def lseries_dokchitser(E, prec=53):
"""
Return the Dokchitser L-series object associated to the elliptic
curve E, which may be defined over the rational numbers or a
number field. Also prec is the number of bits of precision to
which evaluation of the L-series occurs.
INPUT:
- E -- elliptic curve over a number field (or QQ)
- prec -- integer (default: 53) precision in *bits*
OUTPUT:
- Dokchitser L-function object
EXAMPLES::
A curve over Q(sqrt(5)), for which we have an optimized implementation::
sage: from psage.ellcurve.lseries.lseries_nf import lseries_dokchitser
sage: K.<a> = NumberField(x^2-x-1); E = EllipticCurve([0,-a,a,0,0])
sage: L = lseries_dokchitser(E); L
Dokchitser L-function of Elliptic Curve defined by y^2 + a*y = x^3 + (-a)*x^2 over Number Field in a with defining polynomial x^2 - x - 1
sage: L(1)
0.422214159001667
sage: L.taylor_series(1,5)
0.422214159001667 + 0.575883864741340*z - 0.102163426876721*z^2 - 0.158119743123727*z^3 + 0.120350687595265*z^4 + O(z^5)
Higher precision::
sage: L = lseries_dokchitser(E, 200)
sage: L(1)
0.42221415900166715092967967717023093014455137669360598558872
A curve over Q(i)::
sage: K.<i> = NumberField(x^2 + 1)
sage: E = EllipticCurve(K, [1,0])
sage: E.conductor().norm()
256
sage: L = lseries_dokchitser(E, 10)
sage: L.taylor_series(1,5)
0.86 + 0.58*z - 0.62*z^2 + 0.19*z^3 + 0.18*z^4 + O(z^5)
More examples::
sage: lseries_dokchitser(EllipticCurve([0,-1,1,0,0]), 10)(1)
0.25
sage: K.<i> = NumberField(x^2+1)
sage: lseries_dokchitser(EllipticCurve(K, [0,-1,1,0,0]), 10)(1)
0.37
sage: K.<a> = NumberField(x^2-x-1)
sage: lseries_dokchitser(EllipticCurve(K, [0,-1,1,0,0]), 10)(1)
0.72
sage: E = EllipticCurve([0,-1,1,0,0])
sage: E.quadratic_twist(2).rank()
1
sage: K.<d> = NumberField(x^2-2)
sage: L = lseries_dokchitser(EllipticCurve(K, [0,-1,1,0,0]), 10)
sage: L(1)
0
sage: L.taylor_series(1, 5)
0.58*z + 0.20*z^2 - 0.50*z^3 + 0.28*z^4 + O(z^5)
You can use this function as an algorithm to compute the sign of the functional
equation (global root number)::
sage: E = EllipticCurve([1..5])
sage: E.root_number()
-1
sage: L = lseries_dokchitser(E,32); L
Dokchitser L-function of Elliptic Curve defined by y^2 + x*y = x^3 - x^2 + 4*x + 3 over Rational Field
sage: L.eps
-1
Over QQ, this isn't so useful (since Sage has a root_number
method), but over number fields it is very useful::
sage: K.<a> = NumberField(x^2 - x - 1)
sage: E1=EllipticCurve([0,-a-1,1,a,0]); E0 = EllipticCurve([0,-a,a,0,0])
sage: lseries_dokchitser(E1, 16).eps
-1
sage: E1.rank()
1
sage: lseries_dokchitser(E0, 16).eps
1
sage: E0.rank()
0
"""
# The code assumes in various places that we have a global minimal model,
# for example, in anlist_sqrt5 above.
E = E.global_minimal_model()
# Check that we're over a number field.
K = E.base_field()
if not is_NumberField(K):
raise TypeError("base field must be a number field")
# Compute norm of the conductor -- awkward because QQ elements have no norm method (they should).
N = E.conductor()
if K != QQ:
N = N.norm()
# We guess the sign epsilon in the functional equation to be +1
# first. If our guess is wrong then we just choose the other
# possibility.
epsilon = 1
# Define the Dokchitser L-function object with all parameters set:
L = Dokchitser(conductor=N * K.discriminant()**2,
gammaV=[0] * K.degree() + [1] * K.degree(),
weight=2,
eps=epsilon,
poles=[],
prec=prec)
# Find out how many coefficients of the Dirichlet series are needed
# to compute to the requested precision.
n = L.num_coeffs()
# print "num coeffs = %s"%n
# Compute the Dirichlet series coefficients
coeffs = anlist(E, n)[1:]
# Define a string that when evaluated in PARI defines a function
# a(k), which returns the Dirichlet coefficient a_k.
s = 'v=%s; a(k)=v[k];' % coeffs
# Actually tell the L-series / PARI about the coefficients.
L.init_coeffs('a(k)', pari_precode=s)
# Test that the functional equation is satisfied. This will very,
# very, very likely if we chose the sign of the functional
# equation incorrectly, or made any mistake in computing the
# Dirichlet series coefficients.
tiny = max(1e-8, old_div(1.0, 2**(prec - 1)))
if abs(L.check_functional_equation()) > tiny:
# The test failed, so we try the other choice of functional equation.
epsilon *= -1
L.eps = epsilon
# It is not necessary to recreate L -- just tell PARI the different sign.
L._gp_eval('sgn = %s' % epsilon)
# Once again, verify that the functional equation is
# satisfied. If it is, then we've got it. If it isn't, then
# there is definitely some other subtle bug, probably in computed
# the Dirichlet series coefficients.
if abs(L.check_functional_equation()) > tiny:
raise RuntimeError(
"Functional equation not numerically satisfied for either choice of sign"
)
L.rename('Dokchitser L-function of %s' % E)
return L
def lseries_dokchitser(E, prec=53):
"""
Return the Dokchitser L-series object associated to the elliptic
curve E, which may be defined over the rational numbers or a
number field. Also prec is the number of bits of precision to
which evaluation of the L-series occurs.
