def quadratic_L_function__exact(n, d):
r"""
Returns the exact value of a quadratic twist of the Riemann Zeta function
by `\chi_d(x) = \left(\frac{d}{x}\right)`.
The input `n` must be a critical value.
EXAMPLES::
sage: quadratic_L_function__exact(1, -4)
1/4*pi
sage: quadratic_L_function__exact(-4, -4)
5/2
TESTS::
sage: quadratic_L_function__exact(2, -4)
Traceback (most recent call last):
...
TypeError: n must be a critical value (i.e. odd > 0 or even <= 0)
REFERENCES:
- [Iwasawa]_, pp 16-17, Special values of `L(1-n, \chi)` and `L(n, \chi)`
- [IreRos]_
- [WashCyc]_
"""
from sage.all import SR, sqrt
if n <= 0:
return QuadraticBernoulliNumber(1 - n, d) / (n - 1)
elif n >= 1:
# Compute the kind of critical values (p10)
if kronecker_symbol(fundamental_discriminant(d), -1) == 1:
delta = 0
else:
delta = 1
# Compute the positive special values (p17)
if ((n - delta) % 2 == 0):
f = abs(fundamental_discriminant(d))
if delta == 0:
GS = sqrt(f)
else:
GS = I * sqrt(f)
ans = SR(ZZ(-1)**(1 + (n - delta) / 2))
ans *= (2 * pi / f)**n
ans *= GS # Evaluate the Gauss sum here! =0
ans *= 1 / (2 * I**delta)
ans *= QuadraticBernoulliNumber(n, d) / factorial(n)
return ans
else:
if delta == 0:
raise TypeError(
"n must be a critical value (i.e. even > 0 or odd < 0)")
if delta == 1:
raise TypeError(
"n must be a critical value (i.e. odd > 0 or even <= 0)")
def quadratic_L_function__exact(n, d):
r"""
Returns the exact value of a quadratic twist of the Riemann Zeta function
by `\chi_d(x) = \left(\frac{d}{x}\right)`.
The input `n` must be a critical value.
EXAMPLES::
sage: quadratic_L_function__exact(1, -4)
1/4*pi
sage: quadratic_L_function__exact(-4, -4)
5/2
TESTS::
sage: quadratic_L_function__exact(2, -4)
Traceback (most recent call last):
...
TypeError: n must be a critical value (i.e. odd > 0 or even <= 0)
REFERENCES:
- [Iwasawa]_, pp 16-17, Special values of `L(1-n, \chi)` and `L(n, \chi)`
- [IreRos]_
- [WashCyc]_
"""
from sage.all import SR, sqrt
if n <= 0:
return QuadraticBernoulliNumber(1 - n, d) / (n - 1)
elif n >= 1:
# Compute the kind of critical values (p10)
if kronecker_symbol(fundamental_discriminant(d), -1) == 1:
delta = 0
else:
delta = 1
# Compute the positive special values (p17)
if (n - delta) % 2 == 0:
f = abs(fundamental_discriminant(d))
if delta == 0:
GS = sqrt(f)
else:
GS = I * sqrt(f)
ans = SR(ZZ(-1) ** (1 + (n - delta) / 2))
ans *= (2 * pi / f) ** n
ans *= GS # Evaluate the Gauss sum here! =0
ans *= 1 / (2 * I ** delta)
ans *= QuadraticBernoulliNumber(n, d) / factorial(n)
return ans
else:
if delta == 0:
raise TypeError("n must be a critical value (i.e. even > 0 or odd < 0)")
if delta == 1:
raise TypeError("n must be a critical value (i.e. odd > 0 or even <= 0)")
def duke_imamoglu_lift(self,
f,
f_k,
precision,
half_integral_weight=False):
"""
INPUT:
- ``half_integral_weight`` -- If ``False`` we assume that `f` is the Fourier expansion of a
Jacobi form. Otherwise we assume it is the Fourier expansion
of a half integral weight elliptic modular form.
"""
if half_integral_weight:
coeff_index = lambda d: d
else:
coeff_index = lambda d: ((d + (-d % 4)) // 4, (-d) % 4)
coeffs = dict()
for t in precision.iter_positive_forms():
dt = (-1)**(precision.genus() // 2) * t.det()
d = fundamental_discriminant(dt)
eps = Integer(isqrt(dt / d))
coeffs[t] = 0
for a in eps.divisors():
d_a = abs(d * (eps // a)**2)
coeffs[t] = coeffs[t] + a**(f_k - 1) * self._kohnen_phi(a, t) \
* f[coeff_index(d_a)]
return coeffs
def QuadraticBernoulliNumber(k, d):
r"""
Compute `k`-th Bernoulli number for the primitive
quadratic character associated to `\chi(x) = \left(\frac{d}{x}\right)`.
