def __init__(self, E, A, prec=53):
r"""
Class for the Gross-Zagier L-series.
This is attached to a pair `(E,A)` where `E` is an elliptic curve over
`\QQ` and `A` is an ideal class in an imaginary quadratic number field.
For the exact definition, in the more general setting of modular forms
instead of elliptic curves, see section IV of [GrossZagier]_.
INPUT:
- ``E`` -- an elliptic curve over `\QQ`
- ``A`` -- an ideal class in an imaginary quadratic number field
- ``prec`` -- an integer (default 53) giving the required precision
EXAMPLES::
sage: e = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0)
sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries
sage: G = GrossZagierLseries(e, A)
TESTS::
sage: K.<b> = QuadraticField(131)
sage: A = K.class_group().one()
sage: G = GrossZagierLseries(e, A)
Traceback (most recent call last):
...
ValueError: A is not an ideal class in an imaginary quadratic field
"""
self._E = E
self._N = N = E.conductor()
self._A = A
ideal = A.ideal()
K = A.gens()[0].parent()
D = K.disc()
if not (K.degree() == 2 and D < 0):
raise ValueError("A is not an ideal class in an"
" imaginary quadratic field")
Q = ideal.quadratic_form().reduced_form()
epsilon = -kronecker_character(D)(N)
self._dokchister = Dokchitser(N**2 * D**2, [0, 0, 1, 1],
weight=2,
eps=epsilon,
prec=prec)
self._nterms = nterms = Integer(self._dokchister.gp()('cflength()'))
if nterms > 1e6:
# just takes way to long
raise ValueError("Too many terms: {}".format(nterms))
zeta_ord = ideal.number_field().zeta_order()
an_list = gross_zagier_L_series(E.anlist(nterms + 1), Q, N, zeta_ord)
self._dokchister.gp().set('a', an_list[1:])
self._dokchister.init_coeffs('a[k]', 1)
def __init__(self, E, A, prec=53):
r"""
Class for the Gross-Zagier L-series.
This is attached to a pair `(E,A)` where `E` is an elliptic curve over
`\QQ` and `A` is an ideal class in an imaginary quadratic number field.
For the exact definition, in the more general setting of modular forms
instead of elliptic curves, see section IV of [GrossZagier]_.
INPUT:
- ``E`` -- an elliptic curve over `\QQ`
- ``A`` -- an ideal class in an imaginary quadratic number field
- ``prec`` -- an integer (default 53) giving the required precision
EXAMPLES::
sage: e = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0)
sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries
sage: G = GrossZagierLseries(e, A)
TESTS::
sage: K.<b> = QuadraticField(131)
sage: A = K.class_group().one()
sage: G = GrossZagierLseries(e, A)
Traceback (most recent call last):
...
ValueError: A is not an ideal class in an imaginary quadratic field
"""
self._E = E
self._N = N = E.conductor()
self._A = A
ideal = A.ideal()
K = A.gens()[0].parent()
D = K.disc()
if not(K.degree() == 2 and D < 0):
raise ValueError("A is not an ideal class in an"
" imaginary quadratic field")
Q = ideal.quadratic_form().reduced_form()
epsilon = - kronecker_character(D)(N)
self._dokchister = Dokchitser(N ** 2 * D ** 2,
[0, 0, 1, 1],
weight=2, eps=epsilon, prec=prec)
self._nterms = nterms = Integer(self._dokchister.num_coeffs())
if nterms > 1e6:
# just takes way to long
raise ValueError("Too many terms: {}".format(nterms))
zeta_ord = ideal.number_field().zeta_order()
an_list = gross_zagier_L_series(E.anlist(nterms + 1), Q, N, zeta_ord)
self._dokchister.gp().set('a', an_list[1:])
self._dokchister.init_coeffs('a[k]', 1)
def dokchitser(self, prec):
# NOTE: In order to get the Dokchitser parameters of an L-function,
# it is useful to know that
#
# gamma(s) = 2^s*gamma(s/2)*gamma((s+1)/2) / (2*sqrt(pi))
#
conductor = self.level()**10
gammaV = [-1,-1,-1,0,0,0,0,1]
weight = 4
eps = self.epsilon()
poles = []
residues = []
