def coerce_to_Qp(self,p,M):
r"""
If `K` is the base_ring of self, this function takes all maps
`K-->Q_p` and applies them to self return a vector of
<modular symbol,map: `K-->Q_p`> as map varies over all such maps.
`M` is the accuracy
INPUT:
- ``p`` --
- ``M`` --
OUTPUT:
EXAMPLES:
"""
K = self.base_ring()
f = K.defining_polynomial()
R = pAdicField(p,M+10)['x']
x = R.gen()
v = R(f).roots()
if len(v) == 0:
print "No coercion possible -- no prime over p has degree 1"
return []
else:
ans = []
for j in range(len(v)):
root = v[j][0]
psi = K.hom([root],pAdicField(p,M))
ans = ans+[[self.map(psi),psi]]
return ans
def interpolation_factor(self, ap, chip=1, psi=None):
r"""
Return the interpolation factor associated to self.
This is the `p`-adic multiplier that which appears in
the interpolation formula of the `p`-adic `L`-function. It
has the form `(1-\alpha_p^{-1})^2`, where `\alpha_p` is the
unit root of `X^2 - \psi(a_p) \chi(p) X + p`.
INPUT:
- ``ap`` -- the eigenvalue of the Up operator
- ``chip`` -- the value of the nebentype at p (default: 1)
- ``psi`` -- a twisting character (default: None)
OUTPUT: a `p`-adic number
EXAMPLES::
sage: E = EllipticCurve('19a2')
sage: L = E.padic_lseries(3,implementation="pollackstevens",precision=6) # long time
sage: ap = E.ap(3) # long time
sage: L.interpolation_factor(ap) # long time
3^2 + 3^3 + 2*3^5 + 2*3^6 + O(3^7)
Comparing against a different implementation::
sage: L = E.padic_lseries(3)
sage: (1-1/L.alpha(prec=4))^2
3^2 + 3^3 + O(3^5)
"""
M = self.symbol().precision_relative()
p = self.prime()
if p == 2:
R = pAdicField(2, M + 1)
else:
R = pAdicField(p, M)
if psi is not None:
ap = psi(ap)
ap = ap * chip
sdisc = R(ap**2 - 4 * p).sqrt()
v0 = (R(ap) + sdisc) / 2
v1 = (R(ap) - sdisc) / 2
if v0.valuation() > 0:
v0, v1 = v1, v0
alpha = v0
return (1 - 1 / alpha)**2
def interpolation_factor(self, ap, chip=1, psi=None):
r"""
Return the interpolation factor associated to self.
This is the `p`-adic multiplier that which appears in
the interpolation formula of the `p`-adic `L`-function. It
has the form `(1-\alpha_p^{-1})^2`, where `\alpha_p` is the
unit root of `X^2 - \psi(a_p) \chi(p) X + p`.
INPUT:
- ``ap`` -- the eigenvalue of the Up operator
- ``chip`` -- the value of the nebentype at p (default: 1)
- ``psi`` -- a twisting character (default: None)
OUTPUT: a `p`-adic number
EXAMPLES::
sage: E = EllipticCurve('19a2')
sage: L = E.padic_lseries(3,implementation="pollackstevens",precision=6) # long time
sage: ap = E.ap(3) # long time
sage: L.interpolation_factor(ap) # long time
3^2 + 3^3 + 2*3^5 + 2*3^6 + O(3^7)
Comparing against a different implementation::
sage: L = E.padic_lseries(3)
sage: (1-1/L.alpha(prec=4))^2
3^2 + 3^3 + O(3^5)
"""
M = self.symbol().precision_relative()
p = self.prime()
if p == 2:
R = pAdicField(2, M + 1)
else:
R = pAdicField(p, M)
if psi is not None:
ap = psi(ap)
ap = ap * chip
sdisc = R(ap ** 2 - 4 * p).sqrt()
v0 = (R(ap) + sdisc) / 2
v1 = (R(ap) - sdisc) / 2
if v0.valuation() > 0:
v0, v1 = v1, v0
alpha = v0
return (1 - 1 / alpha) ** 2
def p_stabilize_ordinary(self,p,ap,M):
r"""
Returns the unique `p`-ordinary `p`-stabilization of self
INPUT:
- ``p`` -- good ordinary prime
- ``ap`` -- Hecke eigenvalue
- ``M`` -- precision of `Q_p`
OUTPUT:
The unique modular symbol of level `N*p` with the same Hecke-eigenvalues as self away from `p`
and unit eigenvalue at `p`.
