gh-38142: `sage.schemes`: Modularization fixes (imports)
    
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- Cherry-picked from #35095.

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### ⌛ Dependencies

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URL: #38142
Reported by: Matthias Köppe
Reviewer(s): Kwankyu Lee, Matthias Köppe

Author: Release Manager

src/sage/schemes/affine/affine_homset.py

@@ -36,8 +36,6 @@
 
 from sage.misc.verbose import verbose
 from sage.rings.integer_ring import ZZ
-from sage.rings.real_mpfr import RR
-from sage.rings.cc import CC
 from sage.rings.rational_field import RationalField
 from sage.categories.fields import Fields
 from sage.categories.number_fields import NumberFields
@@ -268,6 +266,7 @@ def points(self, **kwds):
             if hasattr(X.base_ring(), 'precision'):
                 numerical = True
                 verbose("Warning: computations in the numerical fields are inexact;points may be computed partially or incorrectly.", level=0)
+                from sage.rings.real_mpfr import RR
                 zero_tol = RR(kwds.pop('zero_tolerance', 10**(-10)))
                 if zero_tol <= 0:
                     raise ValueError("tolerance must be positive")
@@ -438,7 +437,7 @@ def numerical_points(self, F=None, **kwds):
         """
         from sage.schemes.affine.affine_space import AffineSpace_generic
         if F is None:
-            F = CC
+            from sage.rings.cc import CC as F
         if F not in Fields() or not hasattr(F, 'precision'):
             raise TypeError('F must be a numerical field')
         X = self.codomain()
@@ -455,6 +454,7 @@ def numerical_points(self, F=None, **kwds):
             return []
 
         # if X zero-dimensional
+        from sage.rings.real_mpfr import RR
         zero_tol = RR(kwds.pop('zero_tolerance', 10**(-10)))
         if zero_tol <= 0:
             raise ValueError("tolerance must be positive")

src/sage/schemes/affine/affine_morphism.py

@@ -49,8 +49,6 @@
 
 import sage.rings.abc
 
-from sage.calculus.functions import jacobian
-
 from sage.categories.homset import Hom, End
 from sage.categories.fields import Fields
 
@@ -896,6 +894,9 @@ def jacobian(self):
             return self.__jacobian
         except AttributeError:
             pass
+
+        from sage.calculus.functions import jacobian
+
         self.__jacobian = jacobian(list(self), self.domain().ambient_space().gens())
         return self.__jacobian
 

src/sage/schemes/affine/affine_point.py

@@ -202,7 +202,7 @@ def global_height(self, prec=None):
             else:
                 R = RealField(prec)
             H = max([self[i].abs() for i in range(self.codomain().ambient_space().dimension_relative())])
-            return R(max(H,1)).log()
+            return R(max(H, 1)).log()
         if self.domain().base_ring() in _NumberFields or isinstance(self.domain().base_ring(), sage.rings.abc.Order):
             return max([self[i].global_height(prec) for i in range(self.codomain().ambient_space().dimension_relative())])
         else:

src/sage/schemes/affine/affine_subscheme.py

@@ -20,8 +20,6 @@
 # ****************************************************************************
 
 from sage.categories.fields import Fields
-from sage.interfaces.singular import singular
-from sage.modules.free_module_element import vector
 from sage.schemes.generic.algebraic_scheme import AlgebraicScheme_subscheme
 
 from .affine_morphism import SchemeMorphism_polynomial_affine_subscheme_field
@@ -418,6 +416,7 @@ def intersection_multiplicity(self, X, P):
         Iloc = R.ideal([f(chng_coords) for f in I.gens()])
         Jloc = R.ideal([f(chng_coords) for f in J.gens()])
         # compute the intersection multiplicity with Serre's Tor formula using Singular
+        from sage.interfaces.singular import singular
         singular.lib("homolog.lib")
         i = 0
         s = 0
@@ -492,6 +491,8 @@ def multiplicity(self, P):
         except TypeError:
             raise TypeError("(=%s) is not a point on (=%s)" % (P, self))
 