from sage.lfunctions.dokchitser import Dokchitser
L = Dokchitser(conductor = conductor,
gammaV = gammaV,
weight = weight,
eps = eps,
poles = poles, residues=[])
#s = 'v=%s; a(k)=if(k>%s,0,v[k])'%(self.dirichlet_series_coeffs(prec), prec)
s = 'v=%s; a(k)=v[k]'%(self.dirichlet_series_coeffs(prec))
L.init_coeffs('a(k)', pari_precode=s)
return L
def dokchitser(self, prec):
# NOTE: In order to get the Dokchitser parameters of an L-function,
# it is useful to know that
#
# gamma(s) = 2^s*gamma(s/2)*gamma((s+1)/2) / (2*sqrt(pi))
#
conductor = self.level()**10
gammaV = [-1, -1, -1, 0, 0, 0, 0, 1]
weight = 4
eps = self.epsilon()
poles = []
residues = []
from sage.lfunctions.dokchitser import Dokchitser
L = Dokchitser(conductor=conductor,
gammaV=gammaV,
weight=weight,
eps=eps,
poles=poles,
residues=[])
#s = 'v=%s; a(k)=if(k>%s,0,v[k])'%(self.dirichlet_series_coeffs(prec), prec)
s = 'v=%s; a(k)=v[k]' % (self.dirichlet_series_coeffs(prec))
L.init_coeffs('a(k)', pari_precode=s)
return L
class GrossZagierLseries(SageObject):
def __init__(self, E, A, prec=53):
r"""
Class for the Gross-Zagier L-series.
This is attached to a pair `(E,A)` where `E` is an elliptic curve over
`\QQ` and `A` is an ideal class in an imaginary quadratic number field.
For the exact definition, in the more general setting of modular forms
instead of elliptic curves, see section IV of [GrossZagier]_.
INPUT:
- ``E`` -- an elliptic curve over `\QQ`
- ``A`` -- an ideal class in an imaginary quadratic number field
- ``prec`` -- an integer (default 53) giving the required precision
EXAMPLES::
sage: e = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0)
sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries
sage: G = GrossZagierLseries(e, A)
TESTS::
sage: K.<b> = QuadraticField(131)
sage: A = K.class_group().one()
sage: G = GrossZagierLseries(e, A)
Traceback (most recent call last):
...
ValueError: A is not an ideal class in an imaginary quadratic field
"""
self._E = E
self._N = N = E.conductor()
self._A = A
ideal = A.ideal()
K = A.gens()[0].parent()
D = K.disc()
if not(K.degree() == 2 and D < 0):
raise ValueError("A is not an ideal class in an"
" imaginary quadratic field")
Q = ideal.quadratic_form().reduced_form()
epsilon = - kronecker_character(D)(N)
self._dokchister = Dokchitser(N ** 2 * D ** 2,
[0, 0, 1, 1],
weight=2, eps=epsilon, prec=prec)
self._nterms = nterms = Integer(self._dokchister.num_coeffs())
if nterms > 1e6:
# just takes way to long
raise ValueError("Too many terms: {}".format(nterms))
zeta_ord = ideal.number_field().zeta_order()
an_list = gross_zagier_L_series(E.anlist(nterms + 1), Q, N, zeta_ord)
self._dokchister.gp().set('a', an_list[1:])
self._dokchister.init_coeffs('a[k]', 1)
def __call__(self, s, der=0):
r"""
Return the value at `s`.
INPUT:
- `s` -- complex number
- ``der`` -- ? (default 0)
EXAMPLES::
sage: e = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0)
sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries
sage: G = GrossZagierLseries(e, A)
sage: G(3)
-0.272946890617590
"""
return self._dokchister(s, der)
def taylor_series(self, s=1, series_prec=6, var='z'):
r"""
Return the Taylor series at `s`.
INPUT:
- `s` -- complex number (default 1)
- ``series_prec`` -- number of terms (default 6) in the Taylor series
- ``var`` -- variable (default 'z')
EXAMPLES::
sage: e = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0)
sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries
sage: G = GrossZagierLseries(e, A)
sage: G.taylor_series(2,3)
-0.613002046122894 + 0.490374999263514*z - 0.122903033710382*z^2 + O(z^3)
"""
return self._dokchister.taylor_series(s, series_prec, var)
def _repr_(self):
"""
Return the string representation.
