def _basic_integral(self, a, j):
r"""
Return `\int_{a+pZ_p} (z-{a})^j d\Phi(0-infty)`.
See formula in section 9.2 of [PS2011]_
INPUT:
- ``a`` -- integer in range(p)
- ``j`` -- integer in range(self.symbol().precision_relative())
EXAMPLES::
sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
sage: E = EllipticCurve('11a3')
sage: L = E.padic_lseries(5, implementation="pollackstevens", precision=4) #long time
sage: L._basic_integral(1,2) # long time
2*5^2 + 5^3 + O(5^4)
"""
symb = self.symbol()
M = symb.precision_relative()
if j > M:
raise PrecisionError("Too many moments requested")
p = self.prime()
ap = symb.Tq_eigenvalue(p)
D = self._quadratic_twist
ap = ap * kronecker(D, p)
K = pAdicField(p, M)
symb_twisted = symb.evaluate_twisted(a, D)
return sum(
binomial(j, r) * ((a - ZZ(K.teichmuller(a)))**(j - r)) *
(p**r) * symb_twisted.moment(r) for r in range(j + 1)) / ap
def _basic_integral(self, a, j):
r"""
Return `\int_{a+pZ_p} (z-{a})^j d\Phi(0-infty)`
-- see formula [Pollack-Stevens, sec 9.2]
INPUT:
- ``a`` -- integer in range(p)
- ``j`` -- integer in range(self.symbol().precision_relative())
EXAMPLES::
sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
sage: E = EllipticCurve('11a3')
sage: L = E.padic_lseries(5, implementation="pollackstevens", precision=4) #long time
sage: L._basic_integral(1,2) # long time
2*5^2 + 5^3 + O(5^4)
"""
symb = self.symbol()
M = symb.precision_relative()
if j > M:
raise PrecisionError("Too many moments requested")
p = self.prime()
ap = symb.Tq_eigenvalue(p)
D = self._quadratic_twist
ap = ap * kronecker(D, p)
K = pAdicField(p, M)
symb_twisted = symb.evaluate_twisted(a, D)
return (
sum(
binomial(j, r) * ((a - ZZ(K.teichmuller(a))) ** (j - r)) * (p ** r) * symb_twisted.moment(r)
for r in range(j + 1)
)
/ ap
)
def supersingular_D(prime):
r"""
Return a fundamental discriminant `D` of an
imaginary quadratic field, where the given prime does not split.
See Silverman's Advanced Topics in the Arithmetic of Elliptic
Curves, page 184, exercise 2.30(d).
INPUT:
- prime -- integer, prime
OUTPUT:
- D -- integer, negative
EXAMPLES:
These examples return *supersingular discriminants* for 7,
15073 and 83401::
sage: supersingular_D(7)
-4
sage: supersingular_D(15073)
-15
sage: supersingular_D(83401)
-7
AUTHORS:
- David Kohel - [email protected]
- Iftikhar Burhanuddin - [email protected]
"""
if not Integer(prime).is_prime():
raise ValueError("%s is not a prime" % prime)
# Making picking D more intelligent
D = -1
while True:
Dmod4 = D % 4
if Dmod4 in (0, 1) and kronecker(D, prime) != 1:
return D
D -= 1
def supersingular_D(prime):
r"""
This function returns a fundamental discriminant `D` of an
imaginary quadratic field, where the given prime does not split
(see Silverman's Advanced Topics in the Arithmetic of Elliptic
Curves, page 184, exercise 2.30(d).)
INPUT:
- prime -- integer, prime
OUTPUT:
D -- integer, negative
EXAMPLES:
These examples return *supersingular discriminants* for 7,
15073 and 83401::
sage: supersingular_D(7)
-4
sage: supersingular_D(15073)
-15
sage: supersingular_D(83401)
-7
AUTHORS:
- David Kohel - [email protected]
- Iftikhar Burhanuddin - [email protected]
"""
if not(rings.Integer(prime).is_prime()):
raise ValueError("%s is not a prime"%prime)
#Making picking D more intelligent
D = -1
while True:
Dmod4 = rings.Mod(D,4)
if Dmod4 in (0,1) and (kronecker(D,prime) != 1):
return D
D = D - 1
def mass__by_Siegel_densities(self,
odd_algorithm="Pall",
even_algorithm="Watson"):
"""
Gives the mass of transformations (det 1 and -1).