EXAMPLES:
Let us create a list of some odd negative fundamental discriminants::
sage: test_set = [d for d in range(-163, -3, 4) if is_fundamental_discriminant(d)]
In general, we have `B_{1, \chi_d} = -2 h/w` for odd negative fundamental
discriminants::
sage: all(QuadraticBernoulliNumber(1, d) == -len(BinaryQF_reduced_representatives(d)) for d in test_set)
True
REFERENCES:
- [Iwasawa]_, pp 7-16.
"""
# Ensure the character is primitive
d1 = fundamental_discriminant(d)
f = abs(d1)
# Make the (usual) k-th Bernoulli polynomial
x = PolynomialRing(QQ, 'x').gen()
bp = bernoulli_polynomial(x, k)
# Make the k-th quadratic Bernoulli number
total = sum([kronecker_symbol(d1, i) * bp(i/f) for i in range(f)])
total *= (f ** (k-1))
return total
def QuadraticBernoulliNumber(k, d):
r"""
Compute k-th Bernoulli number for the primitive
quadratic character associated to `\chi(x) = \left(\frac{d}{x}\right)`.
Reference: Iwasawa's "Lectures on p-adic L-functions", pp7-16.
EXAMPLES::
sage: ## Makes a set of odd fund discriminants < -3
sage: Fund_odd_test_set = [D for D in range(-163, -3, 4) if is_fundamental_discriminant(D)]
sage: ## In general, we have B_{1, \chi_d} = -2h/w for odd fund disc < 0
sage: for D in Fund_odd_test_set:
... if len(BinaryQF_reduced_representatives(D)) != -QuadraticBernoulliNumber(1, D):
... print "Oops! There is an error at D = ", D
"""
## Ensure the character is primitive
d1 = fundamental_discriminant(d)
f = abs(d1)
## Make the (usual) k-th Bernoulli polynomial
x = PolynomialRing(QQ, 'x').gen()
bp = bernoulli_polynomial(x, k)
## Make the k-th quadratic Bernoulli number
total = sum([kronecker_symbol(d1, i) * bp(i/f) for i in range(f)])
total *= (f ** (k-1))
return total
def QuadraticBernoulliNumber(k, d):
r"""
Compute k-th Bernoulli number for the primitive
quadratic character associated to `\chi(x) = \left(\frac{d}{x}\right)`.
Reference: Iwasawa's "Lectures on p-adic L-functions", pp7-16.
EXAMPLES::
sage: ## Makes a set of odd fund discriminants < -3
sage: Fund_odd_test_set = [D for D in range(-163, -3, 4) if is_fundamental_discriminant(D)]
sage: ## In general, we have B_{1, \chi_d} = -2h/w for odd fund disc < 0
sage: for D in Fund_odd_test_set:
... if len(BinaryQF_reduced_representatives(D)) != -QuadraticBernoulliNumber(1, D):
... print "Oops! There is an error at D = ", D
"""
## Ensure the character is primitive
d1 = fundamental_discriminant(d)
f = abs(d1)
## Make the (usual) k-th Bernoulli polynomial
x = PolynomialRing(QQ, 'x').gen()
bp = bernoulli_polynomial(x, k)
## Make the k-th quadratic Bernoulli number
total = sum([kronecker_symbol(d1, i) * bp(i / f) for i in range(f)])
total *= (f**(k - 1))
return total
def QuadraticBernoulliNumber(k, d):
r"""
Compute `k`-th Bernoulli number for the primitive
quadratic character associated to `\chi(x) = \left(\frac{d}{x}\right)`.
EXAMPLES:
Let us create a list of some odd negative fundamental discriminants::
sage: test_set = [d for d in range(-163, -3, 4) if is_fundamental_discriminant(d)]
In general, we have `B_{1, \chi_d} = -2 h/w` for odd negative fundamental
discriminants::
sage: all(QuadraticBernoulliNumber(1, d) == -len(BinaryQF_reduced_representatives(d)) for d in test_set)
True
REFERENCES:
- [Iwasawa]_, pp 7-16.
"""
# Ensure the character is primitive
d1 = fundamental_discriminant(d)
f = abs(d1)
# Make the (usual) k-th Bernoulli polynomial
x = PolynomialRing(QQ, 'x').gen()
bp = bernoulli_polynomial(x, k)
# Make the k-th quadratic Bernoulli number
total = sum([kronecker_symbol(d1, i) * bp(i / f) for i in range(f)])
total *= (f**(k - 1))
return total
def duke_imamoglu_lift(self, f, f_k, precision, half_integral_weight = False) :
"""
INPUT:
- ``half_integral_weight`` -- If ``False`` we assume that `f` is the Fourier expansion of a
Jacobi form. Otherwise we assume it is the Fourier expansion
of a half integral weight elliptic modular form.
