sagemathgh-36870: some ruff auto-fixes in schemes + error links in doc

    
Some fixes made by ruff in the schemes folder (codes C4, UP02, UP009)

plus adding manually some links to errors in the doc

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
    
URL: sagemath#36870
Reported by: Frédéric Chapoton
Reviewer(s): Matthias Köppe

src/sage/schemes/affine/affine_space.py

@@ -270,10 +270,10 @@ def rational_points(self, F=None):
         if F is None:
             if not isinstance(self.base_ring(), FiniteField):
                 raise TypeError("base ring (= %s) must be a finite field" % self.base_ring())
-            return [P for P in self]
+            return list(self)
         elif not isinstance(F, FiniteField):
             raise TypeError("second argument (= %s) must be a finite field" % F)
-        return [P for P in self.base_extend(F)]
+        return list(self.base_extend(F))
 
     def __eq__(self, right):
         """
@@ -1226,7 +1226,7 @@ def translation(self, p, q=None):
         if q is not None:
             v = [cp - cq for cp, cq in zip(p, q)]
         else:
-            v = [cp for cp in p]
+            v = list(p)
 
         return self._morphism(self.Hom(self), [x - c for x, c in zip(gens, v)])
 

src/sage/schemes/curves/affine_curve.py

@@ -684,13 +684,13 @@ def tangents(self, P, factor=True):
             if t > 0:
                 fact.append(vars[0])
                 # divide T by that power of vars[0]
-                T = self.ambient_space().coordinate_ring()(dict([((v[0] - t, v[1]), h) for (v, h) in T.dict().items()]))
+                T = self.ambient_space().coordinate_ring()({(v[0] - t, v[1]): h for (v, h) in T.dict().items()})
             t = min([e[1] for e in T.exponents()])
             # vars[1] divides T
             if t > 0:
                 fact.append(vars[1])
                 # divide T by that power of vars[1]
-                T = self.ambient_space().coordinate_ring()(dict([((v[0], v[1] - t), h) for (v, h) in T.dict().items()]))
+                T = self.ambient_space().coordinate_ring()({(v[0], v[1] - t): h for (v, h) in T.dict().items()})
             # T is homogeneous in var[0], var[1] if nonconstant, so dehomogenize
             if T not in self.base_ring():
                 if T.degree(vars[0]) > 0:
@@ -1032,7 +1032,7 @@ def projection(self, indices, AS=None):
             C = AA2.curve(G)
         except (TypeError, ValueError):
             C = AA2.subscheme(G)
-        return tuple([psi, C])
+        return (psi, C)
 
     def plane_projection(self, AP=None):
         r"""
@@ -1397,7 +1397,7 @@ def blowup(self, P=None):
             homvars.insert(i, 1)
             coords = [(p_A.gens()[0] - P[i])*homvars[j] + P[j] for j in range(n)]
             proj_maps.append(H(coords))
-        return tuple([tuple(patches), tuple(t_maps), tuple(proj_maps)])
+        return (tuple(patches), tuple(t_maps), tuple(proj_maps))
 
     def resolution_of_singularities(self, extend=False):
         r"""
@@ -1688,7 +1688,7 @@ def extension(self):
         patches = [res[i][0] for i in range(len(res))]
         t_maps = [tuple(res[i][1]) for i in range(len(res))]
         p_maps = [res[i][2] for i in range(len(res))]
-        return tuple([tuple(patches), tuple(t_maps), tuple(p_maps)])
+        return (tuple(patches), tuple(t_maps), tuple(p_maps))
 
     def tangent_line(self, p):
         """
@@ -2600,7 +2600,7 @@ def places_at_infinity(self):
             sage: C.places_at_infinity()
             [Place (1/x, 1/x*z^2)]
         """
-        return list(set(p for f in self._coordinate_functions if f for p in f.poles()))
+        return list({p for f in self._coordinate_functions if f for p in f.poles()})
 
     def places_on(self, point):
         """

src/sage/schemes/curves/projective_curve.py

@@ -495,7 +495,7 @@ def projection(self, P=None, PS=None):
         phi = K(l)
         G = [phi(f) for f in J.gens()]
         C = PP2.curve(G)
-        return tuple([psi, C])
+        return (psi, C)
 
     def plane_projection(self, PP=None):
         r"""
@@ -577,7 +577,7 @@ def plane_projection(self, PP=None):
             psi = K(phi.defining_polynomials())
             H = Hom(self, L[1].ambient_space())
             phi = H([psi(L[0].defining_polynomials()[i]) for i in range(len(L[0].defining_polynomials()))])
-        return tuple([phi, C])
+        return (phi, C)
 
