def weil_representation(self) :
r"""
OUTPUT:
- A pair of matrices corresponding to T and S.
"""
disc_bilinear = lambda a, b: (self._dual_basis * vector(QQ, a.lift())) * self._L * (self._dual_basis * vector(QQ, b.lift()))
disc_quadratic = lambda a: disc_bilinear(a, a) / ZZ(2)
zeta_order = ZZ(lcm([8, 12, prod(self.invariants())] + map(lambda ed: 2 * ed, self.invariants())))
K = CyclotomicField(zeta_order); zeta = K.gen()
R = PolynomialRing(K, 'x'); x = R.gen()
# sqrt2s = (x**2 - 2).factor()
# if sqrt2s[0][0][0].complex_embedding().real() > 0 :
# sqrt2 = sqrt2s[0][0][0]
# else :
# sqrt2 = sqrt2s[0][1]
Ldet_rts = (x**2 - prod(self.invariants())).factor()
if Ldet_rts[0][0][0].complex_embedding().real() > 0 :
Ldet_rt = Ldet_rts[0][0][0]
else :
Ldet_rt = Ldet_rts[0][0][0]
Tmat = diagonal_matrix( K, [zeta**(zeta_order*disc_quadratic(a)) for a in self] )
Smat = zeta**(zeta_order / 8 * self._L.nrows()) / Ldet_rt \
* matrix( K, [ [ zeta**ZZ(-zeta_order * disc_bilinear(gamma,delta))
for delta in self ]
for gamma in self ])
return (Tmat, Smat)
def _test__jacobi_torsion_point(phi, weight, torsion_point):
r"""
Given a list of power series, which are the corrected Taylor coefficients
of a Jacobi form, return the specialization to ``torsion_point``.
INPUT:
- ``phi`` -- A Fourier expansion of a Jacobi form.
- ``weight`` -- An integer.
- ``torsion_point`` -- A rational.
OUPUT:
- A power series.
TESTS:
See jacobi_form_by_taylor_expansion.
sage: from psage.modform.jacobiforms.jacobiformd1nn_fegenerators import *
sage: from psage.modform.jacobiforms.jacobiformd1nn_types import *
sage: precision = 50
sage: weight = 10; index = 7
sage: phis = [jacobi_form_by_taylor_expansion(i, JacobiFormD1NNFilter(precision, index), weight) for i in range(JacobiFormD1NN_Gamma(weight, index)._rank(QQ))]
sage: fs = [JacobiFormD1NNFactory_class._test__jacobi_torsion_point(phi, weight, 2/3) for phi in phis]
sage: fs_vec = [vector(f.padded_list(precision)) for f in fs]
sage: mf_span = span([vector(b.qexp(precision).padded_list(precision)) for b in ModularForms(GammaH(9, [4]), weight).basis()])
sage: all(f_vec in mf_span for f_vec in fs_vec)
True
FIXME: The case of torsion points of order 5, which should lead to forms for Gamma1(25) fails even in the simplest case.
"""
from sage.rings.all import CyclotomicField
K = CyclotomicField(QQ(torsion_point).denominator())
zeta = K.gen()
R = PowerSeriesRing(K, 'q')
q = R.gen(0)
ch = JacobiFormD1WeightCharacter(weight)
coeffs = dict((n, QQ(0)) for n in range(phi.precision().index()))
for (n, r) in phi.precision().monoid_filter():
coeffs[n] += zeta**r * phi[(ch, (n, r))]
return PowerSeriesRing(K, 'q')(coeffs)
def _test__jacobi_torsion_point(phi, weight, torsion_point) :
r"""
Given a list of power series, which are the corrected Taylor coefficients
of a Jacobi form, return the specialization to ``torsion_point``.
INPUT:
- ``phi`` -- A Fourier expansion of a Jacobi form.
- ``weight`` -- An integer.
- ``torsion_point`` -- A rational.
OUPUT:
- A power series.
TESTS:
See jacobi_form_by_taylor_expansion.
sage: from psage.modform.jacobiforms.jacobiformd1nn_fegenerators import *
sage: from psage.modform.jacobiforms.jacobiformd1nn_types import *
sage: precision = 50
sage: weight = 10; index = 7
sage: phis = [jacobi_form_by_taylor_expansion(i, JacobiFormD1NNFilter(precision, index), weight) for i in range(JacobiFormD1NN_Gamma(weight, index)._rank(QQ))]
sage: fs = [JacobiFormD1NNFactory_class._test__jacobi_torsion_point(phi, weight, 2/3) for phi in phis]
sage: fs_vec = [vector(f.padded_list(precision)) for f in fs]
sage: mf_span = span([vector(b.qexp(precision).padded_list(precision)) for b in ModularForms(GammaH(9, [4]), weight).basis()])
sage: all(f_vec in mf_span for f_vec in fs_vec)
True
FIXME: The case of torsion points of order 5, which should lead to forms for Gamma1(25) fails even in the simplest case.
"""
from sage.rings.all import CyclotomicField
K = CyclotomicField(QQ(torsion_point).denominator()); zeta = K.gen()
R = PowerSeriesRing(K, 'q'); q = R.gen(0)
ch = JacobiFormD1WeightCharacter(weight)
coeffs = dict( (n, QQ(0)) for n in range(phi.precision().index()) )
for (n, r) in phi.precision().monoid_filter() :
coeffs[n] += zeta**r * phi[(ch, (n,r))]
return PowerSeriesRing(K, 'q')(coeffs)
def weil_representation(self):
r"""
OUTPUT:
- A pair of matrices corresponding to T and S.
"""
disc_bilinear = lambda a, b: (self._dual_basis * vector(QQ, a.lift(
))) * self._L * (self._dual_basis * vector(QQ, b.lift()))
disc_quadratic = lambda a: disc_bilinear(a, a) / ZZ(2)
zeta_order = ZZ(
lcm([8, 12, prod(self.invariants())] +
map(lambda ed: 2 * ed, self.invariants())))
K = CyclotomicField(zeta_order)
zeta = K.gen()
R = PolynomialRing(K, 'x')
x = R.gen()
# sqrt2s = (x**2 - 2).factor()
# if sqrt2s[0][0][0].complex_embedding().real() > 0 :
# sqrt2 = sqrt2s[0][0][0]
# else :
# sqrt2 = sqrt2s[0][1]
Ldet_rts = (x**2 - prod(self.invariants())).factor()
if Ldet_rts[0][0][0].complex_embedding().real() > 0:
Ldet_rt = Ldet_rts[0][0][0]
else:
Ldet_rt = Ldet_rts[0][0][0]
Tmat = diagonal_matrix(
K, [zeta**(zeta_order * disc_quadratic(a)) for a in self])
Smat = zeta**(zeta_order / 8 * self._L.nrows()) / Ldet_rt \
* matrix( K, [ [ zeta**ZZ(-zeta_order * disc_bilinear(gamma,delta))
for delta in self ]
for gamma in self ])
return (Tmat, Smat)
