def parameter(self, prec=20):
r"""
Return the Tate parameter `q` such that the curve is isomorphic
over the algebraic closure of `\QQ_p` to the curve
`\QQ_p^{\times}/q^{\ZZ}`.
INPUT:
- ``prec`` - the `p`-adic precision, default is 20.
EXAMPLES::
sage: eq = EllipticCurve('130a1').tate_curve(5)
sage: eq.parameter(prec=5)
3*5^3 + 3*5^4 + 2*5^5 + 2*5^6 + 3*5^7 + O(5^8)
"""
try:
qE = self._q
if qE.absolute_precision() >= prec:
return qE
except AttributeError:
pass
E4 = EisensteinForms(weight=4).basis()[0]
Delta = CuspForms(weight=12).basis()[0]
j = (E4.q_expansion(prec + 3))**3 / Delta.q_expansion(prec + 3)
jinv = (1 / j).power_series()
q_in_terms_of_jinv = jinv.reverse()
R = Qp(self._p, prec=prec)
qE = q_in_terms_of_jinv(R(1 / self._E.j_invariant()))
self._q = qE
return qE
def parameter(self, prec=20):
r"""
Return the Tate parameter `q` such that the curve is isomorphic
over the algebraic closure of `\QQ_p` to the curve
`\QQ_p^{\times}/q^{\ZZ}`.
INPUT:
- ``prec`` - the `p`-adic precision, default is 20.
EXAMPLES::
sage: eq = EllipticCurve('130a1').tate_curve(5)
sage: eq.parameter(prec=5)
3*5^3 + 3*5^4 + 2*5^5 + 2*5^6 + 3*5^7 + O(5^8)
"""
try:
qE = self._q
if qE.absolute_precision() >= prec:
return qE
except AttributeError:
pass
E4 = EisensteinForms(weight=4).basis()[0]
Delta = CuspForms(weight=12).basis()[0]
j = (E4.q_expansion(prec + 3)) ** 3 / Delta.q_expansion(prec + 3)
jinv = (1 / j).power_series()
q_in_terms_of_jinv = jinv.reverse()
R = Qp(self._p, prec=prec)
qE = q_in_terms_of_jinv(R(1 / self._E.j_invariant()))
self._q = qE
return qE
def tate_parameter(E, p, prec = 20, R = None):
if R is None:
R = Qp(p,prec)
jE = E.j_invariant()
E4 = EisensteinForms(weight=4).basis()[0]
Delta = CuspForms(weight=12).basis()[0]
j = (E4.q_expansion(prec+3))**3/Delta.q_expansion(prec+3)
jinv = (1/j).power_series()
q_in_terms_of_jinv = jinv.reversion()
return q_in_terms_of_jinv(R(1/E.j_invariant()))
def getcoords(E, u, prec=20, R=None):
if R is None:
R = u.parent()
u = R(u)
p = R.prime()
jE = E.j_invariant()
# Calculate the Tate parameter
E4 = EisensteinForms(weight=4).basis()[0]
Delta = CuspForms(weight=12).basis()[0]
j = (E4.q_expansion(prec + 7)) ** 3 / Delta.q_expansion(prec + 7)
qE = (1 / j).power_series().reversion()(R(1 / jE))
# Normalize the period by appropriate powers of qE
un = u * qE ** (-(u.valuation() / qE.valuation()).floor())
precn = (prec / qE.valuation()).floor() + 4
# formulas in Silverman II (Advanced Topics in the Arithmetic of Elliptic curves, p. 425)
xx = un / (1 - un) ** 2 + sum(
[
qE ** n * un / (1 - qE ** n * un) ** 2
+ qE ** n / un / (1 - qE ** n / un) ** 2
- 2 * qE ** n / (1 - qE ** n) ** 2
for n in range(1, precn)
]
)
yy = un ** 2 / (1 - un) ** 3 + sum(
[
qE ** (2 * n) * un ** 2 / (1 - qE ** n * un) ** 3
- qE ** n / un / (1 - qE ** n / un) ** 3
+ qE ** n / (1 - qE ** n) ** 2
for n in range(1, precn)
]
)
sk = lambda q, k, pprec: sum([n ** k * q ** n / (1 - q ** n) for n in range(1, pprec + 1)])
n = qE.valuation()
precp = ((prec + 4) / n).floor() + 2
tate_a4 = -5 * sk(qE, 3, precp)
tate_a6 = (tate_a4 - 7 * sk(qE, 5, precp)) / 12
Eq = EllipticCurve([R(1), R(0), R(0), tate_a4, tate_a6])
C2 = (Eq.c4() * E.c6()) / (Eq.c6() * E.c4())
C = R(C2).square_root()
s = (C - R(E.a1())) / R(2)
r = (s * (C - s) - R(E.a2())) / 3
t = (r * (2 * s - C) - R(E.a3())) / 2
return (r + C2 * xx, t + s * C2 * xx + C * C2 * yy)
def getcoords(E, u, prec=20, R=None):
if R is None:
R = u.parent()
u = R(u)
p = R.prime()
jE = E.j_invariant()
# Calculate the Tate parameter
E4 = EisensteinForms(weight=4).basis()[0]
Delta = CuspForms(weight=12).basis()[0]
j = (E4.q_expansion(prec + 7))**3 / Delta.q_expansion(prec + 7)