EXAMPLES::
sage: e = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0)
sage: from sage.modular.modform.l_series_gross_zagier import GrossZagierLseries
sage: GrossZagierLseries(e, A)
Gross Zagier L-series attached to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field with ideal class Fractional ideal class (2, 1/2*a)
"""
msg = "Gross Zagier L-series attached to {} with ideal class {}"
return msg.format(self._E, self._A)
def lseries(self,
num_prec=None,
max_imaginary_part=0,
max_asymp_coeffs=40):
r"""
Return the L-series of ``self`` if ``self`` is modular and holomorphic.
Note: This relies on the (pari) based function ``Dokchitser``.
INPUT:
- ``num_prec`` -- An integer denoting the to-be-used numerical precision.
If integer ``num_prec=None`` (default) the default
numerical precision of the parent of ``self`` is used.
- ``max_imaginary_part`` -- A real number (default: 0), indicating up to which
imaginary part the L-series is going to be studied.
- ``max_asymp_coeffs`` -- An integer (default: 40).
OUTPUT:
An interface to Tim Dokchitser's program for computing L-series, namely
the series given by the Fourier coefficients of ``self``.
EXAMPLES::
sage: from sage.modular.modform.eis_series import eisenstein_series_lseries
sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: f = ModularForms(n=3, k=4).E4()/240
sage: L = f.lseries()
sage: L
L-series associated to the modular form 1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + O(q^5)
sage: L.conductor
1
sage: L(1).prec()
53
sage: L.check_functional_equation() < 2^(-50)
True
sage: L(1)
-0.0304484570583...
sage: abs(L(1) - eisenstein_series_lseries(4)(1)) < 2^(-53)
True
sage: L.derivative(1, 1)
-0.0504570844798...
sage: L.derivative(1, 2)/2
-0.0350657360354...
sage: L.taylor_series(1, 3)
-0.0304484570583... - 0.0504570844798...*z - 0.0350657360354...*z^2 + O(z^3)
sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True)
sage: sum([coeffs[k]*k^(-10) for k in range(1,len(coeffs))]).n(53)
1.00935215408...
sage: L(10)
1.00935215649...
sage: f = ModularForms(n=6, k=4).E4()
sage: L = f.lseries(num_prec=200)
sage: L.conductor
3
sage: L.check_functional_equation() < 2^(-180)
True
sage: L(1)
-2.92305187760575399490414692523085855811204642031749788...
sage: L(1).prec()
200
sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True)
sage: sum([coeffs[k]*k^(-10) for k in range(1,len(coeffs))]).n(53)
24.2281438789...
sage: L(10).n(53)
24.2281439447...
sage: f = ModularForms(n=8, k=6, ep=-1).E6()
sage: L = f.lseries()
sage: L.check_functional_equation() < 2^(-45)
True
sage: L.taylor_series(3, 3)
0.000000000000... + 0.867197036668...*z + 0.261129628199...*z^2 + O(z^3)
sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True)
sage: sum([coeffs[k]*k^(-10) for k in range(1,len(coeffs))]).n(53)
-13.0290002560...
sage: L(10).n(53)
-13.0290184579...
sage: f = (ModularForms(n=17, k=24).Delta()^2) # long time
sage: L = f.lseries() # long time
sage: L.check_functional_equation() < 2^(-50) # long time
True
sage: L.taylor_series(12, 3) # long time
0.000683924755280... - 0.000875942285963...*z + 0.000647618966023...*z^2 + O(z^3)
sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True) # long time
sage: sum([coeffs[k]*k^(-30) for k in range(1,len(coeffs))]).n(53) # long time
9.31562890589...e-10
sage: L(30).n(53) # long time
9.31562890589...e-10
sage: f = ModularForms(n=infinity, k=2, ep=-1).f_i()
sage: L = f.lseries()
sage: L.check_functional_equation() < 2^(-50)
True
sage: L.taylor_series(1, 3)
0.000000000000... + 5.76543616701...*z + 9.92776715593...*z^2 + O(z^3)
sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True)
sage: sum([coeffs[k]*k^(-10) for k in range(1,len(coeffs))]).n(53)
-23.9781792831...
sage: L(10).n(53)
-23.9781792831...
