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padics.py
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padics.py
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# -*- coding: utf-8 -*-
"""
Miscellaneous `p`-adic functions
`p`-adic functions from ell_rational_field.py, moved here to reduce
crowding in that file.
"""
######################################################################
# Copyright (C) 2007 William Stein <wstein@gmail.com>
#
# Distributed under the terms of the GNU General Public License (GPL)
#
# This code is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# General Public License for more details.
#
# The full text of the GPL is available at:
#
# http://www.gnu.org/licenses/
######################################################################
from __future__ import absolute_import
import sage.rings.all as rings
from . import padic_lseries as plseries
import sage.arith.all as arith
from sage.rings.all import (
Qp, Zp,
Integers,
Integer,
O,
PowerSeriesRing,
LaurentSeriesRing,
RationalField)
import math
import sage.misc.misc as misc
import sage.matrix.all as matrix
sqrt = math.sqrt
import sage.schemes.hyperelliptic_curves.monsky_washnitzer
import sage.schemes.hyperelliptic_curves.hypellfrob
from sage.misc.all import cached_method
def __check_padic_hypotheses(self, p):
r"""
Helper function that determines if `p`
is an odd prime of good ordinary reduction.
EXAMPLES::
sage: E = EllipticCurve('11a1')
sage: from sage.schemes.elliptic_curves.padics import __check_padic_hypotheses
sage: __check_padic_hypotheses(E,5)
5
sage: __check_padic_hypotheses(E,29)
Traceback (most recent call last):
...
ArithmeticError: p must be a good ordinary prime
"""
p = rings.Integer(p)
if not p.is_prime():
raise ValueError("p = (%s) must be prime"%p)
if p == 2:
raise ValueError("p must be odd")
if self.conductor() % p == 0 or self.ap(p) % p == 0:
raise ArithmeticError("p must be a good ordinary prime")
return p
def _normalize_padic_lseries(self, p, normalize, use_eclib, implementation, precision):
r"""
Normalize parameters for :meth:`padic_lseries`.
TESTS::
sage: from sage.schemes.elliptic_curves.padics import _normalize_padic_lseries
sage: u = _normalize_padic_lseries(None, 5, None, None, 'sage', 10)
sage: v = _normalize_padic_lseries(None, 5, "L_ratio", None, 'sage', 10)
sage: u == v
True
"""
if use_eclib is not None:
from sage.misc.superseded import deprecation
deprecation(812,"Use the option 'implementation' instead of 'use_eclib'")
if implementation == 'pollackstevens':
raise ValueError
if use_eclib:
implementation = 'eclib'
else:
implementation = 'sage'
if implementation == 'eclib':
if normalize is None:
normalize = "L_ratio"
elif implementation == 'sage':
if normalize is None:
normalize = "L_ratio"
elif implementation == 'pollackstevens':
if precision is None:
raise ValueError("Must specify precision when using 'pollackstevens'")
if normalize is not None:
raise ValueError("The 'normalize' parameter is not used for Pollack-Stevens' overconvergent modular symbols")
else:
raise ValueError("Implementation should be one of 'sage', 'eclib' or 'pollackstevens'")
#if precision is not None and implementation != 'pollackstevens':
# raise ValueError("Must *not* specify precision unless using 'pollackstevens'")
return (p, normalize, implementation, precision)
@cached_method(key=_normalize_padic_lseries)
def padic_lseries(self, p, normalize = None, use_eclib = None, implementation = 'eclib', precision = None):
r"""
Return the `p`-adic `L`-series of self at
`p`, which is an object whose approx method computes
approximation to the true `p`-adic `L`-series to
any desired precision.
INPUT:
- ``p`` - prime
- ``normalize`` - 'L_ratio' (default), 'period' or 'none';
this is describes the way the modular symbols
are normalized. See modular_symbol for
more details.