INPUT:
- E -- elliptic curve over a number field (or QQ)
- prec -- integer (default: 53) precision in *bits*
OUTPUT:
- Dokchitser L-function object
EXAMPLES::
A curve over Q(sqrt(5)), for which we have an optimized implementation::
sage: from psage.ellcurve.lseries.lseries_nf import lseries_dokchitser
sage: K.<a> = NumberField(x^2-x-1); E = EllipticCurve([0,-a,a,0,0])
sage: L = lseries_dokchitser(E); L
Dokchitser L-function of Elliptic Curve defined by y^2 + a*y = x^3 + (-a)*x^2 over Number Field in a with defining polynomial x^2 - x - 1
sage: L(1)
0.422214159001667
sage: L.taylor_series(1,5)
0.422214159001667 + 0.575883864741340*z - 0.102163426876721*z^2 - 0.158119743123727*z^3 + 0.120350687595265*z^4 + O(z^5)
Higher precision::
sage: L = lseries_dokchitser(E, 200)
sage: L(1)
0.42221415900166715092967967717023093014455137669360598558872
A curve over Q(i)::
sage: K.<i> = NumberField(x^2 + 1)
sage: E = EllipticCurve(K, [1,0])
sage: E.conductor().norm()
256
sage: L = lseries_dokchitser(E, 10)
sage: L.taylor_series(1,5)
0.86 + 0.58*z - 0.62*z^2 + 0.19*z^3 + 0.18*z^4 + O(z^5)
More examples::
sage: lseries_dokchitser(EllipticCurve([0,-1,1,0,0]), 10)(1)
0.25
sage: K.<i> = NumberField(x^2+1)
sage: lseries_dokchitser(EllipticCurve(K, [0,-1,1,0,0]), 10)(1)
0.37
sage: K.<a> = NumberField(x^2-x-1)
sage: lseries_dokchitser(EllipticCurve(K, [0,-1,1,0,0]), 10)(1)
0.72
sage: E = EllipticCurve([0,-1,1,0,0])
sage: E.quadratic_twist(2).rank()
1
sage: K.<d> = NumberField(x^2-2)
sage: L = lseries_dokchitser(EllipticCurve(K, [0,-1,1,0,0]), 10)
sage: L(1)
0
sage: L.taylor_series(1, 5)
0.58*z + 0.20*z^2 - 0.50*z^3 + 0.28*z^4 + O(z^5)
You can use this function as an algorithm to compute the sign of the functional
equation (global root number)::
sage: E = EllipticCurve([1..5])
sage: E.root_number()
-1
sage: L = lseries_dokchitser(E,32); L
Dokchitser L-function of Elliptic Curve defined by y^2 + x*y = x^3 - x^2 + 4*x + 3 over Rational Field
sage: L.eps
-1
Over QQ, this isn't so useful (since Sage has a root_number
method), but over number fields it is very useful::
sage: K.<a> = NumberField(x^2 - x - 1)
sage: E1=EllipticCurve([0,-a-1,1,a,0]); E0 = EllipticCurve([0,-a,a,0,0])
sage: lseries_dokchitser(E1, 16).eps
-1
sage: E1.rank()
1
sage: lseries_dokchitser(E0, 16).eps
1
sage: E0.rank()
0
"""
# The code asssumes in various places that we have a global minimal model,
# for example, in anlist_sqrt5 above.
E = E.global_minimal_model()
# Check that we're over a number field.
K = E.base_field()
if not is_NumberField(K):
raise TypeError, "base field must be a number field"
# Compute norm of the conductor -- awkward because QQ elements have no norm method (they should).
N = E.conductor()
if K != QQ:
N = N.norm()
# We guess the sign epsilon in the functional equation to be +1
# first. If our guess is wrong then we just choose the other
# possibility.
epsilon = 1
# Define the Dokchitser L-function object with all parameters set:
L = Dokchitser(conductor = N * K.discriminant()**2,
gammaV = [0]*K.degree() + [1]*K.degree(),
weight = 2, eps = epsilon, poles = [], prec = prec)
# Find out how many coefficients of the Dirichlet series are needed
# to compute to the requested precision.
n = L.num_coeffs()
# print "num coeffs = %s"%n
# Compute the Dirichlet series coefficients
coeffs = anlist(E, n)[1:]
# Define a string that when evaluated in PARI defines a function
# a(k), which returns the Dirichlet coefficient a_k.
s = 'v=%s; a(k)=v[k];'%coeffs
# Actually tell the L-series / PARI about the coefficients.
L.init_coeffs('a(k)', pari_precode = s)
# Test that the functional equation is satisfied. This will very,
# very, very likely if we chose the sign of the functional
# equation incorrectly, or made any mistake in computing the
# Dirichlet series coefficients.
tiny = max(1e-8, 1.0/2**(prec-1))
if abs(L.check_functional_equation()) > tiny:
# The test failed, so we try the other choice of functional equation.
epsilon *= -1
L.eps = epsilon
# It is not necessary to recreate L -- just tell PARI the different sign.
L._gp_eval('sgn = %s'%epsilon)
# Once again, verify that the functional equation is
# satisfied. If it is, then we've got it. If it isn't, then
# there is definitely some other subtle bug, probably in computed
# the Dirichlet series coefficients.
if abs(L.check_functional_equation()) > tiny:
raise RuntimeError, "Functional equation not numerically satisfied for either choice of sign"
L.rename('Dokchitser L-function of %s'%E)
return L
