def construct_from_generators_indices(generators, filtration, base_ring, check):
"""
Construct a filtered vector space.
INPUT:
- ``generators`` -- a list/tuple/iterable of vectors, or something
convertible to them. The generators spanning various
subspaces.
- ``filtration`` -- a list or iterable of filtration steps. Each
filtration step is a pair ``(degree, ray_indices)``. The
``ray_indices`` are a list or iterable of ray indices, which
span a subspace of the vector space. The integer ``degree``
stipulates that all filtration steps of degree higher or equal
than ``degree`` (up to the next filtration step) are said
subspace.
EXAMPLES::
sage: from sage.modules.filtered_vector_space import construct_from_generators_indices
sage: gens = [(1,0), (0,1), (-1,-1)]
sage: V = construct_from_generators_indices(gens, {1:[0,1], 3:[1]}, QQ, True); V
QQ^2 >= QQ^1 >= QQ^1 >= 0
TESTS::
sage: gens = [(int(1),int(0)), (0,1), (-1,-1)]
sage: construct_from_generators_indices(iter(gens), {int(0):[0, int(1)], 2:[2]}, QQ, True)
QQ^2 >= QQ^1 >= QQ^1 >= 0
"""
# normalize generators
generators = map(list, generators)
# deduce dimension
if len(generators) == 0:
dim = ZZ(0)
else:
dim = ZZ(len(generators[0]))
ambient = VectorSpace(base_ring, dim)
# complete generators to a generating set
if matrix(base_ring, generators).rank() < dim:
complement = ambient.span(generators).complement()
generators = generators + list(complement.gens())
# normalize generators II
generators = tuple(ambient(v) for v in generators)
for v in generators:
v.set_immutable()
# normalize filtration data
normalized = dict()
for deg, gens in iteritems(filtration):
deg = normalize_degree(deg)
gens = map(ZZ, gens)
if any(i < 0 or i >= len(generators) for i in gens):
raise ValueError('generator index out of bounds')
normalized[deg] = tuple(sorted(gens))
try:
del normalized[minus_infinity]
except KeyError:
pass
filtration = normalized
return FilteredVectorSpace_class(base_ring, dim, generators, filtration, check=check)
def _possible_normalizers(E, SA):
r"""Find a list containing all primes `l` such that the Galois image at `l`
is contained in the normalizer of a Cartan subgroup, such that the
corresponding quadratic character is ramified only at the given primes.
INPUT:
- ``E`` - EllipticCurve - over a number field K.
- ``SA`` - list - a list of primes of K.
OUTPUT:
- list - A list of primes, which contains all primes `l` such that the
Galois image at `l` is contained in the normalizer of a Cartan
subgroup, such that the corresponding quadratic character is
ramified only at primes in SA.
- If `E` has geometric CM that is not defined over its ground field, a
ValueError is raised.
EXAMPLES::
sage: E = EllipticCurve([0,0,0,-56,4848])
sage: 5 in sage.schemes.elliptic_curves.gal_reps_number_field._possible_normalizers(E, [ZZ.ideal(2)])
True
"""
E = _over_numberfield(E)
K = E.base_field()
SA = [K.ideal(I.gens()) for I in SA]
x = K['x'].gen()
selmer_group = K.selmer_group(SA, 2) # Generators of the selmer group.
if selmer_group == []:
return []
V = VectorSpace(GF(2), len(selmer_group))
# We think of this as the character group of the selmer group.
traces_list = []
W = V.zero_subspace()
deg_one_primes = K.primes_of_degree_one_iter()
while W.dimension() < V.dimension() - 1:
P = deg_one_primes.next()
k = P.residue_field()
defines_valid_character = True
# A prime P defines a quadratic residue character
# on the Selmer group. This variable will be set
# to zero if any elements of the selmer group are
# zero mod P (i.e. the character is ramified).
splitting_vector = [] # This will be the values of this
# character on the generators of the Selmer group.
for a in selmer_group:
abar = k(a)
if abar == 0:
# Ramification.
defines_valid_character = False
break
if abar.is_square():
splitting_vector.append(GF(2)(0))
else:
splitting_vector.append(GF(2)(1))
if not defines_valid_character:
continue
if splitting_vector in W:
continue
try:
Etilde = E.change_ring(k)
except ArithmeticError: # Bad reduction.
continue
tr = Etilde.trace_of_frobenius()
if tr == 0:
continue
traces_list.append(tr)
W = W + V.span([splitting_vector])
bad_primes = set([])
for i in traces_list:
for p in i.prime_factors():
bad_primes.add(p)
# We find the unique vector v in V orthogonal to W:
v = W.matrix().transpose().kernel().basis()[0]
# We find the element a of the selmer group corresponding to v:
a = 1
for i in xrange(len(selmer_group)):
if v[i] == 1:
a *= selmer_group[i]
# Since we've already included the above bad primes, we can assume
# that the quadratic character corresponding to the exceptional primes
# we're looking for is given by mapping into Gal(K[\sqrt{a}]/K).
patience = 5 * K.degree()
# Number of Frobenius elements to check before suspecting that E
# has CM and computing the set of CM j-invariants of K to check.
# TODO: Is this the best value for this parameter?
while True:
P = deg_one_primes.next()
k = P.residue_field()
if not k(a).is_square():
try:
tr = E.change_ring(k).trace_of_frobenius()
except ArithmeticError: # Bad reduction.
continue
if tr == 0:
patience -= 1
if patience == 0:
# We suspect E has CM, so we check:
if E.j_invariant() in cm_j_invariants(K):
raise ValueError("The curve E should not have CM.")
else:
for p in tr.prime_factors():
bad_primes.add(p)
bad_primes = sorted(bad_primes)
return bad_primes
def construct_from_generators_indices(generators, filtration, base_ring,
check):
"""
Construct a filtered vector space.