"""
if half_integral_weight :
coeff_index = lambda d : d
else :
coeff_index = lambda d : ((d + (-d % 4)) // 4, (-d) % 4)
coeffs = dict()
for t in precision.iter_positive_forms() :
dt = (-1)**(precision.genus() // 2) * t.det()
d = fundamental_discriminant(dt)
eps = Integer(isqrt(dt / d))
coeffs[t] = 0
for a in eps.divisors() :
d_a = abs(d * (eps // a)**2)
coeffs[t] = coeffs[t] + a**(f_k - 1) * self._kohnen_phi(a, t) \
* f[coeff_index(d_a)]
return coeffs
def quadratic_L_function__exact(n, d):
r"""
Returns the exact value of a quadratic twist of the Riemann Zeta function
by `\chi_d(x) = \left(\frac{d}{x}\right)`.
References:
- Iwasawa's "Lectures on p-adic L-functions", p16-17, "Special values of
`L(1-n, \chi)` and `L(n, \chi)`
- Ireland and Rosen's "A Classical Introduction to Modern Number Theory"
- Washington's "Cyclotomic Fields"
EXAMPLES::
sage: bool(quadratic_L_function__exact(1, -4) == pi/4)
True
"""
from sage.all import SR, sqrt
if n <= 0:
k = 1 - n
return -QuadraticBernoulliNumber(k, d) / k
elif n >= 1:
## Compute the kind of critical values (p10)
if kronecker_symbol(fundamental_discriminant(d), -1) == 1:
delta = 0
else:
delta = 1
## Compute the positive special values (p17)
if ((n - delta) % 2 == 0):
f = abs(fundamental_discriminant(d))
if delta == 0:
GS = sqrt(f)
else:
GS = I * sqrt(f)
ans = SR(ZZ(-1)**(1 + (n - delta) / 2))
ans *= (2 * pi / f)**n
ans *= GS ## Evaluate the Gauss sum here! =0
ans *= 1 / (2 * I**delta)
ans *= QuadraticBernoulliNumber(n, d) / factorial(n)
return ans
else:
if delta == 0:
raise TypeError, "n must be a critical value!\n" + "(I.e. even > 0 or odd < 0.)"
if delta == 1:
raise TypeError, "n must be a critical value!\n" + "(I.e. odd > 0 or even <= 0.)"
def quadratic_L_function__exact(n, d):
r"""
Returns the exact value of a quadratic twist of the Riemann Zeta function
by `\chi_d(x) = \left(\frac{d}{x}\right)`.
References:
- Iwasawa's "Lectures on p-adic L-functions", p16-17, "Special values of
`L(1-n, \chi)` and `L(n, \chi)`
- Ireland and Rosen's "A Classical Introduction to Modern Number Theory"
- Washington's "Cyclotomic Fields"
EXAMPLES::
sage: bool(quadratic_L_function__exact(1, -4) == pi/4)
True
"""
from sage.all import SR, sqrt
if n<=0:
k = 1-n
return -QuadraticBernoulliNumber(k,d)/k
elif n>=1:
## Compute the kind of critical values (p10)
if kronecker_symbol(fundamental_discriminant(d), -1) == 1:
delta = 0
else:
delta = 1
## Compute the positive special values (p17)
if ((n - delta) % 2 == 0):
f = abs(fundamental_discriminant(d))
if delta == 0:
GS = sqrt(f)
else:
GS = I * sqrt(f)
ans = SR(ZZ(-1)**(1+(n-delta)/2))
ans *= (2*pi/f)**n
ans *= GS ## Evaluate the Gauss sum here! =0
ans *= 1/(2 * I**delta)
ans *= QuadraticBernoulliNumber(n,d)/factorial(n)
return ans
else:
if delta == 0:
raise TypeError, "n must be a critical value!\n" + "(I.e. even > 0 or odd < 0.)"
if delta == 1:
raise TypeError, "n must be a critical value!\n" + "(I.e. odd > 0 or even <= 0.)"
def _kohnen_rho(self, t, a) :
dt = (-1)**(t.nrows() // 2) * t.det()
d = fundamental_discriminant(dt)
eps = isqrt(dt // d)
res = 1
for (p,e) in gcd(a, eps).factor() :
pol = self._kohnen_rho_polynomial(t, p).coefficients()
if e < len(pol) :
res = res * pol[e]
return res
def _kohnen_rho(self, t, a):
dt = (-1)**(t.nrows() // 2) * t.det()
d = fundamental_discriminant(dt)
eps = isqrt(dt // d)
res = 1
for (p, e) in gcd(a, eps).factor():
pol = self._kohnen_rho_polynomial(t, p).coefficients()
if e < len(pol):
res = res * pol[e]
return res
def quadratic_L_function__numerical(n, d, num_terms=1000):
"""
Evaluate the Dirichlet L-function (for quadratic character) numerically
(in a very naive way).