EXAMPLES:
::
sage: E = EllipticCurve('11a')
sage: phi = form_modsym_from_elliptic_curve(E); phi
[-1/5, 3/2, -1/2]
sage: phi_ord = phi.p_stabilize_ordinary(3,E.ap(3),10); phi_ord
[-47611/238060, 47615/95224, 95221/95224, 47611/238060, -95217/476120, 95225/95224, -5951/11903, -95229/95224, 95237/476120]
sage: phi_ord.hecke(2) - phi_ord.scale(E.ap(2))
[0, 0, 0, 0, 0, 0, 0, 0, 0]
sage: phi_ord.hecke(5) - phi_ord.scale(E.ap(5))
[0, 0, 0, 0, 0, 0, 0, 0, 0]
sage: p = 3
sage: M = 100
sage: R = pAdicField(p,M)['y']
sage: y = R.gen()
sage: f = y**2-ap*y+p
sage: v = f.roots()
sage: alpha = v[0][0]
sage: alpha
2 + 3^2 + 2*3^3 + 2*3^4 + 2*3^6 + 3^8 + 2*3^9 + 2*3^11 + 2*3^13 + 3^14 + 2*3^15 + 3^18 + 3^19 + 2*3^20 + 2*3^23 + 2*3^25 + 3^28 + 3^29 + 2*3^32 + 3^33 + 3^34 + 3^36 + 3^37 + 2*3^38 + 3^39 + 3^40 + 2*3^43 + 3^44 + 3^45 + 3^46 + 2*3^48 + 3^49 + 2*3^50 + 3^51 + 3^52 + 3^53 + 3^55 + 3^56 + 3^58 + 3^60 + 3^62 + 3^63 + 2*3^65 + 3^67 + 3^70 + 3^71 + 2*3^73 + 2*3^75 + 2*3^78 + 2*3^79 + 3^82 + 2*3^84 + 3^86 + 3^87 + 3^88 + 3^89 + 2*3^91 + 2*3^92 + 3^93 + 3^94 + 3^95 + O(3^100)
sage: alpha^2 - E.ap(3)*alpha + 3
O(3^100)
sage: alpha=ZZ(alpha)
sage: (phi_ord.hecke(3) - phi_ord.scale(alpha)).valuation(3)
11
"""
N = self.level()
k = self.data(0).weight()
assert N%p!=0, "The level isn't prime to p"
assert (ap%p)!=0, "Not ordinary!"
# Makes alpha the unit root of the Hecke polynomial x^2-a_p*x+p^(k+1)
R = pAdicField(p,M)['y']
y = R.gen()
f = y**2-ap*y+p**(k+1)
v = f.roots()
if ZZ(v[0][0])%p<>0:
alpha = ZZ(v[0][0])
else:
alpha = ZZ(v[1][0])
if self.full_data() == 0:
self.compute_full_data_from_gen_data()
return self.p_stabilize(p,alpha)
def series(self, prec=5):
r"""
Return the ``prec``-th approximation to the `p`-adic `L`-series
associated to self, as a power series in `T` (corresponding to
`\gamma-1` with `\gamma` the chosen generator of `1+p\ZZ_p`).
