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