EXAMPLES:
First, let us test several values for a given character::
sage: RR = RealField(100)
sage: for i in range(5):
... print "L(" + str(1+2*i) + ", (-4/.)): ", RR(quadratic_L_function__exact(1+2*i, -4)) - quadratic_L_function__numerical(RR(1+2*i),-4, 10000)
L(1, (-4/.)): 0.000049999999500000024999996962707
L(3, (-4/.)): 4.99999970000003...e-13
L(5, (-4/.)): 4.99999922759382...e-21
L(7, (-4/.)): ...e-29
L(9, (-4/.)): ...e-29
This procedure fails for negative special values, as the Dirichlet
series does not converge here::
sage: quadratic_L_function__numerical(-3,-4, 10000)
Traceback (most recent call last):
...
ValueError: the Dirichlet series does not converge here
Test for several characters that the result agrees with the exact
value, to a given accuracy ::
sage: for d in range(-20,0): # long time (2s on sage.math 2014)
....: if abs(RR(quadratic_L_function__numerical(1, d, 10000) - quadratic_L_function__exact(1, d))) > 0.001:
....: print "Oops! We have a problem at d = ", d, " exact = ", RR(quadratic_L_function__exact(1, d)), " numerical = ", RR(quadratic_L_function__numerical(1, d))
"""
# Set the correct precision if it is given (for n).
if is_RealField(n.parent()):
R = n.parent()
else:
R = RealField()
if n < 0:
raise ValueError('the Dirichlet series does not converge here')
d1 = fundamental_discriminant(d)
ans = R(0)
for i in range(1, num_terms):
ans += R(kronecker_symbol(d1, i) / R(i)**n)
return ans
def quadratic_L_function__numerical(n, d, num_terms=1000):
"""
Evaluate the Dirichlet L-function (for quadratic character) numerically
(in a very naive way).
EXAMPLES:
First, let us test several values for a given character::
sage: RR = RealField(100)
sage: for i in range(5):
... print "L(" + str(1+2*i) + ", (-4/.)): ", RR(quadratic_L_function__exact(1+2*i, -4)) - quadratic_L_function__numerical(RR(1+2*i),-4, 10000)
L(1, (-4/.)): 0.000049999999500000024999996962707
L(3, (-4/.)): 4.99999970000003...e-13
L(5, (-4/.)): 4.99999922759382...e-21
L(7, (-4/.)): ...e-29
L(9, (-4/.)): ...e-29
This procedure fails for negative special values, as the Dirichlet
series does not converge here::
sage: quadratic_L_function__numerical(-3,-4, 10000)
Traceback (most recent call last):
...
ValueError: the Dirichlet series does not converge here
Test for several characters that the result agrees with the exact
value, to a given accuracy ::
sage: for d in range(-20,0): # long time (2s on sage.math 2014)
....: if abs(RR(quadratic_L_function__numerical(1, d, 10000) - quadratic_L_function__exact(1, d))) > 0.001:
....: print "Oops! We have a problem at d = ", d, " exact = ", RR(quadratic_L_function__exact(1, d)), " numerical = ", RR(quadratic_L_function__numerical(1, d))
"""
# Set the correct precision if it is given (for n).
if is_RealField(n.parent()):
R = n.parent()
else:
R = RealField()
if n < 0:
raise ValueError('the Dirichlet series does not converge here')
d1 = fundamental_discriminant(d)
ans = R(0)
for i in range(1,num_terms):
ans += R(kronecker_symbol(d1,i) / R(i)**n)
return ans
def quadratic_L_function__numerical(n, d, num_terms=1000):
"""
Evaluate the Dirichlet L-function (for quadratic character) numerically
(in a very naive way).