INPUT:
- ``prec`` -- (default 5) is the precision of the power series
EXAMPLES::
sage: E = EllipticCurve('14a2')
sage: p = 3
sage: prec = 6
sage: L = E.padic_lseries(p,implementation="pollackstevens",precision=prec) # long time
sage: L.series(4) # long time
2*3 + 3^4 + 3^5 + O(3^6) + (2*3 + 3^2 + O(3^4))*T + (2*3 + O(3^2))*T^2 + (3 + O(3^2))*T^3 + O(T^4)
sage: E = EllipticCurve("15a3")
sage: L = E.padic_lseries(5,implementation="pollackstevens",precision=15) # long time
sage: L.series(3) # long time
O(5^15) + (2 + 4*5^2 + 3*5^3 + 5^5 + 2*5^6 + 3*5^7 + 3*5^8 + 2*5^9 + 2*5^10 + 3*5^11 + 5^12 + O(5^13))*T + (4*5 + 4*5^3 + 3*5^4 + 4*5^5 + 3*5^6 + 2*5^7 + 5^8 + 4*5^9 + 3*5^10 + O(5^11))*T^2 + O(T^3)
sage: E = EllipticCurve("79a1")
sage: L = E.padic_lseries(2,implementation="pollackstevens",precision=10) # not tested
sage: L.series(4) # not tested
O(2^9) + (2^3 + O(2^4))*T + O(2^0)*T^2 + (O(2^-3))*T^3 + O(T^4)
"""
p = self.prime()
M = self.symbol().precision_relative()
K = pAdicField(p, M)
R = PowerSeriesRing(K, names="T")
T = R.gens()[0]
return R([self[i] for i in range(prec)]).add_bigoh(prec)
def _basic_integral(self, a, j):
r"""
Return `\int_{a+pZ_p} (z-{a})^j d\Phi(0-infty)`.
See formula in section 9.2 of [PS2011]_
INPUT:
- ``a`` -- integer in range(p)
- ``j`` -- integer in range(self.symbol().precision_relative())
EXAMPLES::
sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
sage: E = EllipticCurve('11a3')
sage: L = E.padic_lseries(5, implementation="pollackstevens", precision=4) #long time
sage: L._basic_integral(1,2) # long time
2*5^2 + 5^3 + O(5^4)
"""
symb = self.symbol()
M = symb.precision_relative()
if j > M:
raise PrecisionError("Too many moments requested")
p = self.prime()
ap = symb.Tq_eigenvalue(p)
D = self._quadratic_twist
ap = ap * kronecker(D, p)
K = pAdicField(p, M)
symb_twisted = symb.evaluate_twisted(a, D)
return sum(
binomial(j, r) * ((a - ZZ(K.teichmuller(a)))**(j - r)) *
(p**r) * symb_twisted.moment(r) for r in range(j + 1)) / ap
def series(self, prec=5):
r"""
Return the ``prec``-th approximation to the `p`-adic `L`-series
associated to ``self``, as a power series in `T` (corresponding to
`\gamma-1` with `\gamma` the chosen generator of `1+p\ZZ_p`).