- ``use_eclib`` - deprecated, use ``implementation`` instead
- ``implementation`` - 'eclib' (default), 'sage', 'pollackstevens';
Whether to use John Cremona's eclib, the Sage implementation,
or Pollack-Stevens' implementation of overconvergent
modular symbols.
EXAMPLES::
sage: E = EllipticCurve('37a')
sage: L = E.padic_lseries(5); L
5-adic L-series of Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: type(L)
<class 'sage.schemes.elliptic_curves.padic_lseries.pAdicLseriesOrdinary'>
We compute the `3`-adic `L`-series of two curves of
rank `0` and in each case verify the interpolation property
for their leading coefficient (i.e., value at 0)::
sage: e = EllipticCurve('11a')
sage: ms = e.modular_symbol()
sage: [ms(1/11), ms(1/3), ms(0), ms(oo)]
[0, -3/10, 1/5, 0]
sage: ms(0)
1/5
sage: L = e.padic_lseries(3)
sage: P = L.series(5)
sage: P(0)
2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + O(3^7)
sage: alpha = L.alpha(9); alpha
2 + 3^2 + 2*3^3 + 2*3^4 + 2*3^6 + 3^8 + O(3^9)
sage: R.<x> = QQ[]
sage: f = x^2 - e.ap(3)*x + 3
sage: f(alpha)
O(3^9)
sage: r = e.lseries().L_ratio(); r
1/5
sage: (1 - alpha^(-1))^2 * r
2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + 3^7 + O(3^9)
sage: P(0)
2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + O(3^7)
Next consider the curve 37b::
sage: e = EllipticCurve('37b')
sage: L = e.padic_lseries(3)
sage: P = L.series(5)
sage: alpha = L.alpha(9); alpha
1 + 2*3 + 3^2 + 2*3^5 + 2*3^7 + 3^8 + O(3^9)
sage: r = e.lseries().L_ratio(); r
1/3
sage: (1 - alpha^(-1))^2 * r
3 + 3^2 + 2*3^4 + 2*3^5 + 2*3^6 + 3^7 + O(3^9)
sage: P(0)
3 + 3^2 + 2*3^4 + 2*3^5 + O(3^6)
We can use Sage modular symbols instead to compute the `L`-series::
sage: e = EllipticCurve('11a')
sage: L = e.padic_lseries(3, implementation = 'sage')
sage: L.series(5,prec=10)
2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + O(3^7) + (1 + 3 + 2*3^2 + 3^3 + O(3^4))*T + (1 + 2*3 + O(3^4))*T^2 + (3 + 2*3^2 + O(3^3))*T^3 + (2*3 + 3^2 + O(3^3))*T^4 + (2 + 2*3 + 2*3^2 + O(3^3))*T^5 + (1 + 3^2 + O(3^3))*T^6 + (2 + 3^2 + O(3^3))*T^7 + (2 + 2*3 + 2*3^2 + O(3^3))*T^8 + (2 + O(3^2))*T^9 + O(T^10)
Finally, we can use the overconvergent method of Pollack-Stevens.::
sage: e = EllipticCurve('11a')
sage: L = e.padic_lseries(3, implementation = 'pollackstevens', precision = 5)
sage: L.series(3)
2 + 3 + 3^2 + 2*3^3 + O(3^5) + (1 + 3 + 2*3^2 + O(3^3))*T + (1 + 2*3 + O(3^2))*T^2 + O(3)*T^3 + O(3^0)*T^4 + O(T^5)
sage: L[3]
O(3)
Another example with a semistable prime.::
sage: E = EllipticCurve("11a1")
sage: L = E.padic_lseries(11, implementation = 'pollackstevens', precision=3)
sage: L[1]
10 + 3*11 + O(11^2)
sage: L[3]
O(11^0)
"""
p, normalize, implementation, precision = self._normalize_padic_lseries(p,\
normalize, use_eclib, implementation, precision)
if implementation in ['sage', 'eclib']:
if self.ap(p) % p != 0:
Lp = plseries.pAdicLseriesOrdinary(self, p,
normalize = normalize, implementation = implementation)
else:
Lp = plseries.pAdicLseriesSupersingular(self, p,
normalize = normalize, implementation = implementation)
else:
phi = self.pollack_stevens_modular_symbol(sign=0)
if phi.parent().level() % p == 0:
Phi = phi.lift(p, precision, eigensymbol = True)
else:
Phi = phi.p_stabilize_and_lift(p, precision, eigensymbol = True)
Lp = Phi.padic_lseries() #mm TODO should this pass precision on too ?