+        from sage.interfaces.singular import singular
+
         # Apply a linear change of coordinates to self so that P is sent to the origin
         # and then compute the multiplicity of the local ring of the translated subscheme
         # corresponding to the point (0,...,0)
@@ -578,6 +579,8 @@ def tangent_space(self, p):
             sage: _.dimension()
             1
         """
+        from sage.modules.free_module_element import vector
+
         A = self.ambient_space()
         R = A.coordinate_ring()
         gens = R.gens()

src/sage/schemes/berkovich/berkovich_cp_element.py

@@ -33,17 +33,19 @@
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
-from sage.structure.element import Element
-from sage.structure.element import Expression
 import sage.rings.abc
+
+from sage.misc.lazy_import import lazy_import
+from sage.rings.infinity import Infinity
+from sage.rings.integer_ring import ZZ
+from sage.rings.rational_field import QQ
 from sage.rings.real_mpfr import RR, RealNumber
-from sage.rings.padics.padic_generic_element import pAdicGenericElement
-from sage.rings.padics.padic_base_generic import pAdicBaseGeneric
-from sage.schemes.projective.projective_space import ProjectiveSpace
 from sage.schemes.projective.projective_point import SchemeMorphism_point_projective_field
-from sage.rings.rational_field import QQ
-from sage.rings.integer_ring import ZZ
-from sage.rings.infinity import Infinity
+from sage.schemes.projective.projective_space import ProjectiveSpace
+from sage.structure.element import Element, Expression
+
+lazy_import('sage.rings.padics.padic_generic_element', 'pAdicGenericElement')
+lazy_import('sage.rings.padics.padic_base_generic', 'pAdicBaseGeneric')
 
 
 class Berkovich_Element(Element):

src/sage/schemes/berkovich/berkovich_space.py

@@ -31,18 +31,21 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
+import sage.rings.abc
+
+from sage.categories.number_fields import NumberFields
+from sage.categories.topological_spaces import TopologicalSpaces
+from sage.misc.lazy_import import lazy_import
+from sage.rings.integer_ring import ZZ
+from sage.rings.rational_field import QQ
+from sage.schemes.affine.affine_space import AffineSpace_generic
 from sage.schemes.berkovich.berkovich_cp_element import (Berkovich_Element_Cp_Affine,
                                                          Berkovich_Element_Cp_Projective)
-from sage.structure.parent import Parent
-from sage.schemes.affine.affine_space import AffineSpace_generic
 from sage.schemes.projective.projective_space import ProjectiveSpace_ring, ProjectiveSpace
+from sage.structure.parent import Parent
 from sage.structure.unique_representation import UniqueRepresentation
-from sage.categories.number_fields import NumberFields
-import sage.rings.abc
-from sage.rings.integer_ring import ZZ
-from sage.rings.rational_field import QQ
-from sage.rings.number_field.number_field_ideal import NumberFieldFractionalIdeal
-from sage.categories.topological_spaces import TopologicalSpaces
+
+lazy_import('sage.rings.number_field.number_field_ideal', 'NumberFieldFractionalIdeal')
 
 
 def is_Berkovich(space) -> bool:

src/sage/schemes/curves/affine_curve.py

@@ -124,26 +124,21 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
-import sage.rings.abc
+from builtins import sum as add
 
-from sage.misc.lazy_attribute import lazy_attribute
-from sage.misc.cachefunc import cached_method
+import sage.rings.abc
 
 from sage.arith.misc import binomial
-from sage.interfaces.singular import singular
-from builtins import sum as add
-
 from sage.categories.fields import Fields
 from sage.categories.finite_fields import FiniteFields
 from sage.categories.homset import Hom, End, hom
 from sage.categories.number_fields import NumberFields
-
 from sage.matrix.constructor import matrix
-
+from sage.misc.cachefunc import cached_method
+from sage.misc.lazy_attribute import lazy_attribute
 from sage.misc.lazy_import import lazy_import
-
+from sage.rings.infinity import infinity
 from sage.rings.polynomial.multi_polynomial_element import degree_lowest_rational_function
-from sage.rings.number_field.number_field import NumberField
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 from sage.rings.rational_field import RationalField
 from sage.rings.infinity import infinity
@@ -152,6 +147,10 @@
 from sage.schemes.affine.affine_subscheme import (AlgebraicScheme_subscheme_affine,
                                                   AlgebraicScheme_subscheme_affine_field)
 