INPUT:
- ``prec`` -- (default 5) is the precision of the power series
EXAMPLES::
sage: E = EllipticCurve('14a2')
sage: p = 3
sage: prec = 6
sage: L = E.padic_lseries(p,implementation="pollackstevens",precision=prec) # long time
sage: L.series(4) # long time
2*3 + 3^4 + 3^5 + O(3^6) + (2*3 + 3^2 + O(3^4))*T + (2*3 + O(3^2))*T^2 + (3 + O(3^2))*T^3 + O(T^4)
sage: E = EllipticCurve("15a3")
sage: L = E.padic_lseries(5,implementation="pollackstevens",precision=15) # long time
sage: L.series(3) # long time
O(5^15) + (2 + 4*5^2 + 3*5^3 + 5^5 + 2*5^6 + 3*5^7 + 3*5^8 + 2*5^9 + 2*5^10 + 3*5^11 + 5^12 + O(5^13))*T + (4*5 + 4*5^3 + 3*5^4 + 4*5^5 + 3*5^6 + 2*5^7 + 5^8 + 4*5^9 + 3*5^10 + O(5^11))*T^2 + O(T^3)
sage: E = EllipticCurve("79a1")
sage: L = E.padic_lseries(2,implementation="pollackstevens",precision=10) # not tested
sage: L.series(4) # not tested
O(2^9) + (2^3 + O(2^4))*T + O(2^0)*T^2 + (O(2^-3))*T^3 + O(T^4)
"""
p = self.prime()
M = self.symbol().precision_relative()
K = pAdicField(p, M)
R = PowerSeriesRing(K, names='T')
return R([self[i] for i in range(prec)]).add_bigoh(prec)
def series(self, n, prec):
r"""
Returns the `n`-th approximation to the `p`-adic `L`-series
associated to self, as a power series in `T` (corresponding to
`\gamma-1` with `\gamma= 1 + p` as a generator of `1+p\ZZ_p`).
EXAMPLES::
sage: from sage.modular.pollack_stevens.space import ps_modsym_from_elliptic_curve
sage: E = EllipticCurve('57a')
sage: p = 5
sage: prec = 4
sage: phi = ps_modsym_from_elliptic_curve(E)
sage: phi_stabilized = phi.p_stabilize(p,M = prec+3)
sage: Phi = phi_stabilized.lift(p,prec,None,algorithm='stevens',eigensymbol=True)
sage: L = pAdicLseries(Phi)
sage: L.series(3,4)
O(5^3) + (3*5 + 5^2 + O(5^3))*T + (5 + O(5^2))*T^2
sage: L1 = E.padic_lseries(5)
sage: L1.series(4)
O(5^6) + (3*5 + 5^2 + O(5^3))*T + (5 + 4*5^2 + O(5^3))*T^2 + (4*5^2 + O(5^3))*T^3 + (2*5 + 4*5^2 + O(5^3))*T^4 + O(T^5)
"""
p = self.prime()
M = self.symb().precision_absolute()
K = pAdicField(p, M)
R = PowerSeriesRing(K, names = 'T')
T = R.gens()[0]
R.set_default_prec(prec)
return sum(self[i] * T**i for i in range(n))
def _basic_integral(self, a, j):
r"""
Return `\int_{a+pZ_p} (z-{a})^j d\Phi(0-infty)`
-- see formula [Pollack-Stevens, sec 9.2]
INPUT:
- ``a`` -- integer in range(p)
- ``j`` -- integer in range(self.symbol().precision_relative())
EXAMPLES::
sage: from sage.modular.pollack_stevens.padic_lseries import pAdicLseries
sage: E = EllipticCurve('11a3')
sage: L = E.padic_lseries(5, implementation="pollackstevens", precision=4) #long time
sage: L._basic_integral(1,2) # long time
2*5^2 + 5^3 + O(5^4)
"""
symb = self.symbol()
M = symb.precision_relative()
if j > M:
raise PrecisionError("Too many moments requested")
p = self.prime()
ap = symb.Tq_eigenvalue(p)
D = self._quadratic_twist
ap = ap * kronecker(D, p)
K = pAdicField(p, M)
symb_twisted = symb.evaluate_twisted(a, D)
return (
sum(
binomial(j, r) * ((a - ZZ(K.teichmuller(a))) ** (j - r)) * (p ** r) * symb_twisted.moment(r)
for r in range(j + 1)
)
/ ap
)
def interpolation_factor(self, ap, chip=1, psi=None):
r"""
Return the interpolation factor associated to self.
This is the `p`-adic multiplier that which appears in
the interpolation formula of the `p`-adic `L`-function.
