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 _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
def _semistable_reducible_primes(E):
r"""Find a list containing all semistable primes l unramified in K/QQ
for which the Galois image for E could be reducible.
INPUT:
- ``E`` - EllipticCurve - over a number field.
OUTPUT: list - A list of primes, which contains all primes l unramified
in K/QQ, such that E is semistable at all primes lying
over l, and the Galois image at l is reducible. If E has
CM defined over its ground field, a ValueError is raised.
EXAMPLES::
sage: E = EllipticCurve([0, -1, 1, -10, -20]) # X_0(11)
sage: 5 in sage.schemes.elliptic_curves.gal_reps_number_field._semistable_reducible_primes(E)
True
"""
E = _over_numberfield(E)
K = E.base_field()
deg_one_primes = K.primes_of_degree_one_iter()
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.
# Both of these primes will give us a nontrivial divisibility constraint
# on the exceptional primes l. For both of these primes P, we precompute
# a generator and the trace of Frob_P^12.
precomp = []
last_char = 0 # The residue characteristic of the most recent prime.
while len(precomp) < 2:
P = deg_one_primes.next()
if not P.is_principal():
continue
det = P.norm()
if det == last_char:
continue
if P.ramification_index() != 1:
continue
try:
tr = E.change_ring(P.residue_field()).trace_of_frobenius()
except ArithmeticError: # Bad reduction at P.
continue
x = P.gens_reduced()[0]
precomp.append((x, _tr12(tr, det)))
last_char = det
x, tx = precomp[0]
y, ty = precomp[1]
Kgal = K.galois_closure('b')
maps = K.embeddings(Kgal)
for i in xrange(2 ** (K.degree() - 1)):
## We iterate through all possible characters. ##
# Here, if i = i_{l-1} i_{l-2} cdots i_1 i_0 in binary, then i
# corresponds to the character prod sigma_j^{i_j}.
phi1x = 1
phi2x = 1
phi1y = 1
phi2y = 1
# We compute the two algebraic characters at x and y:
for j in xrange(K.degree()):
if i % 2 == 1:
phi1x *= maps[j](x)
phi1y *= maps[j](y)
else:
phi2x *= maps[j](x)
phi2y *= maps[j](y)
i = int(i/2)
# Any prime with reducible image must divide both of:
gx = phi1x**12 + phi2x**12 - tx
gy = phi1y**12 + phi2y**12 - ty
if (gx != 0) or (gy != 0):
for prime in Integer(Kgal.ideal([gx, gy]).norm()).prime_factors():
bad_primes.add(prime)
continue
## It is possible that our curve has CM. ##
# Our character must be of the form Nm^K_F for an imaginary
# quadratic subfield F of K (which is the CM field if E has CM).
# We compute F:
a = (Integer(phi1x + phi2x)**2 - 4 * x.norm()).squarefree_part()
y = QQ['y'].gen()
F = NumberField(y**2 - a, 'a')
# Next, we turn K into relative number field over F.
K = K.relativize(F.embeddings(K)[0], 'b')
E = E.change_ring(K.structure()[1])
## We try to find a nontrivial divisibility condition. ##
patience = 5 * K.absolute_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()
if not P.is_principal():
continue
try:
tr = E.change_ring(P.residue_field()).trace_of_frobenius()
except ArithmeticError: # Bad reduction at P.
continue
x = P.gens_reduced()[0].norm(F)
div = (x**12).trace() - _tr12(tr, x.norm())
patience -= 1
if div != 0:
# We found our divisibility constraint.
for prime in Integer(div).prime_factors():
bad_primes.add(prime)
# Turn K back into an absolute number field.
E = E.change_ring(K.structure()[0])
K = K.structure()[0].codomain()
break
if patience == 0:
# We suspect that E has CM, so we check:
f = K.structure()[0]
if f(E.j_invariant()) in cm_j_invariants(f.codomain()):
raise ValueError("The curve E should not have CM.")