qE = (1 / j).power_series().reversion()(R(1 / jE))
# Normalize the period by appropriate powers of qE
un = u * qE**(-(u.valuation() / qE.valuation()).floor())
precn = (prec / qE.valuation()).floor() + 4
# formulas in Silverman II (Advanced Topics in the Arithmetic of Elliptic curves, p. 425)
xx = un / (1 - un)**2 + sum([
qE**n * un / (1 - qE**n * un)**2 + qE**n / un /
(1 - qE**n / un)**2 - 2 * qE**n / (1 - qE**n)**2
for n in range(1, precn)
])
yy = un**2 / (1 - un)**3 + sum([
qE**(2 * n) * un**2 / (1 - qE**n * un)**3 - qE**n / un /
(1 - qE**n / un)**3 + qE**n / (1 - qE**n)**2 for n in range(1, precn)
])
sk = lambda q, k, pprec: sum(
[n**k * q**n / (1 - q**n) for n in range(1, pprec + 1)])
n = qE.valuation()
precp = ((prec + 4) / n).floor() + 2
tate_a4 = -5 * sk(qE, 3, precp)
tate_a6 = (tate_a4 - 7 * sk(qE, 5, precp)) / 12
Eq = EllipticCurve([R(1), R(0), R(0), tate_a4, tate_a6])
C2 = (Eq.c4() * E.c6()) / (Eq.c6() * E.c4())
C = R(C2).square_root()
s = (C - R(E.a1())) / R(2)
r = (s * (C - s) - R(E.a2())) / 3
t = (r * (2 * s - C) - R(E.a3())) / 2
return (r + C2 * xx, t + s * C2 * xx + C * C2 * yy)
def modular_ratio_space(chi):
r"""
Compute the space of 'modular ratios', i.e. meromorphic modular forms f
level N and character chi such that f * E is a holomorphic cusp form for
every Eisenstein series E of weight 1 and character 1/chi.
Elements are returned as q-expansions up to precision R, where R is one
greater than the weight 3 Sturm bound.
EXAMPLES::
sage: chi = DirichletGroup(31,QQ).0
sage: sage.modular.modform.weight1.modular_ratio_space(chi)
[q - 8/3*q^3 + 13/9*q^4 + 43/27*q^5 - 620/81*q^6 + 1615/243*q^7 + 3481/729*q^8 + O(q^9),
q^2 - 8/3*q^3 + 13/9*q^4 + 70/27*q^5 - 620/81*q^6 + 1858/243*q^7 + 2752/729*q^8 + O(q^9)]
"""
from sage.modular.modform.constructor import EisensteinForms, CuspForms
if chi(-1) == 1:
return []
N = chi.modulus()
chi = chi.minimize_base_ring()
K = chi.base_ring()
R = Gamma0(N).sturm_bound(3) + 1
verbose("Working precision is %s" % R, level=1)
verbose("Coeff field is %s" % K, level=1)
V = K**R
I = V
d = I.rank()
t = verbose("Calculating Eisenstein forms in weight 1...", level=1)
B0 = EisensteinForms(~chi, 1).q_echelon_basis(prec=R)
B = [b + B0[0] for b in B0]
verbose("Done (dimension %s)" % len(B), level=1, t=t)
t = verbose("Calculating in weight 2...", level=1)
C = CuspForms(Gamma0(N), 2).q_echelon_basis(prec=R)
verbose("Done (dimension %s)" % len(C), t=t, level=1)
t = verbose("Computing candidate space", level=1)
for b in B:
quots = (c / b for c in C)
W = V.span(V(x.padded_list(R)) for x in quots)
I = I.intersection(W)
if I.rank() < d:
verbose(" Cut down to dimension %s" % I.rank(), level=1)
d = I.rank()
if I.rank() == 0:
break
verbose("Done: intersection is %s-dimensional" % I.dimension(),
t=t,
level=1)
A = PowerSeriesRing(K, 'q')
return [A(x.list()).add_bigoh(R) for x in I.gens()]
def modular_ratio_to_prec(chi, qexp, prec):
r"""
Given a q-expansion of a modular ratio up to sufficient precision to
determine it uniquely, compute it to greater precision.
EXAMPLES::
sage: from sage.modular.modform.weight1 import modular_ratio_to_prec
sage: R.<q> = QQ[[]]
sage: modular_ratio_to_prec(DirichletGroup(31,QQ).0, q-q^2-q^5-q^7+q^8+O(q^9), 20)
q - q^2 - q^5 - q^7 + q^8 + q^9 + q^10 + q^14 - q^16 - q^18 - q^19 + O(q^20)
"""
if prec <= qexp.prec():
return qexp.add_bigoh(prec)
from sage.modular.modform.constructor import EisensteinForms, CuspForms
B = EisensteinForms(~chi, 1).gen(0).qexp(prec)
fB = qexp * B
fB_elt = CuspForms(chi.level(), 2, base_ring=qexp.base_ring())(fB)
return fB_elt.qexp(prec) / B
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