Lp._cinf = self.real_components()
return Lp
def padic_regulator(self, p, prec=20, height=None, check_hypotheses=True):
r"""
Computes the cyclotomic `p`-adic regulator of this curve.
INPUT:
- ``p`` - prime = 5
- ``prec`` - answer will be returned modulo
`p^{\mathrm{prec}}`
- ``height`` - precomputed height function. If not
supplied, this function will call padic_height to compute it.
- ``check_hypotheses`` - boolean, whether to check
that this is a curve for which the p-adic height makes sense
OUTPUT: The p-adic cyclotomic regulator of this curve, to the
requested precision.
If the rank is 0, we output 1.
TODO: - remove restriction that curve must be in minimal
Weierstrass form. This is currently required for E.gens().
AUTHORS:
- Liang Xiao: original implementation at the 2006 MSRI
graduate workshop on modular forms
- David Harvey (2006-09-13): cleaned up and integrated into Sage,
removed some redundant height computations
- Chris Wuthrich (2007-05-22): added multiplicative and
supersingular cases
- David Harvey (2007-09-20): fixed some precision loss that was
occurring
EXAMPLES::
sage: E = EllipticCurve("37a")
sage: E.padic_regulator(5, 10)
5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + O(5^10)
An anomalous case::
sage: E.padic_regulator(53, 10)
26*53^-1 + 30 + 20*53 + 47*53^2 + 10*53^3 + 32*53^4 + 9*53^5 + 22*53^6 + 35*53^7 + 30*53^8 + O(53^9)
An anomalous case where the precision drops some::
sage: E = EllipticCurve("5077a")
sage: E.padic_regulator(5, 10)
5 + 5^2 + 4*5^3 + 2*5^4 + 2*5^5 + 2*5^6 + 4*5^7 + 2*5^8 + 5^9 + O(5^10)
Check that answers agree over a range of precisions::
sage: max_prec = 30 # make sure we get past p^2 # long time
sage: full = E.padic_regulator(5, max_prec) # long time
sage: for prec in range(1, max_prec): # long time
....: assert E.padic_regulator(5, prec) == full # long time
A case where the generator belongs to the formal group already
(trac #3632)::
sage: E = EllipticCurve([37,0])
sage: E.padic_regulator(5,10)
2*5^2 + 2*5^3 + 5^4 + 5^5 + 4*5^6 + 3*5^8 + 4*5^9 + O(5^10)
The result is not dependent on the model for the curve::
sage: E = EllipticCurve([0,0,0,0,2^12*17])
sage: Em = E.minimal_model()
sage: E.padic_regulator(7) == Em.padic_regulator(7)
True
Allow a Python int as input::
sage: E = EllipticCurve('37a')
sage: E.padic_regulator(int(5))
5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + 5^10 + 3*5^11 + 3*5^12 + 5^13 + 4*5^14 + 5^15 + 2*5^16 + 5^17 + 2*5^18 + 4*5^19 + O(5^20)
"""
p = Integer(p) # this is assumed in code below
if check_hypotheses:
if not p.is_prime():
raise ValueError("p = (%s) must be prime"%p)
if p == 2:
raise ValueError("p must be odd") # todo
if self.conductor() % (p**2) == 0:
raise ArithmeticError("p must be a semi-stable prime")
if self.conductor() % p == 0:
Eq = self.tate_curve(p)
reg = Eq.padic_regulator(prec=prec)
return reg
elif self.ap(p) % p == 0:
lp = self.padic_lseries(p)
reg = lp.Dp_valued_regulator(prec=prec)
return reg
else:
if self.rank() == 0:
return Qp(p,prec)(1)
if height is None:
height = self.padic_height(p, prec, check_hypotheses=False)
d = self.padic_height_pairing_matrix(p=p, prec=prec, height=height, check_hypotheses=False)
return d.determinant()
def padic_height_pairing_matrix(self, p, prec=20, height=None, check_hypotheses=True):
r"""
Computes the cyclotomic `p`-adic height pairing matrix of
this curve with respect to the basis self.gens() for the
Mordell-Weil group for a given odd prime p of good ordinary
reduction.
