)
-        Z = sage.rings.all.Integer
         s = self('--rank-compute %s' % L)
         i = s.find('equals')
-        return Z(s[i + 6:])
+        return ZZ(s[i + 6:])
 
 
 # An instance

src/sage/lfunctions/sympow.py

@@ -51,7 +51,7 @@
 from sage.structure.sage_object import SageObject
 from sage.misc.pager import pager
 from sage.misc.verbose import verbose
-import sage.rings.all
+from sage.rings.integer import Integer
 
 
 class Sympow(SageObject):
@@ -237,7 +237,7 @@ def modular_degree(self, E):
         if i == -1:
             print(self._fix_err(v))
             raise RuntimeError("failed to compute modular degree")
-        return sage.rings.all.Integer(v[i + len(s):])
+        return Integer(v[i + len(s):])
 
     def analytic_rank(self, E):
         r"""
@@ -296,7 +296,7 @@ def analytic_rank(self, E):
             print(self._fix_err(v))
             raise RuntimeError("failed to compute analytic rank")
         j = v.rfind(':')
-        r = sage.rings.all.Integer(v[i + len(s):j])
+        r = Integer(v[i + len(s):j])
         i = v.rfind(' ')
         L = v[i + 1:]
         return r, L

src/sage/misc/replace_dot_all.py

@@ -385,7 +385,8 @@ def make_replacements_in_file(location, package_regex=None, verbose=False, outpu
         write_file.write(replaced_content)  # overwriting the old file contents with the new/replaced content
 
 
-def walkdir_replace_dot_all(dir, file_regex=r'.*[.](py|pyx|pxi)$', package_regex=None, verbose=False):
+def walkdir_replace_dot_all(dir, file_regex=r'.*[.](py|pyx|pxi)$', package_regex=None, verbose=False, *,
+                            excluded_file_regex=r'auto-methods|replace_dot_all'):
     r"""
     Replace ``import`` statements in the files in directory ``dir`` matching the regex pattern ``file_regex``.
 
@@ -396,6 +397,7 @@ def walkdir_replace_dot_all(dir, file_regex=r'.*[.](py|pyx|pxi)$', package_regex
     - ``package_regex`` -- (default: :obj:`default_package_regex`) a regular expression matching
       the ``sage.PAC.KAGE.all`` package names from which we do not want to import.
     - ``verbose`` -- if True, print statements when interesting examples are found
+    - ``excluded_file_regex`` -- a regular expression matching the file names to exclude
 
     EXAMPLES::
 
@@ -404,14 +406,14 @@ def walkdir_replace_dot_all(dir, file_regex=r'.*[.](py|pyx|pxi)$', package_regex
     """
     global numberFiles, numberFilesMatchingRegex
     file_regex = re.compile(file_regex)
+    excluded_file_regex = re.compile(excluded_file_regex)
     for root, dirs, files in os.walk(dir, topdown=False):
         for name in files:
             numberFiles += 1
-            if file_regex.search(name):
+            if file_regex.search(name) and not excluded_file_regex.search(name):
                 numberFilesMatchingRegex += 1
                 location = os.path.join(root, name)
-                if location.find('replace_dot_all') == -1:  # to avoid changing anything in this file itself
-                    make_replacements_in_file(location, package_regex, verbose)
+                make_replacements_in_file(location, package_regex, verbose)
 
 
 # ******************************************************** EXECUTES MAIN FUNCTION **********************************************************************

src/sage/modular/abvar/abvar.py

@@ -33,6 +33,8 @@
 from copy import copy
 from random import randrange
 
+import sage.rings.abc
+
 from sage.arith.functions import lcm as LCM
 from sage.arith.misc import divisors, next_prime, is_prime
 from sage.categories.modular_abelian_varieties import ModularAbelianVarieties
@@ -55,7 +57,6 @@
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-from sage.rings.qqbar import QQbar
 from sage.rings.rational_field import QQ
 from sage.rings.ring import is_Ring
 from sage.schemes.elliptic_curves.constructor import EllipticCurve
@@ -67,6 +68,7 @@
 lazy_import('sage.databases.cremona',
             ['cremona_letter_code', 'CremonaDatabase'])
 
+
 from . import homspace
 from . import lseries
 from .morphism import HeckeOperator, Morphism, DegeneracyMap
@@ -798,8 +800,8 @@ def _Hom_(self, B, cat=None):
             L = B.base_field()
             if K == L:
                 F = K
-            elif K == QQbar or L == QQbar:
-                F = QQbar
+            elif isinstance(K, sage.rings.abc.AlgebraicField) or isinstance(L, sage.rings.abc.AlgebraicField):
+                from sage.rings.qqbar import QQbar as F
             else:
                 # TODO -- improve this
                 raise ValueError("please specify a category")
@@ -997,7 +999,7 @@ def intersection(self, other):
                 for v in V.coordinate_module(S).basis()]
 
         if A.dimension() > 0:
-            finitegroup_base_field = QQbar
+            from sage.rings.qqbar import QQbar as finitegroup_base_field
         else:
             finitegroup_base_field = self.base_field()
         G = self.finite_subgroup(gens, field_of_definition=finitegroup_base_field)
@@ -3126,7 +3128,7 @@ def finite_subgroup(self, X, field_of_definition=None, check=True):
                     raise ValueError("X must be a subgroup of self.")
 