WARNING: THIS IS BROKEN RIGHT NOW... =(
Optional Arguments:
- When p > 2 -- odd_algorithm = "Pall" (only one choice for now)
- When p = 2 -- even_algorithm = "Kitaoka" or "Watson"
REFERENCES:
- Nipp's Book "Tables of Quaternary Quadratic Forms".
- Papers of Pall (only for p>2) and Watson (for `p=2` -- tricky!).
- Siegel, Milnor-Hussemoller, Conway-Sloane Paper IV, Kitoaka (all of which
have problems...)
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: m = Q.mass__by_Siegel_densities(); m
1/384
sage: m - (2^Q.dim() * factorial(Q.dim()))^(-1)
0
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: m = Q.mass__by_Siegel_densities(); m
1/48
sage: m - (2^Q.dim() * factorial(Q.dim()))^(-1)
0
"""
## Setup
n = self.dim()
s = (n - 1) // 2
if n % 2 != 0:
char_d = squarefree_part(2 *
self.det()) ## Accounts for the det as a QF
else:
char_d = squarefree_part(self.det())
## Form the generic zeta product
generic_prod = ZZ(2) * (pi)**(-ZZ(n) * (n + 1) / 4)
##########################################
generic_prod *= (self.det())**(
ZZ(n + 1) / 2) ## ***** This uses the Hessian Determinant ********
##########################################
#print "gp1 = ", generic_prod
generic_prod *= prod([gamma__exact(ZZ(j) / 2) for j in range(1, n + 1)])
#print "\n---", [(ZZ(j)/2, gamma__exact(ZZ(j)/2)) for j in range(1,n+1)]
#print "\n---", prod([gamma__exact(ZZ(j)/2) for j in range(1,n+1)])
#print "gp2 = ", generic_prod
generic_prod *= prod([zeta__exact(ZZ(j)) for j in range(2, 2 * s + 1, 2)])
#print "\n---", [zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)]
#print "\n---", prod([zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)])
#print "gp3 = ", generic_prod
if (n % 2 == 0):
generic_prod *= quadratic_L_function__exact(n // 2,
ZZ(-1)**(n // 2) * char_d)
#print " NEW = ", ZZ(1) * quadratic_L_function__exact(n/2, (-1)**(n/2) * char_d)
#print
#print "gp4 = ", generic_prod
#print "generic_prod =", generic_prod
## Determine the adjustment factors
adj_prod = ZZ.one()
for p in prime_divisors(2 * self.det()):
## Cancel out the generic factors
p_adjustment = prod([1 - ZZ(p)**(-j) for j in range(2, 2 * s + 1, 2)])
if (n % 2 == 0):
p_adjustment *= (1 - kronecker(
(-1)**(n // 2) * char_d, p) * ZZ(p)**(-n // 2))
#print " EXTRA = ", ZZ(1) * (1 - kronecker((-1)**(n/2) * char_d, p) * ZZ(p)**(-n/2))
#print "Factor to cancel the generic one:", p_adjustment
## Insert the new mass factors
if p == 2:
if even_algorithm == "Kitaoka":
p_adjustment = p_adjustment / self.Kitaoka_mass_at_2()
elif even_algorithm == "Watson":
p_adjustment = p_adjustment / self.Watson_mass_at_2()
else:
raise TypeError(
"There is a problem -- your even_algorithm argument is invalid. Try again. =("
)
else:
if odd_algorithm == "Pall":
p_adjustment = p_adjustment / self.Pall_mass_density_at_odd_prime(
p)
else:
raise TypeError(
"There is a problem -- your optional arguments are invalid. Try again. =("
)
#print "p_adjustment for p =", p, "is", p_adjustment
## Put them together (cumulatively)
adj_prod *= p_adjustment
#print "Cumulative adj_prod =", adj_prod
## Extra adjustment for the case of a 2-dimensional form.
#if (n == 2):
# generic_prod *= 2
## Return the mass
mass = generic_prod * adj_prod
return mass
def supersingular_j(FF):
r"""
Return a supersingular j-invariant over the finite
field FF.