"""
from sage.rings.all import ZZ
from sage.symbolic.all import pi
from sage.functions.other import sqrt
from sage.lfunctions.dokchitser import Dokchitser
if (not (self.is_modular() and self.is_holomorphic())
or self.weight() == 0):
raise NotImplementedError(
"L-series are only implemented for non-trivial holomorphic modular forms."
)
if (num_prec is None):
num_prec = self.parent().default_num_prec()
conductor = self.group().lam()**2
if (self.group().is_arithmetic()):
conductor = ZZ(conductor)
else:
conductor = conductor.n(num_prec)
gammaV = [0, 1]
weight = self.weight()
eps = self.ep()
# L^*(s) = cor_factor * (2*pi)^(-s)gamma(s)*L(f,s),
cor_factor = (2 * sqrt(pi)).n(num_prec)
if (self.is_cuspidal()):
poles = []
residues = []
else:
poles = [weight]
val_inf = self.q_expansion_fixed_d(prec=1, d_num_prec=num_prec)[0]
residue = eps * val_inf * cor_factor
# (pari) BUG?
# The residue of the above L^*(s) differs by a factor -1 from
# the residue pari expects (?!?).
residue *= -1
residues = [residue]
L = Dokchitser(conductor=conductor,
gammaV=gammaV,
weight=weight,
eps=eps,
poles=poles,
residues=residues,
prec=num_prec)
# TODO for later: Figure out the correct coefficient growth and do L.set_coeff_growth(...)
# num_coeffs = L.num_coeffs()
num_coeffs = L.num_coeffs(1.2)
coeff_vector = [
coeff for coeff in self.q_expansion_vector(
min_exp=0, max_exp=num_coeffs + 1, fix_d=True)
]
pari_precode = "coeff = {};".format(coeff_vector)
L.init_coeffs(v="coeff[k+1]",
pari_precode=pari_precode,
max_imaginary_part=max_imaginary_part,
max_asymp_coeffs=max_asymp_coeffs)
L.check_functional_equation()
L.rename("L-series associated to the {} form {}".format(
"cusp" if self.is_cuspidal() else "modular", self))
return L
def lseries(self, num_prec=None, max_imaginary_part=0, max_asymp_coeffs=40):
r"""
Return the L-series of ``self`` if ``self`` is modular and holomorphic.
Note: This relies on the (pari) based function ``Dokchitser``.
INPUT:
- ``num_prec`` -- An integer denoting the to-be-used numerical precision.
If integer ``num_prec=None`` (default) the default
numerical precision of the parent of ``self`` is used.
- ``max_imaginary_part`` -- A real number (default: 0), indicating up to which
imaginary part the L-series is going to be studied.
- ``max_asymp_coeffs`` -- An integer (default: 40).
OUTPUT:
An interface to Tim Dokchitser's program for computing L-series, namely
the series given by the Fourier coefficients of ``self``.
EXAMPLES::
sage: from sage.modular.modform.eis_series import eisenstein_series_lseries
sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: f = ModularForms(n=3, k=4).E4()/240
sage: L = f.lseries()
sage: L
L-series associated to the modular form 1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + O(q^5)
sage: L.conductor
1
sage: L(1).prec()
53
sage: L.check_functional_equation() < 2^(-50)
True
sage: L(1)
-0.0304484570583...
sage: abs(L(1) - eisenstein_series_lseries(4)(1)) < 2^(-53)
True
sage: L.derivative(1, 1)
-0.0504570844798...
sage: L.derivative(1, 2)/2
-0.0350657360354...
sage: L.taylor_series(1, 3)
-0.0304484570583... - 0.0504570844798...*z - 0.0350657360354...*z^2 + O(z^3)
sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True)
sage: sum([coeffs[k]*k^(-10) for k in range(1,len(coeffs))]).n(53)
1.00935215408...
sage: L(10)
1.00935215649...
