def init_tensor_product(self, V, W):
"""
We are given two Galois representations and we
will return their tensor product.
"""
self.original_object = V.original_object + W.original_object
self.object_type = "tensorproduct"
self.V1 = V
self.V2 = W
self.dim = V.dim * W.dim
self.motivic_weight = V.motivic_weight + W.motivic_weight
self.langlands = False # status 2014 :)
self.besancon_bound = min(V.besancon_bound, W.besancon_bound)
bad2 = ZZ(W.conductor).prime_factors()
bad_primes = [x for x in ZZ(V.conductor).prime_factors() if x in bad2]
for p in bad_primes:
if (p not in V.bad_semistable_primes
and p not in W.bad_semistable_primes):
# this condition above only applies to the current type of objects
# for general reps we would have to test the lines below
# to be certain that the formulae are correct.
#if ((p not in V.bad_semistable_primes or p not in W.bad_pot_good) and
#(p not in W.bad_semistable_primes or p not in V.bad_pot_good) and
#(p not in V.bad_semistable_primes or p not in W.bad_semistable_primes)):
raise NotImplementedError(
"Currently tensor products of Galois representations are only implemented under some conditions.",
"The behaviour at %d is too wild (both factors must be semistable)."
% p)
# check for the possibily of getting poles
if V.weight == W.weight and V.conductor == W.conductor:
Vans = V.algebraic_coefficients(50)
Wans = W.algebraic_coefficients(50)
CC = ComplexField()
if ((Vans[2] in ZZ and Wans[2] in ZZ
and all(Vans[n] == Wans[n] for n in range(1, 50)))
or all(CC(Vans[n]) == CC(Wans[n]) for n in range(1, 50))):
raise NotImplementedError(
"It seems you are asking to tensor a " +
"Galois representation with its dual " +
"which results in the L-function having " +
"a pole. This is not implemented here.")
scommon = [
x for x in V.bad_semistable_primes if x in W.bad_semistable_primes
]
N = W.conductor**V.dim
N *= V.conductor**W.dim
for p in bad_primes:
n1_tame = V.dim - V.local_euler_factor(p).degree()
n2_tame = W.dim - W.local_euler_factor(p).degree()
nn = n1_tame * n2_tame
N = N // p**nn
if p in scommon: # both are degree 1 in this case
N = N // p
self.conductor = N
h1 = selberg_to_hodge(V.motivic_weight, V.mu_fe, V.nu_fe)
h2 = selberg_to_hodge(W.motivic_weight, W.mu_fe, W.nu_fe)
h = tensor_hodge(h1, h2)
w, m, n = hodge_to_selberg(h)
self.mu_fe = m
self.nu_fe = n
_, self.gammaV = gamma_factors(h)
# this is used in getting the Dirichlet coefficients.
self.bad_primes_info = []
for p in bad_primes:
# we have to check if this works in all bad cases !
f1 = V.local_euler_factor(p)
f2 = W.local_euler_factor(p)
# might be dodgy if f1 or f2 is an approx to the Euler factor
if p in scommon:
E = tensor_local_factors(f1, f2, V.dim * W.dim)
T = f1.parent().gens()[0] # right answer is E(T)*E(pT)
self.bad_primes_info.append(
[p, E * E(p * T), 1 - T]
) #bad_primes_info.append() with 1-T as the second argument is equivalent to taking the first argument as the results (it does a convolution, as in the next line)
else:
self.bad_primes_info.append([p, f1, f2])
CC = ComplexField()
I = CC.gens()[0]
self.sign = I**root_number_at_oo(h)
self.sign /= I**(root_number_at_oo(h1) * V.dim)
self.sign /= I**(root_number_at_oo(h2) * W.dim)
self.sign *= V.sign**W.dim
self.sign *= W.sign**V.dim
for p in bad_primes:
if p not in V.bad_semistable_primes or p not in V.bad_semistable_primes:
f1 = V.local_euler_factor(p)
f2 = W.local_euler_factor(p)
det1 = f1.leading_coefficient() * (-1)**f1.degree()
det2 = f2.leading_coefficient() * (-1)**f2.degree()
n1_tame = V.dim - f1.degree()
n2_tame = W.dim - f2.degree()
n1_wild = ZZ(V.conductor).valuation(p) - n1_tame
n2_wild = ZZ(W.conductor).valuation(p) - n2_tame
# additionally, we would need to correct this by
# replacing det1 by chi1(p) if p is semistable for V
# however for all the possible input this currently does
# not affect the sign
if p in V.bad_semistable_primes:
chi1p = 1 # here
else:
chi1p = det1
if p in W.bad_semistable_primes:
chi2p = 1 # here
else:
chi2p = det2
corr = chi1p**n2_wild
corr *= det1**n2_tame
corr *= chi2p**n1_wild
corr *= det2**n1_tame
corr *= (-1)**(n1_tame * n2_tame)
self.sign *= corr / corr.abs()
#self.primitive = False
self.set_dokchitser_Lfunction()
# maybe we should change this to take as many coefficients as implemented
# in other Lfunctions
self.set_number_of_coefficients()
someans = self.algebraic_coefficients(50) # why not.
if all(x in ZZ for x in someans):
self.selfdual = True
else:
CC = ComplexField()
self.selfdual = all(CC(an).imag().abs() < 0.0001 for an in someans)
self.coefficient_type = max(V.coefficient_type, W.coefficient_type)
self.coefficient_period = ZZ(V.coefficient_period).lcm(
W.coefficient_period)
self.ld.gp().quit()
def init_tensor_product(self, V, W):
"""
We are given two Galois representations and we
will return their tensor product.