INPUT:
- ``p`` - prime = 5
- ``prec`` - answer will be returned modulo
`p^{\mathrm{prec}}`
- ``height`` - precomputed height function. If not
supplied, this function will call padic_height to compute it.
- ``check_hypotheses`` - boolean, whether to check
that this is a curve for which the p-adic height makes sense
OUTPUT: The p-adic cyclotomic height pairing matrix of this curve
to the given precision.
TODO: - remove restriction that curve must be in minimal
Weierstrass form. This is currently required for E.gens().
AUTHORS:
- David Harvey, Liang Xiao, Robert Bradshaw, Jennifer
Balakrishnan: original implementation at the 2006 MSRI graduate
workshop on modular forms
- David Harvey (2006-09-13): cleaned up and integrated into Sage,
removed some redundant height computations
EXAMPLES::
sage: E = EllipticCurve("37a")
sage: E.padic_height_pairing_matrix(5, 10)
[5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + O(5^10)]
A rank two example::
sage: e =EllipticCurve('389a')
sage: e._set_gens([e(-1, 1), e(1,0)]) # avoid platform dependent gens
sage: e.padic_height_pairing_matrix(5,10)
[ 3*5 + 2*5^2 + 5^4 + 5^5 + 5^7 + 4*5^9 + O(5^10) 5 + 4*5^2 + 5^3 + 2*5^4 + 3*5^5 + 4*5^6 + 5^7 + 5^8 + 2*5^9 + O(5^10)]
[5 + 4*5^2 + 5^3 + 2*5^4 + 3*5^5 + 4*5^6 + 5^7 + 5^8 + 2*5^9 + O(5^10) 4*5 + 2*5^4 + 3*5^6 + 4*5^7 + 4*5^8 + O(5^10)]
An anomalous rank 3 example::
sage: e = EllipticCurve("5077a")
sage: e._set_gens([e(-1,3), e(2,0), e(4,6)])
sage: e.padic_height_pairing_matrix(5,4)
[4 + 3*5 + 4*5^2 + 4*5^3 + O(5^4) 4 + 4*5^2 + 2*5^3 + O(5^4) 3*5 + 4*5^2 + 5^3 + O(5^4)]
[ 4 + 4*5^2 + 2*5^3 + O(5^4) 3 + 4*5 + 3*5^2 + 5^3 + O(5^4) 2 + 4*5 + O(5^4)]
[ 3*5 + 4*5^2 + 5^3 + O(5^4) 2 + 4*5 + O(5^4) 1 + 3*5 + 5^2 + 5^3 + O(5^4)]
"""
if check_hypotheses:
p = __check_padic_hypotheses(self, p)
K = Qp(p, prec=prec)
rank = self.rank()
M = matrix.matrix(K, rank, rank, 0)
if rank == 0:
return M
basis = self.gens()
if height is None:
height = self.padic_height(p, prec, check_hypotheses=False)
# Use <P, Q> =1/2*( h(P + Q) - h(P) - h(Q) )
for i in range(rank):
M[i,i] = height(basis[i])
for i in range(rank):
for j in range(i+1, rank):
M[i, j] = ( height(basis[i] + basis[j]) - M[i,i] - M[j,j] ) / 2
M[j, i] = M[i, j]
return M
def _multiply_point(E, R, P, m):
r"""
Computes coordinates of a multiple of P with entries in a ring.