EXAMPLES::
sage: ## Test several values for a given character
sage: RR = RealField(100)
sage: for i in range(5):
... print "L(" + str(1+2*i) + ", (-4/.)): ", RR(quadratic_L_function__exact(1+2*i, -4)) - quadratic_L_function__numerical(RR(1+2*i),-4, 10000)
L(1, (-4/.)): 0.000049999999500000024999996962707
L(3, (-4/.)): 4.99999970000003...e-13
L(5, (-4/.)): 4.99999922759382...e-21
L(7, (-4/.)): ...e-29
L(9, (-4/.)): ...e-29
sage: ## Testing the accuracy of the negative special values
sage: ## ---- THIS FAILS SINCE THE DIRICHLET SERIES DOESN'T CONVERGE HERE! ----
sage: ## Test several characters agree with the exact value, to a given accuracy.
sage: for d in range(-20,0):
... if abs(RR(quadratic_L_function__numerical(1, d, 10000) - quadratic_L_function__exact(1, d))) > 0.001:
... print "Oops! We have a problem at d = ", d, " exact = ", RR(quadratic_L_function__exact(1, d)), " numerical = ", RR(quadratic_L_function__numerical(1, d))
...
"""
## Set the correct precision if it's given (for n).
if is_RealField(n.parent()):
R = n.parent()
else:
R = RealField()
d1 = fundamental_discriminant(d)
ans = R(0)
for i in range(1,num_terms):
ans += R(kronecker_symbol(d1,i) / R(i)**n)
return ans
def quadratic_L_function__numerical(n, d, num_terms=1000):
"""
Evaluate the Dirichlet L-function (for quadratic character) numerically
(in a very naive way).
EXAMPLES::
sage: ## Test several values for a given character
sage: RR = RealField(100)
sage: for i in range(5):
... print "L(" + str(1+2*i) + ", (-4/.)): ", RR(quadratic_L_function__exact(1+2*i, -4)) - quadratic_L_function__numerical(RR(1+2*i),-4, 10000)
L(1, (-4/.)): 0.000049999999500000024999996962707
L(3, (-4/.)): 4.99999970000003...e-13
L(5, (-4/.)): 4.99999922759382...e-21
L(7, (-4/.)): ...e-29
L(9, (-4/.)): ...e-29
sage: ## Testing the accuracy of the negative special values
sage: ## ---- THIS FAILS SINCE THE DIRICHLET SERIES DOESN'T CONVERGE HERE! ----
sage: ## Test several characters agree with the exact value, to a given accuracy.
sage: for d in range(-20,0):
... if abs(RR(quadratic_L_function__numerical(1, d, 10000) - quadratic_L_function__exact(1, d))) > 0.001:
... print "Oops! We have a problem at d = ", d, " exact = ", RR(quadratic_L_function__exact(1, d)), " numerical = ", RR(quadratic_L_function__numerical(1, d))
...
"""
## Set the correct precision if it's given (for n).
if is_RealField(n.parent()):
R = n.parent()
else:
R = RealField()
d1 = fundamental_discriminant(d)
ans = R(0)
for i in range(1, num_terms):
ans += R(kronecker_symbol(d1, i) / R(i)**n)
return ans
def siegel_product(self, u):
"""
Computes the infinite product of local densities of the quadratic
form for the number `u`.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: Q.theta_series(11)
1 + 8*q + 24*q^2 + 32*q^3 + 24*q^4 + 48*q^5 + 96*q^6 + 64*q^7 + 24*q^8 + 104*q^9 + 144*q^10 + O(q^11)
sage: Q.siegel_product(1)
8
sage: Q.siegel_product(2) ## This one is wrong -- expect 24, and the higher powers of 2 don't work... =(
24
sage: Q.siegel_product(3)
32
sage: Q.siegel_product(5)
48
sage: Q.siegel_product(6)
96
sage: Q.siegel_product(7)
64
sage: Q.siegel_product(9)
104
sage: Q.local_density(2,1)
1
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3
1
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3 # long time (41s on sage.math, 2011)
1
sage: Q.local_density(2,2)
3/2
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3
3/2
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3 # long time (41s on sage.math, 2011)
3/2
TESTS::
sage: [1] + [Q.siegel_product(ZZ(a)) for a in range(1,11)] == Q.theta_series(11).list()
True
"""
## Protect u (since it fails often if it's an just an int!)
u = ZZ(u)
n = self.dim()
d = self.det() ## ??? Warning: This is a factor of 2^n larger than it should be!