+lazy_import('sage.interfaces.singular', 'singular')
+lazy_import('sage.rings.number_field.number_field', 'NumberField')
+lazy_import('sage.rings.qqbar', 'number_field_elements_from_algebraics')
+
 from .curve import Curve_generic
 
 from .point import (AffineCurvePoint_field,
@@ -164,8 +163,6 @@
 
 from .closed_point import IntegralAffineCurveClosedPoint
 
-lazy_import('sage.rings.qqbar', 'number_field_elements_from_algebraics')
-
 
 class AffineCurve(Curve_generic, AlgebraicScheme_subscheme_affine):
     """

src/sage/schemes/curves/projective_curve.py

@@ -136,32 +136,30 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
-from sage.misc.lazy_attribute import lazy_attribute
-from sage.misc.cachefunc import cached_method
+from builtins import sum as add
 
 from sage.categories.fields import Fields
 from sage.categories.homset import hom, Hom, End
 from sage.categories.number_fields import NumberFields
-
-from sage.interfaces.singular import singular
 from sage.matrix.constructor import matrix
-from builtins import sum as add
+from sage.misc.cachefunc import cached_method
+from sage.misc.lazy_attribute import lazy_attribute
+from sage.misc.lazy_import import lazy_import
+from sage.misc.persist import register_unpickle_override
 from sage.misc.sage_eval import sage_eval
-
-from sage.rings.polynomial.multi_polynomial_element import degree_lowest_rational_function
+from sage.rings.integer import Integer
 from sage.rings.integer_ring import IntegerRing
-from sage.rings.number_field.number_field import NumberField
+from sage.rings.polynomial.multi_polynomial_element import degree_lowest_rational_function
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-from sage.rings.qqbar import (number_field_elements_from_algebraics,
-                              QQbar)
 from sage.rings.rational_field import RationalField
-from sage.rings.integer import Integer
-
 from sage.schemes.projective.projective_space import ProjectiveSpace, ProjectiveSpace_ring
-
 from sage.schemes.projective.projective_subscheme import (AlgebraicScheme_subscheme_projective,
                                                           AlgebraicScheme_subscheme_projective_field)
 
+lazy_import('sage.interfaces.singular', 'singular')
+lazy_import('sage.rings.number_field.number_field', 'NumberField')
+lazy_import('sage.rings.qqbar', ['number_field_elements_from_algebraics', 'QQbar'])
+
 from .curve import Curve_generic
 
 from .point import (ProjectiveCurvePoint_field,
@@ -173,7 +171,6 @@
                     IntegralProjectivePlaneCurvePoint_finite_field)
 
 from .closed_point import IntegralProjectiveCurveClosedPoint
-from sage.misc.persist import register_unpickle_override
 
 
 class ProjectiveCurve(Curve_generic, AlgebraicScheme_subscheme_projective):

src/sage/schemes/curves/zariski_vankampen.py

@@ -42,9 +42,10 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 import itertools
-from copy import copy
 
+from copy import copy
 from itertools import combinations
+
 from sage.combinat.permutation import Permutation
 from sage.functions.generalized import sign
 from sage.geometry.voronoi_diagram import VoronoiDiagram
@@ -53,10 +54,10 @@
 from sage.groups.finitely_presented import wrap_FpGroup
 from sage.groups.free_group import FreeGroup
 from sage.groups.perm_gps.permgroup_named import SymmetricGroup
-from sage.libs.braiding import leftnormalform, rightnormalform
 from sage.matrix.constructor import matrix
 from sage.misc.cachefunc import cached_function
 from sage.misc.flatten import flatten
+from sage.misc.lazy_import import lazy_import
 from sage.misc.misc_c import prod
 from sage.parallel.decorate import parallel
 from sage.rings.complex_interval_field import ComplexIntervalField
@@ -67,7 +68,9 @@
 from sage.rings.qqbar import QQbar
 from sage.rings.rational_field import QQ
 from sage.rings.real_mpfr import RealField
-from .constructor import Curve
+from sage.schemes.curves.constructor import Curve
+
+lazy_import('sage.libs.braiding', ['leftnormalform', 'rightnormalform'])
 