INPUT:
- ``E`` - elliptic curve over Q with integer
coefficients
- ``P`` - a rational point on P that reduces to a
non-singular point at all primes
- ``R`` - a ring in which 2 is invertible (typically
`\ZZ/L\ZZ` for some positive odd integer
`L`).
- ``m`` - an integer, = 1
OUTPUT: A triple `(a', b', d')` such that if the point
`mP` has coordinates `(a/d^2, b/d^3)`, then we have
`a' \equiv a`, `b' \equiv \pm b`,
`d' \equiv \pm d` all in `R` (i.e. modulo
`L`).
Note the ambiguity of signs for `b'` and `d'`.
There's not much one can do about this, but at least one can say
that the sign for `b'` will match the sign for
`d'`.
ALGORITHM: Proposition 9 of "Efficient Computation of p-adic
Heights" (David Harvey, to appear in LMS JCM).
Complexity is soft-`O(\log L \log m + \log^2 m)`.
AUTHORS:
- David Harvey (2008-01): replaced _DivPolyContext with
_multiply_point
EXAMPLES:
37a has trivial Tamagawa numbers so all points have nonsingular
reduction at all primes::
sage: E = EllipticCurve("37a")
sage: P = E([0, -1]); P
(0 : -1 : 1)
sage: 19*P
(-59997896/67387681 : 88075171080/553185473329 : 1)
sage: R = Integers(625)
sage: from sage.schemes.elliptic_curves.padics import _multiply_point
sage: _multiply_point(E, R, P, 19)
(229, 170, 541)
sage: -59997896 % 625
229
sage: -88075171080 % 625 # note sign is flipped
170
sage: -67387681.sqrt() % 625 # sign is flipped here too
541
Trivial cases (trac 3632)::
sage: _multiply_point(E, R, P, 1)
(0, 624, 1)
sage: _multiply_point(E, R, 19*P, 1)
(229, 455, 84)
sage: (170 + 455) % 625 # note the sign did not flip here
0
sage: (541 + 84) % 625
0
Test over a range of `n` for a single curve with fairly
random coefficients::
sage: R = Integers(625)
sage: E = EllipticCurve([4, -11, 17, -8, -10])
sage: P = E.gens()[0] * LCM(E.tamagawa_numbers())
sage: from sage.schemes.elliptic_curves.padics import _multiply_point
sage: Q = P
sage: for n in range(1, 25):
....: naive = R(Q[0].numerator()), \
....: R(Q[1].numerator()), \
....: R(Q[0].denominator().sqrt())
....: triple = _multiply_point(E, R, P, n)
....: assert (triple[0] == naive[0]) and ( \
....: (triple[1] == naive[1] and triple[2] == naive[2]) or \
....: (triple[1] == -naive[1] and triple[2] == -naive[2])), \
....: "_multiply_point() gave an incorrect answer"
....: Q = Q + P
"""
assert m >= 1
alpha = R(P[0].numerator())
beta = R(P[1].numerator())
d = R(P[0].denominator().sqrt())
if m == 1:
return alpha, beta, d
a1 = R(E.a1()) * d
a3 = R(E.a3()) * d**3
b2 = R(E.b2()) * d**2
b4 = R(E.b4()) * d**4
b6 = R(E.b6()) * d**6
b8 = R(E.b8()) * d**8
B4 = 6*alpha**2 + b2*alpha + b4
B6 = 4*alpha**3 + b2*alpha**2 + 2*b4*alpha + b6