INPUT:
- ``ap`` -- ## mm TODO
- ``chip`` --
- ``psi`` --
OUTPUT: a `p`-adic number
EXAMPLES::
sage: E = EllipticCurve('19a2')
sage: L = E.padic_lseries(3,implementation="pollackstevens",precision=6) # long time
sage: ap = E.ap(3) # long time
sage: L.interpolation_factor(ap) # long time
3^2 + 3^3 + 2*3^5 + 2*3^6 + O(3^7)
Comparing against a different implementation::
sage: L = E.padic_lseries(3)
sage: (1-1/L.alpha(prec=4))^2
3^2 + 3^3 + O(3^5)
"""
M = self.symbol().precision_relative()
p = self.prime()
if p == 2:
R = pAdicField(2, M + 1)
else:
R = pAdicField(p, M)
if psi is not None:
ap = psi(ap)
ap = ap * chip
sdisc = R(ap ** 2 - 4 * p).sqrt()
v0 = (R(ap) + sdisc) / 2
v1 = (R(ap) - sdisc) / 2
if v0.valuation() > 0:
v0, v1 = v1, v0
alpha = v0
return (1 - 1 / alpha) ** 2
def interpolation_factor(self, ap, chip=1, psi=None):
r"""
Return the interpolation factor associated to self.
This is the `p`-adic multiplier that which appears in
the interpolation formula of the `p`-adic `L`-function.
INPUT:
- ``ap`` -- ## mm TODO
- ``chip`` --
- ``psi`` --
OUTPUT: a `p`-adic number
EXAMPLES::
sage: E = EllipticCurve('19a2')
sage: L = E.padic_lseries(3,implementation="pollackstevens",precision=6) # long time
sage: ap = E.ap(3) # long time
sage: L.interpolation_factor(ap) # long time
3^2 + 3^3 + 2*3^5 + 2*3^6 + O(3^7)
Comparing against a different implementation::
sage: L = E.padic_lseries(3)
sage: (1-1/L.alpha(prec=4))^2
3^2 + 3^3 + O(3^5)
"""
M = self.symbol().precision_relative()
p = self.prime()
if p == 2:
R = pAdicField(2, M + 1)
else:
R = pAdicField(p, M)
if psi is not None:
ap = psi(ap)
ap = ap * chip
sdisc = R(ap**2 - 4 * p).sqrt()
v0 = (R(ap) + sdisc) / 2
v1 = (R(ap) - sdisc) / 2
if v0.valuation() > 0:
v0, v1 = v1, v0
alpha = v0
return (1 - 1 / alpha)**2
def p_stabilize_critical(self,p,ap,M):
r"""
INPUT:
- ``p`` -- good ordinary prime
- ``ap`` -- Hecke eigenvalue
- ``M`` -- precision of `Q_p`
OUTPUT:
The unique modular symbol of level `N*p` with the same Hecke-eigenvalues as self away from `p`
and non-unit eigenvalue at `p`.
EXAMPLES:
::
sage: E = EllipticCurve('11a')
sage: phi = form_modsym_from_elliptic_curve(E); phi
[-1/5, 3/2, -1/2]
sage: phi_ss = phi.p_stabilize_critical(3,E.ap(3),10)
sage: phi_ord.hecke(2) - phi_ord.scale(E.ap(2))
[0, 0, 0, 0, 0, 0, 0, 0, 0]
sage: phi_ord.hecke(5) - phi_ord.scale(E.ap(5))
[0, 0, 0, 0, 0, 0, 0, 0, 0]
sage: p = 3
sage: M = 10
sage: R = pAdicField(p,M)['y']
sage: y = R.gen()
sage: f = y**2-ap*y+p
sage: v = f.roots()
sage: beta = v[1][0]
sage: beta^2 - E.ap(3)*beta + 3
O(3^10)
sage: beta=ZZ(beta)
sage: (phi_ss.hecke(3) - phi_ss.scale(beta)).valuation(3)
9
"""
N = self.level()
k = self.data(0).weight()
assert N%p<>0, "The level isn't prime to p"
assert (ap%p)!=0, "Not ordinary!"