         if field_of_definition is None:
-            field_of_definition = QQbar
+            from sage.rings.qqbar import QQbar as field_of_definition
 
         return FiniteSubgroup_lattice(
             self, X, field_of_definition=field_of_definition, check=check)

src/sage/modular/abvar/finite_subgroup.py

@@ -97,6 +97,9 @@
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
+import sage.rings.abc
+
+from sage.misc.lazy_import import lazy_import
 from sage.modular.abvar.torsion_point import TorsionPoint
 from sage.modules.module import Module
 from sage.modules.free_module import is_FreeModule
@@ -105,7 +108,6 @@
 from sage.structure.richcmp import richcmp_method, richcmp
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
-from sage.rings.qqbar import QQbar
 from sage.rings.rational_field import QQ
 from sage.arith.functions import lcm
 from sage.misc.misc_c import prod
@@ -521,7 +523,7 @@ def _repr_(self):
             'Finite subgroup with invariants [3, 3, 3, 3, 3, 3, 3, 3, 3, 3] over QQ of Abelian variety J0(42) of dimension 5'
         """
         K = self.__field_of_definition
-        if K == QQbar:
+        if isinstance(K, sage.rings.abc.AlgebraicField):
             field = "QQbar"
         elif K == QQ:
             field = "QQ"
@@ -750,6 +752,8 @@ def subgroup(self, gens):
             sage: H == G.subgroup([[1/11,0,0,0]])
             True
         """
+        from sage.rings.qqbar import QQbar
+
         if not isinstance(gens, (tuple, list)):
             raise TypeError("gens must be a list or tuple")
         A = self.abelian_variety()
@@ -815,7 +819,7 @@ def invariants(self):
 
 
 class FiniteSubgroup_lattice(FiniteSubgroup):
-    def __init__(self, abvar, lattice, field_of_definition=QQbar, check=True):
+    def __init__(self, abvar, lattice, field_of_definition=None, check=True):
         """
         A finite subgroup of a modular abelian variety that is defined by a
         given lattice.
@@ -841,6 +845,8 @@ def __init__(self, abvar, lattice, field_of_definition=QQbar, check=True):
             sage: G = J.finite_subgroup([[1/3,0], [0,1/5]]); G
             Finite subgroup with invariants [15] over QQbar of Abelian variety J0(11) of dimension 1
         """
+        if field_of_definition is None:
+            from sage.rings.qqbar import QQbar as field_of_definition
         if check:
             from .abvar import is_ModularAbelianVariety
             if not is_FreeModule(lattice) or lattice.base_ring() != ZZ:

src/sage/modular/abvar/homspace.py

@@ -186,14 +186,14 @@
 from . import morphism
 
 import sage.rings.integer_ring
-import sage.rings.all
+from sage.rings.infinity import Infinity
 
 from sage.rings.ring import Ring
 from sage.matrix.matrix_space import MatrixSpace
 from sage.matrix.constructor import Matrix, identity_matrix
 from sage.structure.element import is_Matrix
 
-ZZ = sage.rings.integer_ring.ZZ
+from sage.rings.integer_ring import ZZ
 
 
 class Homspace(HomsetWithBase):
@@ -882,7 +882,7 @@ def index_in(self, other, check=True):
         M = self.free_module()
         N = other.free_module()
         if M.rank() < N.rank():
-            return sage.rings.all.Infinity
+            return Infinity
         return M.index_in(N)
 
     def index_in_saturation(self):

src/sage/modular/arithgroup/all.py

@@ -9,7 +9,8 @@
 from .congroup_gamma import Gamma_constructor as Gamma, is_Gamma
 from .congroup_sl2z import SL2Z, is_SL2Z
 
-from .arithgroup_perm import ArithmeticSubgroup_Permutation
+from sage.misc.lazy_import import lazy_import
+lazy_import('sage.modular.arithgroup.arithgroup_perm', 'ArithmeticSubgroup_Permutation')
 
 from .congroup import (degeneracy_coset_representatives_gamma0,
                             degeneracy_coset_representatives_gamma1)

src/sage/modular/arithgroup/arithgroup_generic.py

@@ -885,20 +885,21 @@ def index(self):
 
     def generalised_level(self):
         r"""
-        Return the generalised level of self, i.e. the least common multiple of
+        Return the generalised level of ``self``, i.e., the least common multiple of
         the widths of all cusps.
 
-        If self is *even*, Wohlfart's theorem tells us that this is equal to
-        the (conventional) level of self when self is a congruence subgroup.
-        This can fail if self is odd, but the actual level is at most twice the
+        If ``self`` is *even*, Wohlfart's theorem tells us that this is equal to
+        the (conventional) level of ``self`` when ``self`` is a congruence subgroup.
+        This can fail if ``self`` is odd, but the actual level is at most twice the
         generalised level. See the paper by Kiming, Schuett and Verrill for
         more examples.
 
         EXAMPLES::
 
             sage: Gamma0(18).generalised_level()
             18
-            sage: sage.modular.arithgroup.arithgroup_perm.HsuExample18().generalised_level()
+            sage: from sage.modular.arithgroup.arithgroup_perm import HsuExample18
+            sage: HsuExample18().generalised_level()
             24
 
         In the following example, the actual level is twice the generalised