 
 class ProjectivePlaneCurve(ProjectiveCurve):
@@ -1104,7 +1104,7 @@ def quadratic_transform(self):
         T = []
         for item in G.dict().items():
             tup = tuple([item[0][i] - degs[i] for i in range(len(L))])
-            T.append(tuple([tup, item[1]]))
+            T.append((tup, item[1]))
         G = R(dict(T))
         H = Hom(self, PP.curve(G))
         phi = H(coords)

src/sage/schemes/curves/zariski_vankampen.py

@@ -68,7 +68,7 @@
 from sage.rings.real_mpfr import RealField
 # from sage.sets.set import Set
 
-roots_interval_cache = dict()
+roots_interval_cache = {}
 
 
 def braid_from_piecewise(strands):
@@ -793,7 +793,7 @@ def populate_roots_interval_cache(inputs):
 
 
 @parallel
-def braid_in_segment(glist, x0, x1, precision=dict()):
+def braid_in_segment(glist, x0, x1, precision={}):
     """
     Return the braid formed by the `y` roots of ``f`` when `x` moves
     from ``x0`` to ``x1``.
@@ -1711,7 +1711,7 @@ def fundamental_group_arrangement(flist, simplified=True, projective=False, puis
             f = prod(flist1)
     if len(flist1) == 0:
         bm = []
-        dic = dict()
+        dic = {}
     else:
         bm, dic = braid_monodromy(f, flist1)
     g = fundamental_group_from_braid_mon(bm, degree=d, simplified=False, projective=projective, puiseux=puiseux)
@@ -1721,7 +1721,7 @@ def fundamental_group_arrangement(flist, simplified=True, projective=False, puis
         hom = g.hom(codomain=g, im_gens=list(g.gens()), check=False)
     g1 = hom.codomain()
     if len(flist) == 0:
-        return (g1, dict())
+        return (g1, {})
     dic1 = {}
     for i in range(len(flist1)):
         L = [j1 for j1 in dic.keys() if dic[j1] == i]

src/sage/schemes/cyclic_covers/cycliccover_finite_field.py

@@ -921,7 +921,7 @@ def _initialize_fat_vertical(self, s0, max_upper_target):
         L = floor((max_upper_target - self._epsilon) / self._p) + 1
         if s0 not in self._vertical_fat_s:
             (m0, m1), (M0, M1) = self._vertical_matrix_reduction(s0)
-            D0, D1 = map(lambda y: matrix(self._Zq, [y]), [m0, m1])
+            D0, D1 = (matrix(self._Zq, [y]) for y in [m0, m1])
             targets = [0] * (2 * L)
             for l in reversed(range(L)):
                 targets[2 * l] = max_upper_target - self._p * (L - l)
@@ -1268,7 +1268,7 @@ def _denominator():
                 f = x ** self._delta - lc
                 L = f.splitting_field("a")
                 roots = [r for r, _ in f.change_ring(L).roots()]
-                roots_dict = dict([(r, i) for i, r in enumerate(roots)])
+                roots_dict = {r: i for i, r in enumerate(roots)}
                 rootsfrob = [L.frobenius_endomorphism(self._Fq.degree())(r) for r in roots]
                 m = zero_matrix(len(roots))
                 for i, r in enumerate(roots):

src/sage/schemes/elliptic_curves/BSD.py

@@ -1,4 +1,3 @@
-# -*- coding: utf-8 -*-
 "Birch and Swinnerton-Dyer formulas"
 
 from sage.arith.misc import prime_divisors

src/sage/schemes/elliptic_curves/Qcurves.py

@@ -1,4 +1,3 @@
-# -*- coding: utf-8 -*-
 r"""
 Testing whether elliptic curves over number fields are `\QQ`-curves
 

src/sage/schemes/elliptic_curves/cm.py

@@ -973,7 +973,7 @@ def is_cm_j_invariant(j, algorithm='CremonaSutherland', method=None):
 
     if j in ZZ:
         j = ZZ(j)
-        table = dict([(jj,(d,f)) for d,f,jj in cm_j_invariants_and_orders(QQ)])
+        table = {jj: (d,f) for d,f,jj in cm_j_invariants_and_orders(QQ)}
         if j in table:
             return True, table[j]
         return False, None

src/sage/schemes/elliptic_curves/ec_database.py

@@ -136,7 +136,7 @@ def rank(self, rank, tors=0, n=10, labels=False):
         data = os.path.join(ELLCURVE_DATA_DIR, 'rank%s' % rank)
         try:
             f = open(data)
-        except IOError:
+        except OSError:
             return []
         v = []
         tors = int(tors)

src/sage/schemes/elliptic_curves/ell_curve_isogeny.py

@@ -1,4 +1,3 @@
-# -*- coding: utf-8 -*-
 r"""
 Isogenies
 
@@ -1953,7 +1952,7 @@ def __sort_kernel_list(self):
         """
         a1, a2, a3, a4, _ = self._domain.a_invariants()
 