# makes alpha the non-unit root of Hecke poly
R = pAdicField(p,M)['y']
y = R.gen()
f = y**2-ap*y+p**(k+1)
v = f.roots()
if ZZ(v[0][0])%p == 0:
alpha = ZZ(v[0][0])
else:
alpha = ZZ(v[1][0])
if self.full_data == 0:
self.compute_full_data_from_gen_data()
return self.p_stabilize(p,alpha)
def interpolation_factor(self, ap,chip=1, psi = None):
r"""
Returns the interpolation factor associated to self
EXAMPLES::
sage: from sage.modular.pollack_stevens.space import ps_modsym_from_elliptic_curve
sage: E = EllipticCurve('57a')
sage: p = 5
sage: prec = 4
sage: phi = ps_modsym_from_elliptic_curve(E)
sage: phi_stabilized = phi.p_stabilize(p,M = prec)
sage: Phi = phi_stabilized.lift(p,prec,None,algorithm='stevens')
sage: L = pAdicLseries(Phi)
sage: ap = phi.Tq_eigenvalue(p)
sage: L.interpolation_factor(ap)
4 + 2*5 + 4*5^3 + O(5^4)
Comparing against a different implementation:
sage: L = E.padic_lseries(5)
sage: (1-1/L.alpha(prec=4))^2
4 + 2*5 + 4*5^3 + O(5^4)
"""
M = self.symb().precision_absolute()
p = self.prime()
if p == 2:
R = pAdicField(2, M + 1)
else:
R = pAdicField(p, M)
if psi != None:
ap = psi(ap)
ap = ap*chip
sdisc = R(ap**2 - 4*p).sqrt()
v0 = (R(ap) + sdisc) / 2
v1 = (R(ap) - sdisc) / 2
if v0.valuation() > 0:
v0, v1 = v1, v0
alpha = v0
return (1 - 1/alpha)**2
def __getitem__(self, n):
"""
Returns the `n`-th coefficient of the `p`-adic `L`-series
EXAMPLES::
sage: from sage.modular.pollack_stevens.space import ps_modsym_from_elliptic_curve
sage: E = EllipticCurve('57a')
sage: p = 5
sage: prec = 4
sage: phi = ps_modsym_from_elliptic_curve(E)
sage: phi_stabilized = phi.p_stabilize(p,M = prec+3)
sage: Phi = phi_stabilized.lift(p=p,M=prec,alpha=None,algorithm='stevens',eigensymbol=True)
sage: L = pAdicLseries(Phi)
sage: L[1]
3*5 + 5^2 + O(5^3)
sage: L1 = E.padic_lseries(5)
sage: L1.series(4)[1]
3*5 + 5^2 + O(5^3)
"""
if self._coefficients.has_key(n):
return self._coefficients[n]
else:
p = self.prime()
symb = self.symb()
ap = symb.Tq_eigenvalue(p)
gamma = self._gamma
precision = self._precision
S = QQ[['z']]
z = S.gen()
M = symb.precision_absolute()
K = pAdicField(p, M)
dn = 0
if n == 0:
precision = M
lb = [1] + [0 for a in range(M-1)]
else:
lb = log_gamma_binomial(p, gamma, z, n, 2*M)
if precision == None:
precision = min([j + lb[j].valuation(p) for j in range(M, len(lb))])
lb = [lb[a] for a in range(M)]
for j in range(len(lb)):
cjn = lb[j]
temp = sum((ZZ(K.teichmuller(a))**(-j)) * self._basic_integral(a, j) for a in range(1, p))
dn = dn + cjn*temp
self._coefficients[n] = dn + O(p**precision)
return self._coefficients[n]
def __getitem__(self, n):
r"""
Return the `n`-th coefficient of the `p`-adic `L`-series
EXAMPLES::
sage: E = EllipticCurve('14a5')
sage: L = E.padic_lseries(7,implementation="pollackstevens",precision=5) # long time
sage: L[0] # long time
O(7^5)
sage: L[1] # long time
5 + 5*7 + 2*7^2 + 2*7^3 + O(7^4)
"""
if n in self._coefficients:
return self._coefficients[n]
else:
p = self.prime()
symb = self.symbol()
# ap = symb.Tq_eigenvalue(p)
gamma = self._gamma
precision = self._precision