INPUT:
- ``generators`` -- a list/tuple/iterable of vectors, or something
convertible to them. The generators spanning various
subspaces.
- ``filtration`` -- a list or iterable of filtration steps. Each
filtration step is a pair ``(degree, ray_indices)``. The
``ray_indices`` are a list or iterable of ray indices, which
span a subspace of the vector space. The integer ``degree``
stipulates that all filtration steps of degree higher or equal
than ``degree`` (up to the next filtration step) are said
subspace.
EXAMPLES::
sage: from sage.modules.filtered_vector_space import construct_from_generators_indices
sage: gens = [(1,0), (0,1), (-1,-1)]
sage: V = construct_from_generators_indices(gens, {1:[0,1], 3:[1]}, QQ, True); V
QQ^2 >= QQ^1 >= QQ^1 >= 0
TESTS::
sage: gens = [(int(1),int(0)), (0,1), (-1,-1)]
sage: construct_from_generators_indices(iter(gens), {int(0):[0, int(1)], 2:[2]}, QQ, True)
QQ^2 >= QQ^1 >= QQ^1 >= 0
"""
# normalize generators
generators = [list(g) for g in generators]
# deduce dimension
if len(generators) == 0:
dim = ZZ(0)
else:
dim = ZZ(len(generators[0]))
ambient = VectorSpace(base_ring, dim)
# complete generators to a generating set
if matrix(base_ring, generators).rank() < dim:
complement = ambient.span(generators).complement()
generators = generators + list(complement.gens())
# normalize generators II
generators = tuple(ambient(v) for v in generators)
for v in generators:
v.set_immutable()
# normalize filtration data
normalized = dict()
for deg, gens in filtration.items():
deg = normalize_degree(deg)
gens = [ZZ(i) for i in gens]
if any(i < 0 or i >= len(generators) for i in gens):
raise ValueError('generator index out of bounds')
normalized[deg] = tuple(sorted(gens))
try:
del normalized[minus_infinity]
except KeyError:
pass
filtration = normalized
return FilteredVectorSpace_class(base_ring,
dim,
generators,
filtration,
check=check)
def _possible_normalizers(E, SA):
r"""Find a list containing all primes `l` such that the Galois image at `l`
is contained in the normalizer of a Cartan subgroup, such that the
corresponding quadratic character is ramified only at the given primes.
INPUT:
- ``E`` - EllipticCurve - over a number field K.
- ``SA`` - list - a list of primes of K.
OUTPUT:
- list - A list of primes, which contains all primes `l` such that the
Galois image at `l` is contained in the normalizer of a Cartan
subgroup, such that the corresponding quadratic character is
ramified only at primes in SA.
- If `E` has geometric CM that is not defined over its ground field, a
ValueError is raised.
EXAMPLES::
sage: E = EllipticCurve([0,0,0,-56,4848])
sage: 5 in sage.schemes.elliptic_curves.gal_reps_number_field._possible_normalizers(E, [ZZ.ideal(2)])
True
"""
E = _over_numberfield(E)
K = E.base_field()
SA = [K.ideal(I.gens()) for I in SA]
x = K['x'].gen()
selmer_group = K.selmer_group(SA, 2) # Generators of the selmer group.
if selmer_group == []:
return []
V = VectorSpace(GF(2), len(selmer_group))
# We think of this as the character group of the selmer group.
traces_list = []
W = V.zero_subspace()
deg_one_primes = K.primes_of_degree_one_iter()
while W.dimension() < V.dimension() - 1:
P = next(deg_one_primes)
k = P.residue_field()
defines_valid_character = True
# A prime P defines a quadratic residue character
# on the Selmer group. This variable will be set
# to zero if any elements of the selmer group are
# zero mod P (i.e. the character is ramified).
splitting_vector = [] # This will be the values of this
# character on the generators of the Selmer group.
for a in selmer_group:
abar = k(a)
if abar == 0:
# Ramification.
defines_valid_character = False
break
if abar.is_square():
splitting_vector.append(GF(2)(0))
else:
splitting_vector.append(GF(2)(1))
if not defines_valid_character:
continue
if splitting_vector in W:
continue
try:
Etilde = E.change_ring(k)
except ArithmeticError: # Bad reduction.
continue
tr = Etilde.trace_of_frobenius()
if tr == 0:
continue
traces_list.append(tr)
W = W + V.span([splitting_vector])
bad_primes = set([])
for i in traces_list:
for p in i.prime_factors():
bad_primes.add(p)
# We find the unique vector v in V orthogonal to W:
v = W.matrix().transpose().kernel().basis()[0]
# We find the element a of the selmer group corresponding to v:
a = 1
for i in xrange(len(selmer_group)):
if v[i] == 1:
a *= selmer_group[i]
# Since we've already included the above bad primes, we can assume
# that the quadratic character corresponding to the exceptional primes
# we're looking for is given by mapping into Gal(K[\sqrt{a}]/K).
patience = 5 * K.degree()
# Number of Frobenius elements to check before suspecting that E
# has CM and computing the set of CM j-invariants of K to check.
# TODO: Is this the best value for this parameter?
while True:
P = next(deg_one_primes)
k = P.residue_field()
if not k(a).is_square():
try:
tr = E.change_ring(k).trace_of_frobenius()
except ArithmeticError: # Bad reduction.
continue
if tr == 0:
patience -= 1
if patience == 0:
# We suspect E has CM, so we check:
if E.j_invariant() in cm_j_invariants(K):
raise ValueError("The curve E should not have CM.")
else:
for p in tr.prime_factors():
bad_primes.add(p)
bad_primes = sorted(bad_primes)
return bad_primes