## DIAGNOSTIC
verbose("n = " + str(n))
verbose("d = " + str(d))
verbose("In siegel_product: d = ", d, "\n");
## Product of "bad" places to omit
S = 2 * d * u
## DIAGNOSTIC
verbose("siegel_product Break 1. \n")
verbose(" u = ", u, "\n")
## Make the odd generic factors
if ((n % 2) == 1):
m = (n-1) / 2
d1 = fundamental_discriminant(((-1)**m) * 2*d * u) ## Replaced d by 2d here to compensate for the determinant
f = abs(d1) ## gaining an odd power of 2 by using the matrix of 2Q instead
## of the matrix of Q.
## --> Old d1 = CoreDiscriminant((mpz_class(-1)^m) * d * u);
## Make the ratio of factorials factor: [(2m)! / m!] * prod_{i=1}^m (2*i-1)
factor1 = 1
for i in range(1, m+1):
factor1 *= 2*i - 1
for i in range(m+1, 2*m + 1):
factor1 *= i
genericfactor = factor1 * ((u / f) ** m) \
* QQ(sqrt((2 ** n) * f) / (u * d)) \
* abs(QuadraticBernoulliNumber(m, d1) / bernoulli(2*m))
## DIAGNOSTIC
verbose("siegel_product Break 2. \n")
## Make the even generic factor
if ((n % 2) == 0):
m = n / 2
d1 = fundamental_discriminant(((-1)**m) * d)
f = abs(d1)
## DIAGNOSTIC
#cout << " mpz_class(-1)^m = " << (mpz_class(-1)^m) << " and d = " << d << endl;
#cout << " f = " << f << " and d1 = " << d1 << endl;
genericfactor = m / QQ(sqrt(f*d)) \
* ((u/2) ** (m-1)) * (f ** m) \
/ abs(QuadraticBernoulliNumber(m, d1)) \
* (2 ** m) ## This last factor compensates for using the matrix of 2*Q
##return genericfactor
## Omit the generic factors in S and compute them separately
omit = 1
include = 1
S_divisors = prime_divisors(S)
## DIAGNOSTIC
#cout << "\n S is " << S << endl;
#cout << " The Prime divisors of S are :";
#PrintV(S_divisors);
for p in S_divisors:
Q_normal = self.local_normal_form(p)
## DIAGNOSTIC
verbose(" p = " + str(p) + " and its Kronecker symbol (d1/p) = (" + str(d1) + "/" + str(p) + ") is " + str(kronecker_symbol(d1, p)) + "\n")
omit *= 1 / (1 - (kronecker_symbol(d1, p) / (p**m)))
## DIAGNOSTIC
verbose(" omit = " + str(omit) + "\n")
verbose(" Q_normal is \n" + str(Q_normal) + "\n")
verbose(" Q_normal = \n" + str(Q_normal))
verbose(" p = " + str(p) + "\n")
verbose(" u = " +str(u) + "\n")
verbose(" include = " + str(include) + "\n")
include *= Q_normal.local_density(p, u)
## DIAGNOSTIC
#cout << " Including the p = " << p << " factor: " << local_density(Q_normal, p, u) << endl;
## DIAGNSOTIC
verbose(" --- Exiting loop \n")
#// **************** Important *******************
#// Additional fix (only included for n=4) to deal
#// with the power of 2 introduced at the real place
#// by working with Q instead of 2*Q. This needs to
#// be done for all other n as well...
#/*
#if (n==4)
# genericfactor = 4 * genericfactor;
#*/
## DIAGNSOTIC
#cout << endl;
#cout << " generic factor = " << genericfactor << endl;
#cout << " omit = " << omit << endl;
#cout << " include = " << include << endl;
#cout << endl;
## DIAGNSOTIC
#// cout << "siegel_product Break 3. " << endl;
## Return the final factor (and divide by 2 if n=2)
if (n == 2):
return (genericfactor * omit * include / 2)
else:
return (genericfactor * omit * include)
def siegel_product(self, u):
"""
Computes the infinite product of local densities of the quadratic
form for the number `u`.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: Q.theta_series(11)
1 + 8*q + 24*q^2 + 32*q^3 + 24*q^4 + 48*q^5 + 96*q^6 + 64*q^7 + 24*q^8 + 104*q^9 + 144*q^10 + O(q^11)
sage: Q.siegel_product(1)
8
sage: Q.siegel_product(2) ## This one is wrong -- expect 24, and the higher powers of 2 don't work... =(
24
sage: Q.siegel_product(3)
32
sage: Q.siegel_product(5)
48
sage: Q.siegel_product(6)
96
sage: Q.siegel_product(7)
64
sage: Q.siegel_product(9)
104
sage: Q.local_density(2,1)
1
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3
1
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3 # long time (41s on sage.math, 2011)
1
sage: Q.local_density(2,2)
3/2
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3
3/2
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3 # long time (41s on sage.math, 2011)
3/2
TESTS::
sage: [1] + [Q.siegel_product(ZZ(a)) for a in range(1,11)] == Q.theta_series(11).list()
True
"""
## Protect u (since it fails often if it's an just an int!)
u = ZZ(u)
n = self.dim()
d = self.det(
) ## ??? Warning: This is a factor of 2^n larger than it should be!