B6_sqr = B6*B6
B8 = 3*alpha**4 + b2*alpha**3 + 3*b4*alpha**2 + 3*b6*alpha + b8
T = 2*beta + a1*alpha + a3
# make a list of disjoint intervals [a[i], b[i]) such that we need to
# compute g(k) for all a[i] <= k <= b[i] for each i
intervals = []
interval = (m - 2, m + 3)
while interval[0] < interval[1]:
intervals.append(interval)
interval = max((interval[0] - 3) >> 1, 0), \
min((interval[1] + 5) >> 1, interval[0])
# now walk through list and compute g(k)
g = {0 : R(0), 1 : R(1), 2 : R(-1), 3 : B8, 4 : B6**2 - B4*B8}
last = [0, 1, 2, 3, 4] # last few k
for i in reversed(intervals):
k = i[0]
while k < i[1]:
if k > 4:
j = k >> 1
if k & 1:
t1 = g[j]
t2 = g[j+1]
prod1 = g[j+2] * t1*t1*t1
prod2 = g[j-1] * t2*t2*t2
g[k] = prod1 - B6_sqr * prod2 if j & 1 else B6_sqr * prod1 - prod2
else:
t1 = g[j-1]
t2 = g[j+1]
g[k] = g[j] * (g[j-2] * t2*t2 - g[j+2] * t1*t1)
k = k + 1
if m & 1:
psi_m = g[m]
psi_m_m1 = g[m-1] * T
psi_m_p1 = g[m+1] * T
else:
psi_m = g[m] * T
psi_m_m1 = g[m-1]
psi_m_p1 = g[m+1]
theta = alpha * psi_m * psi_m - psi_m_m1 * psi_m_p1
t1 = g[m-2] * g[m+1] * g[m+1] - g[m+2] * g[m-1] * g[m-1]
if m & 1:
t1 = t1 * T
omega = (t1 + (a1 * theta + a3 * psi_m * psi_m) * psi_m) / -2
return theta, omega, psi_m * d
def padic_height(self, p, prec=20, sigma=None, check_hypotheses=True):
r"""
Compute the cyclotomic p-adic height.
The equation of the curve must be minimal at `p`.
INPUT:
- ``p`` - prime = 5 for which the curve has
semi-stable reduction
- ``prec`` - integer = 1, desired precision of result
- ``sigma`` - precomputed value of sigma. If not
supplied, this function will call padic_sigma to compute it.
- ``check_hypotheses`` - boolean, whether to check
that this is a curve for which the p-adic height makes sense
OUTPUT: A function that accepts two parameters:
- a Q-rational point on the curve whose height should be computed
- optional boolean flag 'check': if False, it skips some input
checking, and returns the p-adic height of that point to the
desired precision.
- The normalization (sign and a factor 1/2 with respect to some other
normalizations that appear in the literature) is chosen in such a way
as to make the p-adic Birch Swinnerton-Dyer conjecture hold as stated
in [Mazur-Tate-Teitelbaum].
AUTHORS:
- Jennifer Balakrishnan: original code developed at the 2006 MSRI
graduate workshop on modular forms
- David Harvey (2006-09-13): integrated into Sage, optimised to
speed up repeated evaluations of the returned height function,
addressed some thorny precision questions
- David Harvey (2006-09-30): rewrote to use division polynomials
for computing denominator of `nP`.