 roots_interval_cache = {}
 

src/sage/schemes/cyclic_covers/charpoly_frobenius.py

@@ -8,7 +8,8 @@
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 from sage.rings.integer_ring import ZZ
-from sage.functions.log import log
+from sage.misc.lazy_import import lazy_import
+lazy_import("sage.functions.log", "log")
 
 
 def charpoly_frobenius(frob_matrix, charpoly_prec, p, weight, a=1, known_factor=[1]):

src/sage/schemes/cyclic_covers/cycliccover_finite_field.py

@@ -66,19 +66,20 @@
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
-
 from sage.arith.misc import euler_phi
-from sage.functions.other import ceil, binomial, floor
-from sage.functions.log import log
-from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-from sage.rings.power_series_ring import PowerSeriesRing
-from sage.rings.padics.factory import Zp, Zq, Qq
-from sage.rings.integer_ring import ZZ
-from sage.rings.finite_rings.integer_mod_ring import IntegerModRing
 from sage.matrix.constructor import matrix, zero_matrix
-from sage.modules.free_module_element import vector
-from sage.schemes.hyperelliptic_curves.hypellfrob import interval_products
 from sage.misc.cachefunc import cached_method
+from sage.misc.lazy_import import lazy_import
+from sage.modules.free_module_element import vector
+from sage.rings.finite_rings.integer_mod_ring import IntegerModRing
+from sage.rings.integer_ring import ZZ
+from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+from sage.rings.power_series_ring import PowerSeriesRing
+
+lazy_import("sage.functions.log", "log")
+lazy_import("sage.functions.other", ["ceil", "binomial", "floor"])
+lazy_import('sage.rings.padics.factory', ['Zp', 'Zq', 'Qq'])
+lazy_import('sage.schemes.hyperelliptic_curves.hypellfrob', 'interval_products')
 
 from .charpoly_frobenius import charpoly_frobenius
 from . import cycliccover_generic

src/sage/schemes/elliptic_curves/BSD.py

@@ -1,11 +1,13 @@
 "Birch and Swinnerton-Dyer formulas"
 
 from sage.arith.misc import prime_divisors
+from sage.misc.lazy_import import lazy_import
+from sage.rings.infinity import Infinity
 from sage.rings.integer_ring import ZZ
 from sage.rings.rational_field import QQ
-from sage.rings.infinity import Infinity
-from sage.rings.number_field.number_field import QuadraticField
-from sage.functions.other import ceil
+
+lazy_import("sage.functions.other", "ceil")
+lazy_import("sage.rings.number_field.number_field", "QuadraticField")
 
 
 class BSD_data:

src/sage/schemes/elliptic_curves/cm.py

@@ -34,12 +34,10 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
-from sage.interfaces.magma import magma
 from sage.rings.integer import Integer
 from sage.rings.rational_field import QQ
 from sage.rings.integer_ring import ZZ
 from sage.rings.integer_ring import IntegerRing
-from sage.rings.number_field.number_field import is_fundamental_discriminant
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 
 from sage.misc.cachefunc import cached_function
@@ -121,6 +119,7 @@ def hilbert_class_polynomial(D, algorithm=None):
         return sage.libs.arb.arith.hilbert_class_polynomial(D)
 
     if algorithm == "magma":
+        from sage.interfaces.magma import magma
         magma.eval("R<x> := PolynomialRing(IntegerRing())")
         f = str(magma.eval("HilbertClassPolynomial(%s)" % D))
         return IntegerRing()['x'](f)

src/sage/schemes/elliptic_curves/ell_curve_isogeny.py

@@ -2903,7 +2903,7 @@ def scaling_factor(self):
 
         Check for :issue:`36638`::
 
-            sage: phi.scaling_factor().parent()  # needs sage.rings.finite_rings
+            sage: phi.scaling_factor().parent()                                         # needs sage.rings.finite_rings
             Finite Field in z2 of size 257^2
 
         ALGORITHM: The "inner" isogeny is normalized by construction,

src/sage/schemes/elliptic_curves/ell_generic.py

@@ -272,7 +272,7 @@ def _latex_(self):
         """
         Internal function. Return a latex string for this elliptic curve.
 
-        Users will normally use latex() instead.
+        Users will normally use :func:`latex` instead.
 
         EXAMPLES::