S = QQ[['z']]
z = S.gen()
M = symb.precision_relative()
K = pAdicField(p, M)
dn = 0
if n == 0:
precision = M
lb = [1] + [0 for a in range(M - 1)]
else:
lb = log_gamma_binomial(p, gamma, z, n, 2 * M)
if precision is None:
precision = min(
[j + lb[j].valuation(p) for j in range(M, len(lb))])
lb = [lb[a] for a in range(M)]
for j in range(len(lb)):
cjn = lb[j]
temp = sum(
(ZZ(K.teichmuller(a))**(-j)) * self._basic_integral(a, j)
for a in range(1, p))
dn = dn + cjn * temp
self._coefficients[n] = dn.add_bigoh(precision)
self._coefficients[n] /= self._cinf
return self._coefficients[n]
def __getitem__(self, n):
r"""
Return the `n`-th coefficient of the `p`-adic `L`-series
EXAMPLES::
sage: E = EllipticCurve('14a5')
sage: L = E.padic_lseries(7,implementation="pollackstevens",precision=5) # long time
sage: L[0] # long time
O(7^5)
sage: L[1] # long time
5 + 5*7 + 2*7^2 + 2*7^3 + O(7^4)
"""
if n in self._coefficients:
return self._coefficients[n]
else:
p = self.prime()
symb = self.symbol()
# ap = symb.Tq_eigenvalue(p)
gamma = self._gamma
precision = self._precision
S = QQ[['z']]
z = S.gen()
M = symb.precision_relative()
K = pAdicField(p, M)
dn = 0
if n == 0:
precision = M
lb = [1] + [0 for a in range(M - 1)]
else:
lb = log_gamma_binomial(p, gamma, z, n, 2 * M)
if precision is None:
precision = min([j + lb[j].valuation(p)
for j in range(M, len(lb))])
lb = [lb[a] for a in range(M)]
for j in range(len(lb)):
cjn = lb[j]
temp = sum((ZZ(K.teichmuller(a)) ** (-j))
* self._basic_integral(a, j) for a in range(1, p))
dn = dn + cjn * temp
self._coefficients[n] = dn.add_bigoh(precision)
self._coefficients[n] /= self._cinf
return self._coefficients[n]
def matrix_of_frobenius(self, p, prec=20):
# BUG: should get this method from HyperellipticCurve_generic
def my_chage_ring(self, R):
from .constructor import HyperellipticCurve
f, h = self._hyperelliptic_polynomials
y = self._printing_ring.gen()
x = self._printing_ring.base_ring().gen()
return HyperellipticCurve(f.change_ring(R), h, "%s,%s"%(x,y))
import sage.schemes.hyperelliptic_curves.monsky_washnitzer as monsky_washnitzer
if is_pAdicField(p) or is_pAdicRing(p):
K = p
else:
K = pAdicField(p, prec)
frob_p, forms = monsky_washnitzer.matrix_of_frobenius_hyperelliptic(my_chage_ring(self, K))
return frob_p
def matrix_of_frobenius(self, p, prec=20):
# BUG: should get this method from HyperellipticCurve_generic
def my_chage_ring(self, R):
from constructor import HyperellipticCurve
f, h = self._hyperelliptic_polynomials
y = self._printing_ring.gen()
x = self._printing_ring.base_ring().gen()
return HyperellipticCurve(f.change_ring(R), h, "%s,%s"%(x,y))
import sage.schemes.elliptic_curves.monsky_washnitzer as monsky_washnitzer
if is_pAdicField(p) or is_pAdicRing(p):
K = p
else:
K = pAdicField(p, prec)
frob_p, forms = monsky_washnitzer.matrix_of_frobenius_hyperelliptic(my_chage_ring(self, K))
return frob_p
def pLfunction_coef(Phi,ap,n,D,gam,error=None):
"""
Returns the nth coefficient of the `p`-adic `L`-function in the
`T`-variable of a quadratic twist of `\Phi`.