L = sorted(bad_primes)
return L
def _semistable_reducible_primes(E):
r"""Find a list containing all semistable primes l unramified in K/QQ
for which the Galois image for E could be reducible.
INPUT:
- ``E`` - EllipticCurve - over a number field.
OUTPUT:
A list of primes, which contains all primes `l` unramified in
`K/\mathbb{QQ}`, such that `E` is semistable at all primes lying
over `l`, and the Galois image at `l` is reducible. If `E` has CM
defined over its ground field, a ``ValueError`` is raised.
EXAMPLES::
sage: E = EllipticCurve([0, -1, 1, -10, -20]) # X_0(11)
sage: 5 in sage.schemes.elliptic_curves.gal_reps_number_field._semistable_reducible_primes(E)
True
"""
E = _over_numberfield(E)
K = E.base_field()
deg_one_primes = K.primes_of_degree_one_iter()
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.
# Both of these primes will give us a nontrivial divisibility constraint
# on the exceptional primes l. For both of these primes P, we precompute
# a generator and the trace of Frob_P^12.
precomp = []
last_char = 0 # The residue characteristic of the most recent prime.
while len(precomp) < 2:
P = next(deg_one_primes)
if not P.is_principal():
continue
det = P.norm()
if det == last_char:
continue
if P.ramification_index() != 1:
continue
try:
tr = E.change_ring(P.residue_field()).trace_of_frobenius()
except ArithmeticError: # Bad reduction at P.
continue
x = P.gens_reduced()[0]
precomp.append((x, _tr12(tr, det)))
last_char = det
x, tx = precomp[0]
y, ty = precomp[1]
Kgal = K.galois_closure('b')
maps = K.embeddings(Kgal)
for i in xrange(2**(K.degree() - 1)):
## We iterate through all possible characters. ##
# Here, if i = i_{l-1} i_{l-2} cdots i_1 i_0 in binary, then i
# corresponds to the character prod sigma_j^{i_j}.
phi1x = 1
phi2x = 1
phi1y = 1
phi2y = 1
# We compute the two algebraic characters at x and y:
for j in xrange(K.degree()):
if i % 2 == 1:
phi1x *= maps[j](x)
phi1y *= maps[j](y)
else:
phi2x *= maps[j](x)
phi2y *= maps[j](y)
i = int(i / 2)
# Any prime with reducible image must divide both of:
gx = phi1x**12 + phi2x**12 - tx
gy = phi1y**12 + phi2y**12 - ty
if (gx != 0) or (gy != 0):
for prime in Integer(Kgal.ideal([gx, gy]).norm()).prime_factors():
bad_primes.add(prime)
continue
## It is possible that our curve has CM. ##
# Our character must be of the form Nm^K_F for an imaginary
# quadratic subfield F of K (which is the CM field if E has CM).
# We compute F:
a = (Integer(phi1x + phi2x)**2 - 4 * x.norm()).squarefree_part()
y = QQ['y'].gen()
F = NumberField(y**2 - a, 'a')
# Next, we turn K into relative number field over F.
K = K.relativize(F.embeddings(K)[0], 'b')
E = E.change_ring(K.structure()[1])
## We try to find a nontrivial divisibility condition. ##
patience = 5 * K.absolute_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)
if not P.is_principal():
continue
try:
tr = E.change_ring(P.residue_field()).trace_of_frobenius()
except ArithmeticError: # Bad reduction at P.
continue
x = P.gens_reduced()[0].norm(F)
div = (x**12).trace() - _tr12(tr, x.norm())
patience -= 1
if div != 0:
# We found our divisibility constraint.
for prime in Integer(div).prime_factors():
bad_primes.add(prime)
# Turn K back into an absolute number field.
E = E.change_ring(K.structure()[0])
K = K.structure()[0].codomain()
break
if patience == 0:
# We suspect that E has CM, so we check:
f = K.structure()[0]
if f(E.j_invariant()) in cm_j_invariants(f.codomain()):
raise ValueError("The curve E should not have CM.")
L = sorted(bad_primes)
return L