- David Harvey (2007-02): cleaned up according to algorithms in
"Efficient Computation of p-adic Heights"
- Chris Wuthrich (2007-05): added supersingular and multiplicative heights
EXAMPLES::
sage: E = EllipticCurve("37a")
sage: P = E.gens()[0]
sage: h = E.padic_height(5, 10)
sage: h(P)
5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + O(5^10)
An anomalous case::
sage: h = E.padic_height(53, 10)
sage: h(P)
26*53^-1 + 30 + 20*53 + 47*53^2 + 10*53^3 + 32*53^4 + 9*53^5 + 22*53^6 + 35*53^7 + 30*53^8 + 17*53^9 + O(53^10)
Boundary case::
sage: E.padic_height(5, 3)(P)
5 + 5^2 + O(5^3)
A case that works the division polynomial code a little harder::
sage: E.padic_height(5, 10)(5*P)
5^3 + 5^4 + 5^5 + 3*5^8 + 4*5^9 + O(5^10)
Check that answers agree over a range of precisions::
sage: max_prec = 30 # make sure we get past p^2 # long time
sage: full = E.padic_height(5, max_prec)(P) # long time
sage: for prec in range(1, max_prec): # long time
....: assert E.padic_height(5, prec)(P) == full # long time
A supersingular prime for a curve::
sage: E = EllipticCurve('37a')
sage: E.is_supersingular(3)
True
sage: h = E.padic_height(3, 5)
sage: h(E.gens()[0])
(3 + 3^3 + O(3^6), 2*3^2 + 3^3 + 3^4 + 3^5 + 2*3^6 + O(3^7))
sage: E.padic_regulator(5)
5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + 5^10 + 3*5^11 + 3*5^12 + 5^13 + 4*5^14 + 5^15 + 2*5^16 + 5^17 + 2*5^18 + 4*5^19 + O(5^20)
sage: E.padic_regulator(3, 5)
(3 + 2*3^2 + 3^3 + O(3^4), 3^2 + 2*3^3 + 3^4 + O(3^5))
A torsion point in both the good and supersingular cases::
sage: E = EllipticCurve('11a')
sage: P = E.torsion_subgroup().gen(0).element(); P
(5 : 5 : 1)
sage: h = E.padic_height(19, 5)
sage: h(P)
0
sage: h = E.padic_height(5, 5)
sage: h(P)
0
The result is not dependent on the model for the curve::
sage: E = EllipticCurve([0,0,0,0,2^12*17])
sage: Em = E.minimal_model()
sage: P = E.gens()[0]
sage: Pm = Em.gens()[0]
sage: h = E.padic_height(7)
sage: hm = Em.padic_height(7)
sage: h(P) == hm(Pm)
True
TESTS:
Check that ticket :trac:`20798` is solved::
sage: E = EllipticCurve("91b")
sage: h = E.padic_height(7,10)
sage: P = E.gen(0)
sage: h(P)
2*7 + 7^2 + 5*7^3 + 6*7^4 + 2*7^5 + 3*7^6 + 7^7 + O(7^9)
sage: h(P+P)
7 + 5*7^2 + 6*7^3 + 5*7^4 + 4*7^5 + 6*7^6 + 5*7^7 + O(7^9)
"""
if check_hypotheses:
if not p.is_prime():
raise ValueError("p = (%s) must be prime"%p)
if p == 2:
raise ValueError("p must be odd") # todo
if self.conductor() % (p**2) == 0:
raise ArithmeticError("p must be a semi-stable prime")
prec = int(prec)
if prec < 1:
raise ValueError("prec (=%s) must be at least 1" % prec)
if self.conductor() % p == 0:
Eq = self.tate_curve(p)
return Eq.padic_height(prec=prec)
elif self.ap(p) % p == 0:
lp = self.padic_lseries(p)
return lp.Dp_valued_height(prec=prec)
# else good ordinary case
# For notation and definitions, see "Efficient Computation of
# p-adic Heights", David Harvey (unpublished)
n1 = self.change_ring(rings.GF(p)).cardinality()
n2 = arith.LCM(self.tamagawa_numbers())
n = arith.LCM(n1, n2)
m = int(n / n2)
adjusted_prec = prec + 2 * arith.valuation(n, p) # this is M'
R = rings.Integers(p ** adjusted_prec)
if sigma is None:
sigma = self.padic_sigma(p, adjusted_prec, check_hypotheses=False)
# K is the field for the final result
K = Qp(p, prec=adjusted_prec-1)
E = self
def height(P, check=True):
if P.is_finite_order():
return K(0)
if check:
assert P.curve() == E, "the point P must lie on the curve " \
"from which the height function was created"
Q = n2 * P
alpha, beta, d = _multiply_point(E, R, Q, m)
assert beta.lift() % p != 0, "beta should be a unit!"
assert d.lift() % p == 0, "d should not be a unit!"
t = -d * alpha / beta
total = R(1)
t_power = t
for k in range(2, adjusted_prec + 1):
total = total + t_power * sigma[k].lift()
t_power = t_power * t
total = (-alpha / beta) * total
L = Qp(p, prec=adjusted_prec)
total = L(total.lift(), adjusted_prec) # yuck... get rid of this lift!
