def weil_representation_kernel(self):
r"""
Compute a scalar `r` and a group `\Gamma` which contains the translation `T`
so that `\diag(1, 1/r) \Gamma \diag(1, r) \subseteq \ker( \rho_D )`.
OUTPUT:
- A pair of an integer and an arithmetic group.
"""
if self._L == matrix(
2, [2, 1, 1, 2]) or self._L == -matrix(2, [2, 1, 1, 2]):
return (ZZ(3), GammaH(9, [4]))
if self._L == matrix(
2, [2, 0, 0, 2]) or self._L == -matrix(2, [2, 0, 0, 2]):
return (ZZ(4), GammaH(16, [5]))
if self._L == matrix(
2, [2, 0, 0, 4]) or self._L == -matrix(2, [2, 0, 0, 4]):
return (ZZ(8), GammaH(64, [57]))
level = 2 * self._L.inverse().denominator()
return (level,
GammaH(level**2,
filter(lambda n: n % level == 1, range(level**2))))
def JH(N, H):
"""
Return the Jacobian `J_H(N)` of the modular curve
`X_H(N)`.
EXAMPLES::
sage: JH(389,[16])
Abelian variety JH(389,[16]) of dimension 64
"""
key = 'JH(%s,%s)' % (N, H)
try:
return _get(key)
except ValueError:
from sage.modular.arithgroup.all import GammaH
return _saved(key, GammaH(N, H).modular_abelian_variety())
def group(self):
r"""
Return a `\Gamma_H` group which is the level of all of the relevant
twists of `f`.
EXAMPLE::
sage: from sage.modular.local_comp.type_space import example_type_space
sage: example_type_space().group()
Congruence Subgroup Gamma_H(98) with H generated by [57]
"""
p = self.prime()
r = self.conductor()
d = max(self.character_conductor(), r // 2)
n = self.tame_level()
chi = self.form().character()
tame_H = [i for i in chi.kernel() if (i % p**r) == 1]
wild_H = [crt(1 + p**d, 1, p**r, n)]
return GammaH(n * p**r, tame_H + wild_H)
def group(self):
r"""
Return a `\Gamma_H` group which is the level of all of the relevant
twists of `f`.
EXAMPLES::
sage: from sage.modular.local_comp.type_space import example_type_space
sage: example_type_space().group()
Congruence Subgroup Gamma_H(98) with H generated by [15, 29, 43]
"""
# Implementation here is not the most efficient but this is heavily not
# time-critical, and getting it wrong can lead to subtle bugs.
p = self.prime()
r = self.conductor()
d = max(self.character_conductor(), r // 2)
n = self.tame_level()
chi = self.form().character()
tame_H = [i for i in chi.kernel() if (i % p**r) == 1]
wild_H = [crt(x, 1, p**r, n) for x in range(p**r) if x % (p**d) == 1]
return GammaH(n * p**r, tame_H + wild_H)
def __init__(self, index=20, index_max=50, odd_probability=0.5):
r"""
Create an arithmetic subgroup testing object.
INPUT:
- ``index`` - the index of random subgroup to test
- ``index_max`` - the maximum index for congruence subgroup to test
EXAMPLES::
sage: from sage.modular.arithgroup.tests import Test
sage: Test()
Arithmetic subgroup testing class
"""
self.congroups = []
i = 1
self.odd_probability = odd_probability
if index % 4:
self.odd_probability = 0
while Gamma(i).index() < index_max:
self.congroups.append(Gamma(i))
i += 1
i = 1
while Gamma0(i).index() < index_max:
self.congroups.append(Gamma0(i))
i += 1
i = 2
while Gamma1(i).index() < index_max:
self.congroups.append(Gamma1(i))
M = Zmod(i)
U = [x for x in M if x.is_unit()]
for j in range(1, len(U) - 1):
self.congroups.append(GammaH(i, prandom.sample(U, j)))
i += 1
self.index = index
def hecke_stable_subspace(chi, aux_prime=ZZ(2)):
r"""
Compute a q-expansion basis for S_1(chi).
Results are returned as q-expansions to a certain fixed (and fairly high)
precision. If more precision is required this can be obtained with
:func:`modular_ratio_to_prec`.
EXAMPLES::
sage: from sage.modular.modform.weight1 import hecke_stable_subspace
sage: hecke_stable_subspace(DirichletGroup(59, QQ).0)
[q - q^3 + q^4 - q^5 - q^7 - q^12 + q^15 + q^16 + 2*q^17 - q^19 - q^20 + q^21 + q^27 - q^28 - q^29 + q^35 + O(q^40)]
"""
# Deal quickly with the easy cases.
if chi(-1) == 1: return []
N = chi.modulus()
H = chi.kernel()
G = GammaH(N, H)
try:
if ArithmeticSubgroup.dimension_cusp_forms(G, 1) == 0:
verbose("no wt 1 cusp forms for N=%s, chi=%s by Riemann-Roch" %
(N, chi._repr_short_()),
level=1)
return []
except NotImplementedError:
pass
from sage.modular.modform.constructor import EisensteinForms
chi = chi.minimize_base_ring()
K = chi.base_ring()
# Auxiliary prime for Hecke stability method
l = aux_prime
while l.divides(N):
l = l.next_prime()
verbose("Auxiliary prime: %s" % l, level=1)
# Compute working precision
R = l * Gamma0(N).sturm_bound(l + 2)
t = verbose("Computing modular ratio space", level=1)
mrs = modular_ratio_space(chi)
t = verbose("Computing modular ratios to precision %s" % R, level=1)
qexps = [modular_ratio_to_prec(chi, f, R) for f in mrs]
verbose("Done", t=t, level=1)
# We want to compute the largest subspace of I stable under T_l. To do
# this, we compute I intersect T_l(I) modulo q^(R/l), and take its preimage
# under T_l, which is then well-defined modulo q^R.
from sage.modular.modform.hecke_operator_on_qexp import hecke_operator_on_qexp
t = verbose("Computing Hecke-stable subspace", level=1)
A = PowerSeriesRing(K, 'q')
r = R // l
V = K**R
W = K**r
Tl_images = [hecke_operator_on_qexp(f, l, 1, chi) for f in qexps]
qvecs = [V(x.padded_list(R)) for x in qexps]
qvecs_trunc = [W(x.padded_list(r)) for x in qexps]
Tvecs = [W(x.padded_list(r)) for x in Tl_images]
I = V.submodule(qvecs)
Iimage = W.span(qvecs_trunc)
TlI = W.span(Tvecs)
Jimage = Iimage.intersection(TlI)
J = I.Hom(W)(Tvecs).inverse_image(Jimage)
verbose("Hecke-stable subspace is %s-dimensional" % J.dimension(),
t=t,
level=1)
if J.rank() == 0: return []
# The theory does not guarantee that J is exactly S_1(chi), just that it is
# intermediate between S_1(chi) and M_1(chi). In every example I know of,
# it is equal to S_1(chi), but just for honesty, we check this anyway.
t = verbose("Checking cuspidality", level=1)
JEis = V.span(
V(x.padded_list(R))
for x in EisensteinForms(chi, 1).q_echelon_basis(prec=R))
D = JEis.intersection(J)
if D.dimension() != 0:
raise ArithmeticError("Got non-cuspidal form!")
verbose("Done", t=t, level=1)
qexps = Sequence(A(x.list()).add_bigoh(R) for x in J.gens())
return qexps