If error is not specified, then the correct error bound is computed
and the answer is return modulo that accuracy.
INPUT:
- ``Phi`` -- overconvergent Hecke-eigensymbol
- ``ap`` -- eigenvalue of `U_p`
- ``n`` -- index of desired coefficient
- ``D`` -- discriminant of quadratic twist
- ``gam`` -- topological generator of `1 + pZ_p`
OUTPUT:
The `n`th coefficient of the `p`-adic `L`-function in the `T`-variable
of a quadratic twist of `\Phi`
EXAMPLES:
"""
S = QQ[['z']]
z = S.gen()
p = Phi.p()
M = Phi.num_moments()
R = pAdicField(p,M)
lb = loggam_binom(p,gam,z,n,2*M)
dn = 0
if n == 0:
err = M
else:
if (error == None):
err = min([j+lb[j].valuation(p) for j in range(M,len(lb))])
else:
err = error
lb = [lb[a] for a in range(M)]
for j in range(len(lb)):
cjn = lb[j]
temp = 0
for a in range(1,p):
temp = temp + (teich(a,p,M)**(-j))*basic_integral(Phi,a,j,ap,D)
dn = dn + cjn*temp
return dn + O(p**err)
def _basic_integral(self, a, j):
r"""
Returns `\int_{a+pZ_p} (z-{a})^j d\Phi(0-infty)`
-- see formula [Pollack-Stevens, sec 9.2]
INPUT:
- ``a`` -- integer in range(p)
- ``j`` -- integer in range(self.symb().precision_absolute())
EXAMPLES::
sage: from sage.modular.pollack_stevens.space import ps_modsym_from_elliptic_curve
sage: E = EllipticCurve('57a')
sage: p = 5
sage: prec = 4
sage: phi = ps_modsym_from_elliptic_curve(E)
sage: phi_stabilized = phi.p_stabilize(p,M = prec+3)
sage: Phi = phi_stabilized.lift(p,prec,None,algorithm = 'stevens',eigensymbol = True)
sage: L = pAdicLseries(Phi)
sage: L.eval_twisted_symbol_on_Da(1)
(2 + 2*5 + 2*5^2 + 2*5^3 + O(5^4), 2 + 3*5 + 2*5^2 + O(5^3), 4*5 + O(5^2), 3 + O(5))
sage: L._basic_integral(1,2)
2*5^3 + O(5^4)
"""
symb = self.symb()
M = symb.precision_absolute()
if j > M:
raise PrecisionError ("Too many moments requested")
p = self.prime()
ap = symb.Tq_eigenvalue(p)
D = self._quadratic_twist
ap = ap * kronecker(D, p)
K = pAdicField(p, M)
symb_twisted = self.eval_twisted_symbol_on_Da(a)
return sum(binomial(j, r) * ((a - ZZ(K.teichmuller(a)))**(j - r)) *
(p**r) * symb_twisted.moment(r) for r in range(j + 1)) / ap
def teich(a,p,M):
r"""
Returns the Teichmuller representative of `a` in `Qp` as an element of `Z`
(`Qp` is created with precision `O(p^M)`)
INPUT:
- ``a`` -- element of `Qp`
- ``p`` -- prime
- ``M`` -- precision, `O(p^M)`
OUTPUT:
The Teichmuller representative of a as an element of `Z`
EXAMPLES:
sage: teich(2,5,10)
6139557
sage: teich(3,7,10)
146507973
"""
R = pAdicField(p,M)
return ZZ(R.teichmuller(a))