def basic_integral(Phi,a,j,ap,D):
"""
Returns `\int_{a+pZ_p} (z-{a})^j d\Phi(0-infty)`
-- see formula [Pollack-Stevens, sec 9.2]
INPUT:
- ``Phi`` -- overconvergnt `U_p`-eigensymbol
- ``a`` -- integer in [0..p-1]
- ``j`` -- positive integer
- ``ap`` -- Hecke eigenvalue?
- ``D`` -- conductor of the quadratic twist `\chi`
OUTPUT:
`\int_{a+pZ_p} (z-{a})^j d\Phi(0-\infty)`
EXAMPLES:
"""
M = Phi.num_moments()
p = Phi.p()
ap = ap*kronecker(D,p)
ans = 0
for r in range(j+1):
ans = ans+binomial(j,r)*((a-teich(a,p,M))**(j-r))*(p**r)*phi_on_Da(Phi,a,D).moment(r)
return ans/ap
def eval_twisted_symbol_on_Da(self, a): # rename! should this be in modsym?
"""
Returns `\Phi_{\chi}(\{a/p}-{\infty})` where `Phi` is the OMS and
`\chi` is a the quadratic character corresponding to self
INPUT:
- ``a`` -- integer in range(p)
OUTPUT:
The distribution `\Phi_{\chi}(\{a/p\}-\{\infty\})`.
EXAMPLES:
sage: from sage.modular.pollack_stevens.space import ps_modsym_from_elliptic_curve
sage: E = EllipticCurve('57a')
sage: p = 5
sage: prec = 4
sage: phi = ps_modsym_from_elliptic_curve(E)
sage: ap = phi.Tq_eigenvalue(p,prec)
sage: Phi = phi.p_stabilize_and_lift(p,ap = ap, M = prec, algorithm='stevens')
sage: L = pAdicLseries(Phi)
sage: L.eval_twisted_symbol_on_Da(1)
(2 + 2*5 + 2*5^2 + 2*5^3 + O(5^4), 2 + 3*5 + 2*5^2 + O(5^3), 4*5 + O(5^2), 3 + O(5))
sage: from sage.modular.pollack_stevens.space import ps_modsym_from_elliptic_curve
sage: E = EllipticCurve('40a4')
sage: p = 7
sage: prec = 4
sage: phi = ps_modsym_from_elliptic_curve(E)
sage: ap = phi.Tq_eigenvalue(p,prec)
sage: Phi = phi.p_stabilize_and_lift(p,ap = ap, M = prec, algorithm='stevens')
sage: L = pAdicLseries(Phi)
sage: L.eval_twisted_symbol_on_Da(1)
(4 + 6*7 + 3*7^2 + O(7^4), 2 + 7 + O(7^3), 4 + 6*7 + O(7^2), 6 + O(7))
"""
symb = self.symb()
p = symb.parent().prime()
S0p = Sigma0(p)
Dists = symb.parent().coefficient_module()
M = Dists.precision_cap()
p = Dists.prime()
twisted_dist = Dists.zero_element()
m_map = symb._map
D = self._quadratic_twist
for b in range(1, abs(D) + 1):
if gcd(b, D) == 1:
M1 = S0p([1, (b / abs(D)) % p**M, 0, 1])
new_dist = m_map(M1 * M2Z([a, 1, p, 0]))*M1
new_dist = new_dist.scale(kronecker(D, b)).normalize()
twisted_dist = twisted_dist + new_dist
#ans = ans + self.eval(M1 * M2Z[a, 1, p, 0])._right_action(M1)._lmul_(kronecker(D, b)).normalize()
return twisted_dist.normalize()
def _basic_integral(self, a, j):
r"""
Returns `\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.symb().precision_absolute())
EXAMPLES::
sage: from sage.modular.pollack_stevens.space import ps_modsym_from_elliptic_curve
sage: E = EllipticCurve('57a')
sage: p = 5
sage: prec = 4
sage: phi = ps_modsym_from_elliptic_curve(E)
sage: phi_stabilized = phi.p_stabilize(p,M = prec+3)
sage: Phi = phi_stabilized.lift(p,prec,None,algorithm = 'stevens',eigensymbol = True)
sage: L = pAdicLseries(Phi)
sage: L.eval_twisted_symbol_on_Da(1)
(2 + 2*5 + 2*5^2 + 2*5^3 + O(5^4), 2 + 3*5 + 2*5^2 + O(5^3), 4*5 + O(5^2), 3 + O(5))
sage: L._basic_integral(1,2)
2*5^3 + O(5^4)
"""
symb = self.symb()
M = symb.precision_absolute()
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 = self.eval_twisted_symbol_on_Da(a)
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 phi_on_Da(Phi,a,D):
"""
Returns `\Phi_{\chi}` where `\chi` is a character of conductor `D`
INPUT:
- ``Phi`` -- overconvergent `U_p`-eigensymbol
- ``a`` -- integer in [0..p-1]
- ``D`` -- conductor of the quadratic twist `\chi`
OUTPUT:
`\Phi_{\chi}`
EXAMPLES:
"""
p = Phi.p()
ans = Phi.zero_elt()
for b in range(1,abs(D)+1):
if gcd(b,D)==1:
M1 = Matrix(2,2,[1,b/abs(D),0,1])
ans=ans+Phi.eval(M1*Matrix(2,2,[a,1,p,0])).act_right(M1).scale(kronecker(D,b)).normalize()
return ans.normalize()
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: Q.mass__by_Siegel_densities()
1/384
sage: Q.mass__by_Siegel_densities() - (2^Q.dim() * factorial(Q.dim()))^(-1)
0
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.mass__by_Siegel_densities()
1/48
sage: Q.mass__by_Siegel_densities() - (2^Q.dim() * factorial(Q.dim()))^(-1)
0
"""
## Setup
n = self.dim()
s = floor((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 *= ZZ(1) * quadratic_L_function__exact(
n / 2, (-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 = 1
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 *= ZZ(1) * (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 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: Q.mass__by_Siegel_densities()
1/384
sage: Q.mass__by_Siegel_densities() - (2^Q.dim() * factorial(Q.dim()))^(-1)
0
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.mass__by_Siegel_densities()
1/48
sage: Q.mass__by_Siegel_densities() - (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 *= ZZ(1) * quadratic_L_function__exact(n/2, (-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 = 1
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 *= ZZ(1) * (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