## DIAGNOSTIC
verbose("n = " + str(n))
verbose("d = " + str(d))
verbose("In siegel_product: d = ", d, "\n")
## Product of "bad" places to omit
S = 2 * d * u
## DIAGNOSTIC
verbose("siegel_product Break 1. \n")
verbose(" u = ", u, "\n")
## Make the odd generic factors
if ((n % 2) == 1):
m = (n - 1) / 2
d1 = fundamental_discriminant(
((-1)**m) * 2 * d *
u) ## Replaced d by 2d here to compensate for the determinant
f = abs(
d1) ## gaining an odd power of 2 by using the matrix of 2Q instead
## of the matrix of Q.
## --> Old d1 = CoreDiscriminant((mpz_class(-1)^m) * d * u);
## Make the ratio of factorials factor: [(2m)! / m!] * prod_{i=1}^m (2*i-1)
factor1 = 1
for i in range(1, m + 1):
factor1 *= 2 * i - 1
for i in range(m + 1, 2 * m + 1):
factor1 *= i
genericfactor = factor1 * ((u / f) ** m) \
* QQ(sqrt((2 ** n) * f) / (u * d)) \
* abs(QuadraticBernoulliNumber(m, d1) / bernoulli(2*m))
## DIAGNOSTIC
verbose("siegel_product Break 2. \n")
## Make the even generic factor
if ((n % 2) == 0):
m = n / 2
d1 = fundamental_discriminant(((-1)**m) * d)
f = abs(d1)
## DIAGNOSTIC
#cout << " mpz_class(-1)^m = " << (mpz_class(-1)^m) << " and d = " << d << endl;
#cout << " f = " << f << " and d1 = " << d1 << endl;
genericfactor = m / QQ(sqrt(f*d)) \
* ((u/2) ** (m-1)) * (f ** m) \
/ abs(QuadraticBernoulliNumber(m, d1)) \
* (2 ** m) ## This last factor compensates for using the matrix of 2*Q
##return genericfactor
## Omit the generic factors in S and compute them separately
omit = 1
include = 1
S_divisors = prime_divisors(S)
## DIAGNOSTIC
#cout << "\n S is " << S << endl;
#cout << " The Prime divisors of S are :";
#PrintV(S_divisors);
for p in S_divisors:
Q_normal = self.local_normal_form(p)
## DIAGNOSTIC
verbose(" p = " + str(p) + " and its Kronecker symbol (d1/p) = (" +
str(d1) + "/" + str(p) + ") is " +
str(kronecker_symbol(d1, p)) + "\n")
omit *= 1 / (1 - (kronecker_symbol(d1, p) / (p**m)))
## DIAGNOSTIC
verbose(" omit = " + str(omit) + "\n")
verbose(" Q_normal is \n" + str(Q_normal) + "\n")
verbose(" Q_normal = \n" + str(Q_normal))
verbose(" p = " + str(p) + "\n")
verbose(" u = " + str(u) + "\n")
verbose(" include = " + str(include) + "\n")
include *= Q_normal.local_density(p, u)
## DIAGNOSTIC
#cout << " Including the p = " << p << " factor: " << local_density(Q_normal, p, u) << endl;
## DIAGNSOTIC
verbose(" --- Exiting loop \n")
#// **************** Important *******************
#// Additional fix (only included for n=4) to deal
#// with the power of 2 introduced at the real place
#// by working with Q instead of 2*Q. This needs to
#// be done for all other n as well...
#/*
#if (n==4)
# genericfactor = 4 * genericfactor;
#*/
## DIAGNSOTIC
#cout << endl;
#cout << " generic factor = " << genericfactor << endl;
#cout << " omit = " << omit << endl;
#cout << " include = " << include << endl;
#cout << endl;
## DIAGNSOTIC
#// cout << "siegel_product Break 3. " << endl;
## Return the final factor (and divide by 2 if n=2)
if (n == 2):
return (genericfactor * omit * include / 2)
else:
return (genericfactor * omit * include)
def HermitianModularFormD2Factory( precision, discriminant = None ) :
"""
Create an instance of a factory for Fourier expansions of Hermitian
modular forms.
INPUT:
- ``precision`` -- An integer or an instance of a precision class.
- ``discriminant`` -- A negative integer or ``None`` (default: ``None``);
The fundamental discriminant of an imaginary quadratic field.
Can be omited if the precision is not an integer.
OUTPUT:
An instance of :class:~`.HermitianModularFormD2Factory_class`.
SEE:
:class:~`.HermitianModularFormD2Factory_class`.