"""
self.original_object = V.original_object + W.original_object
self.object_type = "tensorproduct"
self.V1 = V
self.V2 = W
self.dim = V.dim * W.dim
self.motivic_weight = V.motivic_weight + W.motivic_weight
self.langlands = False # status 2014 :)
self.besancon_bound = min(V.besancon_bound, W.besancon_bound)
bad2 = ZZ(W.conductor).prime_factors()
bad_primes = [x for x in ZZ(V.conductor).prime_factors() if x in bad2]
for p in bad_primes:
if ( p not in V.bad_semistable_primes and p not in W.bad_semistable_primes) :
# this condition above only applies to the current type of objects
# for general reps we would have to test the lines below
# to be certain that the formulae are correct.
#if ((p not in V.bad_semistable_primes or p not in W.bad_pot_good) and
#(p not in W.bad_semistable_primes or p not in V.bad_pot_good) and
#(p not in V.bad_semistable_primes or p not in W.bad_semistable_primes)):
raise NotImplementedError("Currently tensor products of Galois representations are only implemented under some conditions.",
"The behaviour at %d is too wild (both factors must be semistable)." % p)
# check for the possibily of getting poles
if V.weight == W.weight and V.conductor == W.conductor :
Vans = V.algebraic_coefficients(50)
Wans = W.algebraic_coefficients(50)
CC = ComplexField()
if ((Vans[2] in ZZ and Wans[2] in ZZ and
all(Vans[n] == Wans[n] for n in range(1,50) ) ) or
all( CC(Vans[n]) == CC(Wans[n]) for n in range(1,50) ) ):
raise NotImplementedError("It seems you are asking to tensor a "+
"Galois representation with its dual " +
"which results in the L-function having "+
"a pole. This is not implemented here.")
scommon = [x for x in V.bad_semistable_primes if x in W.bad_semistable_primes]
N = W.conductor ** V.dim
N *= V.conductor ** W.dim
for p in bad_primes:
n1_tame = V.dim - V.local_euler_factor(p).degree()
n2_tame = W.dim - W.local_euler_factor(p).degree()
nn = n1_tame * n2_tame
N = N // p ** nn
if p in scommon: # both are degree 1 in this case
N = N // p
self.conductor = N
h1 = selberg_to_hodge(V.motivic_weight,V.mu_fe,V.nu_fe)
h2 = selberg_to_hodge(W.motivic_weight,W.mu_fe,W.nu_fe)
h = tensor_hodge(h1, h2)
w,m,n = hodge_to_selberg(h)
self.mu_fe = m
self.nu_fe = n
_, self.gammaV = gamma_factors(h)
# this is used in getting the Dirichlet coefficients.
self.bad_primes_info = []
for p in bad_primes:
# we have to check if this works in all bad cases !
f1 = V.local_euler_factor(p)
f2 = W.local_euler_factor(p)
# might be dodgy if f1 or f2 is an approx to the Euler factor
if p in scommon:
E = tensor_local_factors(f1,f2,V.dim*W.dim)
T = f1.parent().gens()[0] # right answer is E(T)*E(pT)
self.bad_primes_info.append([p,E*E(p*T),1-T]) #bad_primes_info.append() with 1-T as the second argument is equivalent to taking the first argument as the results (it does a convolution, as in the next line)
else:
self.bad_primes_info.append([p,f1,f2])
CC = ComplexField()
I = CC.gens()[0]
self.sign = I ** root_number_at_oo(h)
self.sign /= I ** (root_number_at_oo(h1) * V.dim)
self.sign /= I ** (root_number_at_oo(h2) * W.dim)
self.sign *= V.sign ** W.dim
self.sign *= W.sign ** V.dim
for p in bad_primes:
if p not in V.bad_semistable_primes or p not in V.bad_semistable_primes:
f1 = V.local_euler_factor(p)
f2 = W.local_euler_factor(p)
det1 = f1.leading_coefficient() * (-1) ** f1.degree()
det2 = f2.leading_coefficient() * (-1) ** f2.degree()
n1_tame = V.dim - f1.degree()
n2_tame = W.dim - f2.degree()
n1_wild = ZZ(V.conductor).valuation(p) - n1_tame
n2_wild = ZZ(W.conductor).valuation(p) - n2_tame
# additionally, we would need to correct this by
# replacing det1 by chi1(p) if p is semistable for V
# however for all the possible input this currently does
# not affect the sign
if p in V.bad_semistable_primes:
chi1p = 1 # here
else:
chi1p = det1
if p in W.bad_semistable_primes:
chi2p = 1 # here
else:
chi2p = det2
corr = chi1p ** n2_wild
corr *= det1 ** n2_tame
corr *= chi2p ** n1_wild
corr *= det2 ** n1_tame
corr *= (-1) ** (n1_tame * n2_tame)
self.sign *= corr/corr.abs()
#self.primitive = False
self.set_dokchitser_Lfunction()
# maybe we should change this to take as many coefficients as implemented
# in other Lfunctions
self.set_number_of_coefficients()
someans = self.algebraic_coefficients(50) # why not.
if all( x in ZZ for x in someans):
self.selfdual = True
else:
CC = ComplexField()
self.selfdual = all( CC(an).imag().abs() < 0.0001 for an in someans)
self.coefficient_type = max(V.coefficient_type, W.coefficient_type)
self.coefficient_period = ZZ(V.coefficient_period).lcm(W.coefficient_period)
self.ld.gp().quit()
