def _basic_integral(self, a, j, twist=None):
r"""
Computes the integral
.. MATH::
\int_{a+p\ZZ_p}(z-\omega(a))^jd\mu_\chi.
If ``twist`` is ``None``, `\\chi` is the trivial character. Otherwise, ``twist`` can be a primitive quadratic character of conductor prime to `p`.
"""
#is this the negative of what we want?
#if Phis is fixed for this p-adic L-function, we should make this method cached
p = self._Phis.parent().prime()
if twist is None:
pass
elif twist in ZZ:
twist = kronecker_character(twist)
if twist.is_trivial():
twist = None
else:
D = twist.level()
assert(D.gcd(p) == 1)
else:
if twist.is_trivial():
twist = None
else:
assert((twist**2).is_trivial())
twist = twist.primitive_character()
D = twist.level()
assert(D.gcd(p) == 1)
onDa = self._on_Da(a, twist)#self._Phis(Da)
aminusat = a - self._Phis.parent().base_ring().base_ring().teichmuller(a)
#aminusat = a - self._coefficient_ring.base_ring().teichmuller(a)
try:
ap = self._ap
except AttributeError:
self._ap = self._Phis.Tq_eigenvalue(p) #catch exception if not eigensymbol
ap = self._ap
if not twist is None:
ap *= twist(p)
if j == 0:
return (~ap) * onDa.moment(0)
if a == 1:
#aminusat is 0, so only the j=r term is non-zero
return (~ap) * (p ** j) * onDa.moment(j)
#print "j =", j, "a = ", a
ans = onDa.moment(0) * (aminusat ** j)
#ans = onDa.moment(0)
#print "\tr =", 0, " ans =", ans
for r in range(1, j+1):
if r == j:
ans += binomial(j, r) * (p ** r) * onDa.moment(r)
else:
ans += binomial(j, r) * (aminusat ** (j - r)) * (p ** r) * onDa.moment(r)
#print "\tr =", r, " ans =", ans
#print " "
return (~ap) * ans
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 _evalf_(self, x, D, **kwargs): #numerical value
D = Integer(D)
if D % 4 > 1:
raise ValueError('Not a discriminant')
s = kronecker_character(D).lfunction(algorithm='lcalc').value(x).real()
f = D.squarefree_part()
if f % 4 > 1 and not D % 4:
f *= 4
m = D // f
if m != 1:
return prod(1 - p**(-x) * kronecker(f, p)
for p in prime_divisors(m)) * s
return s
def _basic_integral(self, a, j, twist=None):
r"""
Computes the integral
.. MATH::
\int_{a+p\ZZ_p}(z-\omega(a))^jd\mu_\chi.
If ``twist`` is ``None``, `\\chi` is the trivial character. Otherwise, ``twist`` can be a primitive quadratic character of conductor prime to `p`.
"""
#is this the negative of what we want?
#if Phis is fixed for this p-adic L-function, we should make this method cached
p = self._Phis.parent().prime()
if twist is None:
pass
elif twist in ZZ:
twist = kronecker_character(twist)
if twist.is_trivial():
twist = None
else:
D = twist.level()
assert (D.gcd(p) == 1)
else:
if twist.is_trivial():
twist = None
else:
assert ((twist**2).is_trivial())
twist = twist.primitive_character()
D = twist.level()
assert (D.gcd(p) == 1)
onDa = self._on_Da(a, twist) #self._Phis(Da)
aminusat = a - self._Phis.parent().base_ring().base_ring().teichmuller(
a)
#aminusat = a - self._coefficient_ring.base_ring().teichmuller(a)
try:
ap = self._ap
except AttributeError:
self._ap = self._Phis.Tq_eigenvalue(
p) #catch exception if not eigensymbol
ap = self._ap
if not twist is None:
ap *= twist(p)
if j == 0:
return (~ap) * onDa.moment(0)
if a == 1:
#aminusat is 0, so only the j=r term is non-zero
return (~ap) * (p**j) * onDa.moment(j)
#print "j =", j, "a = ", a
ans = onDa.moment(0) * (aminusat**j)
#ans = onDa.moment(0)
#print "\tr =", 0, " ans =", ans
for r in range(1, j + 1):
if r == j:
ans += binomial(j, r) * (p**r) * onDa.moment(r)
else:
ans += binomial(j, r) * (aminusat
**(j - r)) * (p**r) * onDa.moment(r)
#print "\tr =", r, " ans =", ans
#print " "
return (~ap) * ans