TESTS::
sage: from hermitianmodularforms.hermitianmodularformd2_fegenerators import HermitianModularFormD2Factory
sage: from hermitianmodularforms.hermitianmodularformd2_fourierexpansion import HermitianModularFormD2Filter_diagonal
sage: h = HermitianModularFormD2Factory(3, -3)
sage: h.precision()
Reduced diagonal filter for discriminant -3 with bound 3
sage: HermitianModularFormD2Factory(HermitianModularFormD2Filter_diagonal(2, -3))
Factory for Fourier expansions of hermitian modular forms with precision Reduced diagonal filter for discriminant -3 with bound 2
sage: HermitianModularFormD2Factory(HermitianModularFormD2Filter_diagonal(2, -3), -4)
Traceback (most recent call last):
...
ValueError: Discriminant must coinside with the precision's discriminant.
sage: HermitianModularFormD2Factory(2, -1)
Traceback (most recent call last):
...
ValueError: Discriminant must be a fundamental discriminant.
sage: HermitianModularFormD2Factory(2, -4)
Traceback (most recent call last):
...
NotImplementedError: Only discriminant -3 is implemented.
"""
if not isinstance(precision, HermitianModularFormD2Filter_diagonal) :
if discriminant is None :
raise ValueError( "If precision is not a precision class, discriminant must be set." )
precision = HermitianModularFormD2Filter_diagonal(precision, discriminant)
else :
if not discriminant is None and precision.discriminant() != discriminant :
raise ValueError( "Discriminant must coinside with the precision's discriminant." )
else :
discriminant = precision.discriminant()
if discriminant >= 0 :
raise ValueError( "Discriminant must be negative." )
if fundamental_discriminant(discriminant) != discriminant :
raise ValueError( "Discriminant must be a fundamental discriminant." )
if discriminant != -3 :
raise NotImplementedError( "Only discriminant -3 is implemented." )
key = (precision)
global _hermitianmodularformd2factory_cache
try :
return _hermitianmodularformd2factory_cache[key]
except :
factory = HermitianModularFormD2Factory_class(precision)
_hermitianmodularformd2factory_cache[key] = factory
return factory
def HermitianModularFormD2Factory(precision, discriminant=None):
"""
Create an instance of a factory for Fourier expansions of Hermitian
modular forms.
INPUT:
- ``precision`` -- An integer or an instance of a precision class.
- ``discriminant`` -- A negative integer or ``None`` (default: ``None``);
The fundamental discriminant of an imaginary quadratic field.
Can be omited if the precision is not an integer.
OUTPUT:
An instance of :class:~`.HermitianModularFormD2Factory_class`.
SEE:
:class:~`.HermitianModularFormD2Factory_class`.
TESTS::
sage: from hermitianmodularforms.hermitianmodularformd2_fegenerators import HermitianModularFormD2Factory
sage: from hermitianmodularforms.hermitianmodularformd2_fourierexpansion import HermitianModularFormD2Filter_diagonal
sage: h = HermitianModularFormD2Factory(3, -3)
sage: h.precision()
Reduced diagonal filter for discriminant -3 with bound 3
sage: HermitianModularFormD2Factory(HermitianModularFormD2Filter_diagonal(2, -3))
Factory for Fourier expansions of hermitian modular forms with precision Reduced diagonal filter for discriminant -3 with bound 2
sage: HermitianModularFormD2Factory(HermitianModularFormD2Filter_diagonal(2, -3), -4)
Traceback (most recent call last):
...
ValueError: Discriminant must coinside with the precision's discriminant.
sage: HermitianModularFormD2Factory(2, -1)
Traceback (most recent call last):
...
ValueError: Discriminant must be a fundamental discriminant.
sage: HermitianModularFormD2Factory(2, -4)
Traceback (most recent call last):
...
NotImplementedError: Only discriminant -3 is implemented.
"""
if not isinstance(precision, HermitianModularFormD2Filter_diagonal):
if discriminant is None:
raise ValueError(
"If precision is not a precision class, discriminant must be set."
)
precision = HermitianModularFormD2Filter_diagonal(
precision, discriminant)
else:
if not discriminant is None and precision.discriminant(
) != discriminant:
raise ValueError(
"Discriminant must coinside with the precision's discriminant."
)
else:
discriminant = precision.discriminant()
if discriminant >= 0:
raise ValueError("Discriminant must be negative.")
if fundamental_discriminant(discriminant) != discriminant:
raise ValueError("Discriminant must be a fundamental discriminant.")
if discriminant != -3:
raise NotImplementedError("Only discriminant -3 is implemented.")
key = (precision)
global _hermitianmodularformd2factory_cache
try:
return _hermitianmodularformd2factory_cache[key]
except:
factory = HermitianModularFormD2Factory_class(precision)
_hermitianmodularformd2factory_cache[key] = factory
return factory