INPUT:
- ``FF`` -- finite field with p^2 elements, where p is a prime number
OUTPUT:
- finite field element -- a supersingular j-invariant
defined over the finite field FF
EXAMPLES:
The following examples calculate supersingular j-invariants for a
few finite fields::
sage: supersingular_j(GF(7^2, 'a'))
6
Observe that in this example the j-invariant is not defined over
the prime field::
sage: supersingular_j(GF(15073^2, 'a'))
4443*a + 13964
sage: supersingular_j(GF(83401^2, 'a'))
67977
AUTHORS:
- David Kohel -- [email protected]
- Iftikhar Burhanuddin -- [email protected]
"""
if not (FF.is_field()) or not (FF.is_finite()):
raise ValueError("%s is not a finite field" % FF)
prime = FF.characteristic()
if not (Integer(prime).is_prime()):
raise ValueError("%s is not a prime" % prime)
if not (Integer(FF.cardinality())) == Integer(prime**2):
raise ValueError("%s is not a quadratic extension" % FF)
if kronecker(-1, prime) != 1:
j_invss = 1728 # (2^2 * 3)^3
elif kronecker(-2, prime) != 1:
j_invss = 8000 # (2^2 * 5)^3
elif kronecker(-3, prime) != 1:
j_invss = 0 # 0^3
elif kronecker(-7, prime) != 1:
j_invss = 16581375 # (3 * 5 * 17)^3
elif kronecker(-11, prime) != 1:
j_invss = -32768 # -(2^5)^3
elif kronecker(-19, prime) != 1:
j_invss = -884736 # -(2^5 * 3)^3
elif kronecker(-43, prime) != 1:
j_invss = -884736000 # -(2^6 * 3 * 5)^3
elif kronecker(-67, prime) != 1:
j_invss = -147197952000 # -(2^5 * 3 * 5 * 11)^3
elif kronecker(-163, prime) != 1:
j_invss = -262537412640768000 # -(2^6 * 3 * 5 * 23 * 29)^3
else:
D = supersingular_D(prime)
hc_poly = FF['x'](pari(D).polclass())
root_hc_poly_list = list(hc_poly.roots())
j_invss = root_hc_poly_list[0][0]
return FF(j_invss)
def mass__by_Siegel_densities(self, odd_algorithm="Pall", even_algorithm="Watson"):
"""
Gives the mass of transformations (det 1 and -1).
WARNING: THIS IS BROKEN RIGHT NOW... =(
Optional Arguments:
- When p > 2 -- odd_algorithm = "Pall" (only one choice for now)
- When p = 2 -- even_algorithm = "Kitaoka" or "Watson"
REFERENCES:
- Nipp's Book "Tables of Quaternary Quadratic Forms".
- Papers of Pall (only for p>2) and Watson (for `p=2` -- tricky!).
- Siegel, Milnor-Hussemoller, Conway-Sloane Paper IV, Kitoaka (all of which
have problems...)
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: m = Q.mass__by_Siegel_densities(); m
1/384
sage: m - (2^Q.dim() * factorial(Q.dim()))^(-1)
0
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: m = Q.mass__by_Siegel_densities(); m
1/48
sage: m - (2^Q.dim() * factorial(Q.dim()))^(-1)
0
"""
## Setup
n = self.dim()
s = (n-1) // 2
if n % 2 != 0:
char_d = squarefree_part(2*self.det()) ## Accounts for the det as a QF
else:
char_d = squarefree_part(self.det())
## Form the generic zeta product
generic_prod = ZZ(2) * (pi)**(-ZZ(n) * (n+1) / 4)
##########################################
generic_prod *= (self.det())**(ZZ(n+1)/2) ## ***** This uses the Hessian Determinant ********
##########################################
#print "gp1 = ", generic_prod
generic_prod *= prod([gamma__exact(ZZ(j)/2) for j in range(1,n+1)])
#print "\n---", [(ZZ(j)/2, gamma__exact(ZZ(j)/2)) for j in range(1,n+1)]
#print "\n---", prod([gamma__exact(ZZ(j)/2) for j in range(1,n+1)])
#print "gp2 = ", generic_prod
generic_prod *= prod([zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)])
#print "\n---", [zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)]