sage: f = ModularForms(n=6, k=4).E4()
sage: L = f.lseries(num_prec=200)
sage: L.conductor
3
sage: L.check_functional_equation() < 2^(-180)
True
sage: L(1)
-2.92305187760575399490414692523085855811204642031749788...
sage: L(1).prec()
200
sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True)
sage: sum([coeffs[k]*k^(-10) for k in range(1,len(coeffs))]).n(53)
24.2281438789...
sage: L(10).n(53)
24.2281439447...
sage: f = ModularForms(n=8, k=6, ep=-1).E6()
sage: L = f.lseries()
sage: L.check_functional_equation() < 2^(-45)
True
sage: L.taylor_series(3, 3)
0.000000000000... + 0.867197036668...*z + 0.261129628199...*z^2 + O(z^3)
sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True)
sage: sum([coeffs[k]*k^(-10) for k in range(1,len(coeffs))]).n(53)
-13.0290002560...
sage: L(10).n(53)
-13.0290184579...
sage: f = (ModularForms(n=17, k=24).Delta()^2) # long time
sage: L = f.lseries() # long time
sage: L.check_functional_equation() < 2^(-50) # long time
True
sage: L.taylor_series(12, 3) # long time
0.000683924755280... - 0.000875942285963...*z + 0.000647618966023...*z^2 + O(z^3)
sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True) # long time
sage: sum([coeffs[k]*k^(-30) for k in range(1,len(coeffs))]).n(53) # long time
9.31562890589...e-10
sage: L(30).n(53) # long time
9.31562890589...e-10
sage: f = ModularForms(n=infinity, k=2, ep=-1).f_i()
sage: L = f.lseries()
sage: L.check_functional_equation() < 2^(-50)
True
sage: L.taylor_series(1, 3)
0.000000000000... + 5.76543616701...*z + 9.92776715593...*z^2 + O(z^3)
sage: coeffs = f.q_expansion_vector(min_exp=0, max_exp=20, fix_d=True)
sage: sum([coeffs[k]*k^(-10) for k in range(1,len(coeffs))]).n(53)
-23.9781792831...
sage: L(10).n(53)
-23.9781792831...
"""
from sage.rings.all import ZZ
from sage.symbolic.all import pi
from sage.functions.other import sqrt
from sage.lfunctions.dokchitser import Dokchitser
if (not (self.is_modular() and self.is_holomorphic()) or self.weight() == 0):
raise NotImplementedError("L-series are only implemented for non-trivial holomorphic modular forms.")
if (num_prec is None):
num_prec = self.parent().default_num_prec()
conductor = self.group().lam()**2
if (self.group().is_arithmetic()):
conductor = ZZ(conductor)
else:
conductor = conductor.n(num_prec)
gammaV = [0, 1]
weight = self.weight()
eps = self.ep()
# L^*(s) = cor_factor * (2*pi)^(-s)gamma(s)*L(f,s),
cor_factor = (2*sqrt(pi)).n(num_prec)
if (self.is_cuspidal()):
poles = []
residues = []
else:
poles = [ weight ]
val_inf = self.q_expansion_fixed_d(prec=1, d_num_prec=num_prec)[0]
residue = eps * val_inf * cor_factor
# (pari) BUG?
# The residue of the above L^*(s) differs by a factor -1 from
# the residue pari expects (?!?).
residue *= -1
residues = [ residue ]
L = Dokchitser(conductor = conductor,
gammaV = gammaV,
weight = weight,
eps = eps,
poles = poles,
residues = residues,
prec = num_prec)
# TODO for later: Figure out the correct coefficient growth and do L.set_coeff_growth(...)
# num_coeffs = L.num_coeffs()
num_coeffs = L.num_coeffs(1.2)
coeff_vector = [coeff for coeff in self.q_expansion_vector(min_exp=0, max_exp=num_coeffs + 1, fix_d=True)]
pari_precode = "coeff = {};".format(coeff_vector)
L.init_coeffs(v = "coeff[k+1]", pari_precode = pari_precode, max_imaginary_part = max_imaginary_part, max_asymp_coeffs = max_asymp_coeffs)
L.check_functional_equation()
L.rename("L-series associated to the {} form {}".format("cusp" if self.is_cuspidal() else "modular", self))
return L