-        self.__kernel_mod_sign = dict()
+        self.__kernel_mod_sign = {}
         v = w = 0
 
         for Q in self.__kernel_list:

src/sage/schemes/elliptic_curves/ell_generic.py

@@ -169,7 +169,7 @@ def __init__(self, K, ainvs):
             - (x**3 + a2*x**2*z + a4*x*z**2 + a6*z**3)
         plane_curve.ProjectivePlaneCurve.__init__(self, PP, f)
 
-        self.__divpolys = (dict(), dict(), dict())
+        self.__divpolys = ({}, {}, {})
 
         # See #1975: we deliberately set the class to
         # EllipticCurvePoint_finite_field for finite rings, so that we
@@ -1732,7 +1732,7 @@ def division_polynomial_0(self, n, x=None):
             x = polygen(self.base_ring())
         else:
             # For other inputs, we use a temporary cache.
-            cache = dict()
+            cache = {}
 
         b2, b4, b6, b8 = self.b_invariants()
 
@@ -2119,7 +2119,7 @@ def _multiple_x_numerator(self, n, x=None):
             try:
                 cache = self.__mulxnums
             except AttributeError:
-                cache = self.__mulxnums = dict()
+                cache = self.__mulxnums = {}
             try:
                 return cache[n]
             except KeyError:
@@ -2216,7 +2216,7 @@ def _multiple_x_denominator(self, n, x=None):
             try:
                 cache = self.__mulxdens
             except AttributeError:
-                cache = self.__mulxdens = dict()
+                cache = self.__mulxdens = {}
             try:
                 return cache[n]
             except KeyError:

src/sage/schemes/elliptic_curves/ell_local_data.py

@@ -1,4 +1,3 @@
-# -*- coding: utf-8 -*-
 r"""
 Local data for elliptic curves over number fields
 

src/sage/schemes/elliptic_curves/ell_modular_symbols.py

@@ -1,4 +1,3 @@
-# -*- coding: utf-8 -*-
 r"""
 Modular symbols attached to elliptic curves over `\QQ`
 

src/sage/schemes/elliptic_curves/ell_number_field.py

@@ -672,9 +672,8 @@ def global_integral_model(self):
         """
         K = self.base_field()
         ai = self.a_invariants()
-        Ps = set(ff[0]
-                 for a in ai if not a.is_integral()
-                 for ff in a.denominator_ideal().factor())
+        Ps = {ff[0] for a in ai if not a.is_integral()
+              for ff in a.denominator_ideal().factor()}
         for P in Ps:
             pi = K.uniformizer(P, 'positive')
             e = min((ai[i].valuation(P)/[1,2,3,4,6][i])

src/sage/schemes/elliptic_curves/ell_point.py

@@ -1,4 +1,3 @@
-# -*- coding: utf-8 -*-
 r"""
 Points on elliptic curves
 

src/sage/schemes/elliptic_curves/ell_rational_field.py

@@ -1,4 +1,3 @@
-# -*- coding: utf-8 -*-
 r"""
 Elliptic curves over the rational numbers
 
@@ -1559,10 +1558,11 @@ def analytic_rank(self, algorithm="pari", leading_coefficient=False):
             return self.analytic_rank_upper_bound()
         elif algorithm == 'all':
             if leading_coefficient:
-                S = set([self.analytic_rank('pari', True)])
+                S = {self.analytic_rank('pari', True)}
             else:
-                S = set([self.analytic_rank('pari'),
-                    self.analytic_rank('rubinstein'), self.analytic_rank('sympow')])
+                S = {self.analytic_rank('pari'),
+                     self.analytic_rank('rubinstein'),
+                     self.analytic_rank('sympow')}
             if len(S) != 1:
                 raise RuntimeError("Bug in analytic_rank; algorithms don't agree! (E=%s)" % self)
             return list(S)[0]
@@ -5347,10 +5347,10 @@ def _shortest_paths(self):
         # Take logs here since shortest path minimizes the *sum* of the weights -- not the product.
         M = M.parent()([a.log() if a else 0 for a in M.list()])
         G = Graph(M, format='weighted_adjacency_matrix')
-        G.set_vertices(dict([(v,isocls[v]) for v in G.vertices(sort=False)]))
+        G.set_vertices({v: isocls[v] for v in G.vertices(sort=False)})
         v = G.shortest_path_lengths(0, by_weight=True)
         # Now exponentiate and round to get degrees of isogenies
-        v = dict([(i, j.exp().round() if j else 0) for i,j in v.items()])
+        v = {i: j.exp().round() if j else 0 for i,j in v.items()}
         return isocls.curves, v
 