#print "\n---", prod([zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)])
#print "gp3 = ", generic_prod
if (n % 2 == 0):
generic_prod *= quadratic_L_function__exact(n//2, ZZ(-1)**(n//2) * char_d)
#print " NEW = ", ZZ(1) * quadratic_L_function__exact(n/2, (-1)**(n/2) * char_d)
#print
#print "gp4 = ", generic_prod
#print "generic_prod =", generic_prod
## Determine the adjustment factors
adj_prod = ZZ.one()
for p in prime_divisors(2 * self.det()):
## Cancel out the generic factors
p_adjustment = prod([1 - ZZ(p)**(-j) for j in range(2, 2*s+1, 2)])
if (n % 2 == 0):
p_adjustment *= (1 - kronecker((-1)**(n//2) * char_d, p) * ZZ(p)**(-n//2))
#print " EXTRA = ", ZZ(1) * (1 - kronecker((-1)**(n/2) * char_d, p) * ZZ(p)**(-n/2))
#print "Factor to cancel the generic one:", p_adjustment
## Insert the new mass factors
if p == 2:
if even_algorithm == "Kitaoka":
p_adjustment = p_adjustment / self.Kitaoka_mass_at_2()
elif even_algorithm == "Watson":
p_adjustment = p_adjustment / self.Watson_mass_at_2()
else:
raise TypeError("There is a problem -- your even_algorithm argument is invalid. Try again. =(")
else:
if odd_algorithm == "Pall":
p_adjustment = p_adjustment / self.Pall_mass_density_at_odd_prime(p)
else:
raise TypeError("There is a problem -- your optional arguments are invalid. Try again. =(")
#print "p_adjustment for p =", p, "is", p_adjustment
## Put them together (cumulatively)
adj_prod *= p_adjustment
#print "Cumulative adj_prod =", adj_prod
## Extra adjustment for the case of a 2-dimensional form.
#if (n == 2):
# generic_prod *= 2
## Return the mass
mass = generic_prod * adj_prod
return mass
def supersingular_j(FF):
r"""
This function returns a supersingular j-invariant over the finite
field FF.
INPUT:
- ``FF`` -- finite field with p^2 elements, where p is a prime number
OUTPUT:
finite field element -- a supersingular j-invariant
defined over the finite field FF
EXAMPLES:
The following examples calculate supersingular j-invariants for a
few finite fields::
sage: supersingular_j(GF(7^2, 'a'))
6
Observe that in this example the j-invariant is not defined over
the prime field::
sage: supersingular_j(GF(15073^2, 'a'))
4443*a + 13964
sage: supersingular_j(GF(83401^2, 'a'))
67977
AUTHORS:
- David Kohel -- [email protected]
- Iftikhar Burhanuddin -- [email protected]
"""
if not(FF.is_field()) or not(FF.is_finite()):
raise ValueError("%s is not a finite field"%FF)
prime = FF.characteristic()
if not(rings.Integer(prime).is_prime()):
raise ValueError("%s is not a prime"%prime)
if not(rings.Integer(FF.cardinality())) == rings.Integer(prime**2):
raise ValueError("%s is not a quadratic extension"%FF)
if kronecker(-1, prime) != 1:
j_invss = 1728 #(2^2 * 3)^3
elif kronecker(-2, prime) != 1:
j_invss = 8000 #(2^2 * 5)^3
elif kronecker(-3, prime) != 1:
j_invss = 0 #0^3
elif kronecker(-7, prime) != 1:
j_invss = 16581375 #(3 * 5 * 17)^3
elif kronecker(-11, prime) != 1:
j_invss = -32768 #-(2^5)^3
elif kronecker(-19, prime) != 1:
j_invss = -884736 #-(2^5 * 3)^3
elif kronecker(-43, prime) != 1:
j_invss = -884736000 #-(2^6 * 3 * 5)^3
elif kronecker(-67, prime) != 1:
j_invss = -147197952000 #-(2^5 * 3 * 5 * 11)^3
elif kronecker(-163, prime) != 1:
j_invss = -262537412640768000 #-(2^6 * 3 * 5 * 23 * 29)^3
else:
D = supersingular_D(prime)
hc_poly = FF['x'](pari(D).polclass())
root_hc_poly_list = list(hc_poly.roots())
j_invss = root_hc_poly_list[0][0]
return FF(j_invss)