     def _multiple_of_degree_of_isogeny_to_optimal_curve(self):
@@ -6179,7 +6179,7 @@ def point_preprocessing(free,tor):
             subgroup of index 2.
             """
             r = len(free)
-            newfree = [Q for Q in free] # copy
+            newfree = list(free)  # copy
             tor_egg = [T for T in tor if not T.is_on_identity_component()]
             free_id = [P.is_on_identity_component() for P in free]
             if any(tor_egg):
@@ -6207,7 +6207,7 @@ def point_preprocessing(free,tor):
             int_points = [P for P in tors_points if not P.is_zero()]
             int_points = [P for P in int_points if P[0].is_integral()]
             if not both_signs:
-                xlist = set([P[0] for P in int_points])
+                xlist = {P[0] for P in int_points}
                 int_points = [self.lift_x(x) for x in xlist]
             int_points.sort()
             if verbose:
@@ -6800,8 +6800,8 @@ def S_integral_x_coords_with_abs_bounded_by(abs_bound):
             alpha = [(log_ab/R(log(p,e))).floor() for p in S]
             if all(alpha_i <= 1 for alpha_i in alpha): # so alpha_i must be 0 to satisfy that denominator is a square
                 int_abs_bound = abs_bound.floor()
-                return set(x for x in range(-int_abs_bound, int_abs_bound)
-                           if E.is_x_coord(x))
+                return {x for x in range(-int_abs_bound, int_abs_bound)
+                        if E.is_x_coord(x)}
             else:
                 xs = []
                 alpha_max_even = [y - y % 2 for y in alpha]
@@ -6849,7 +6849,7 @@ def S_integral_x_coords_with_abs_bounded_by(abs_bound):
             int_points = [P for P in tors_points if not P.is_zero()]
             int_points = [P for P in int_points if P[0].is_S_integral(S)]
             if not both_signs:
-                xlist = set([P[0] for P in int_points])
+                xlist = {P[0] for P in int_points}
                 int_points = [E.lift_x(x) for x in xlist]
             int_points.sort()
             if verbose:

src/sage/schemes/elliptic_curves/ell_tate_curve.py

@@ -1,4 +1,3 @@
-# -*- coding: utf-8 -*-
 r"""
 Tate's parametrisation of `p`-adic curves with multiplicative reduction
 

src/sage/schemes/elliptic_curves/ell_torsion.py

@@ -1,4 +1,3 @@
-# -*- coding: utf-8 -*-
 r"""
 Torsion subgroups of elliptic curves over number fields (including `\QQ`)
 

src/sage/schemes/elliptic_curves/ell_wp.py

@@ -1,4 +1,3 @@
-# -*- coding: utf-8 -*-
 r"""
 Weierstrass `\wp`-function for elliptic curves
 

src/sage/schemes/elliptic_curves/formal_group.py

@@ -1,4 +1,3 @@
-# -*- coding: utf-8 -*-
 r"""
 Formal groups of elliptic curves
 

src/sage/schemes/elliptic_curves/gal_reps.py

@@ -1,4 +1,3 @@
-# -*- coding: utf-8 -*-
 r"""
 Galois representations attached to elliptic curves
 

src/sage/schemes/elliptic_curves/gal_reps_number_field.py

@@ -860,7 +860,7 @@ def _semistable_reducible_primes(E, verbose=False):
 
     deg_one_primes = deg_one_primes_iter(K, principal_only=True)
 
-    bad_primes = set([]) # This will store the output.
+    bad_primes = set()  # This will store the output.
 
     # We find two primes (of distinct residue characteristics) which are
     # of degree 1, unramified in K/Q, and at which E has good reduction.
@@ -1098,7 +1098,7 @@ def _possible_normalizers(E, SA):
 
         W = W + V.span([splitting_vector])
 
-    bad_primes = set([])
+    bad_primes = set()
 
     for i in traces_list:
         for p in i.prime_factors():