# changed sign to make it correct for p-adic bsd
answer = -total.log() * 2 / n**2
if check:
assert answer.precision_absolute() >= prec, "we should have got an " \
"answer with precision at least prec, but we didn't."
return K(answer)
# (man... I love python's local function definitions...)
return height
def padic_height_via_multiply(self, p, prec=20, E2=None, check_hypotheses=True):
r"""
Computes the cyclotomic p-adic height.
The equation of the curve must be minimal at `p`.
INPUT:
- ``p`` - prime = 5 for which the curve has good
ordinary reduction
- ``prec`` - integer = 2, desired precision of result
- ``E2`` - precomputed value of E2. If not supplied,
this function will call padic_E2 to compute it. The value supplied
must be correct mod `p^(prec-2)` (or slightly higher in the
anomalous case; see the code for details).
- ``check_hypotheses`` - boolean, whether to check
that this is a curve for which the p-adic height makes sense
OUTPUT: A function that accepts two parameters:
- a Q-rational point on the curve whose height should be computed
- optional boolean flag 'check': if False, it skips some input
checking, and returns the p-adic height of that point to the
desired precision.
- The normalization (sign and a factor 1/2 with respect to some other
normalizations that appear in the literature) is chosen in such a way
as to make the p-adic Birch Swinnerton-Dyer conjecture hold as stated
in [Mazur-Tate-Teitelbaum].
AUTHORS:
- David Harvey (2008-01): based on the padic_height() function,
using the algorithm of"Computing p-adic heights via
point multiplication"
EXAMPLES::
sage: E = EllipticCurve("37a")
sage: P = E.gens()[0]
sage: h = E.padic_height_via_multiply(5, 10)
sage: h(P)
5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + O(5^10)
An anomalous case::
sage: h = E.padic_height_via_multiply(53, 10)
sage: h(P)
26*53^-1 + 30 + 20*53 + 47*53^2 + 10*53^3 + 32*53^4 + 9*53^5 + 22*53^6 + 35*53^7 + 30*53^8 + 17*53^9 + O(53^10)
Supply the value of E2 manually::
sage: E2 = E.padic_E2(5, 8)
sage: E2
2 + 4*5 + 2*5^3 + 5^4 + 3*5^5 + 2*5^6 + O(5^8)
sage: h = E.padic_height_via_multiply(5, 10, E2=E2)
sage: h(P)
5 + 5^2 + 5^3 + 3*5^6 + 4*5^7 + 5^9 + O(5^10)
Boundary case::
sage: E.padic_height_via_multiply(5, 3)(P)
5 + 5^2 + O(5^3)
Check that answers agree over a range of precisions::
sage: max_prec = 30 # make sure we get past p^2 # long time
sage: full = E.padic_height(5, max_prec)(P) # long time
sage: for prec in range(2, max_prec): # long time
....: assert E.padic_height_via_multiply(5, prec)(P) == full # long time
"""
if check_hypotheses:
if not p.is_prime():
raise ValueError("p = (%s) must be prime"%p)
if p == 2:
raise ValueError("p must be odd") # todo
if self.conductor() % p == 0:
raise ArithmeticError("must have good reduction at p")
if self.ap(p) % p == 0:
raise ArithmeticError("must be ordinary at p")
prec = int(prec)
if prec < 1:
raise ValueError("prec (=%s) must be at least 1" % prec)
# For notation and definitions, see ``Computing p-adic heights via point
# multiplication'' (David Harvey, still in draft form)
n1 = self.change_ring(rings.GF(p)).cardinality()
n2 = arith.LCM(self.tamagawa_numbers())
n = arith.LCM(n1, n2)
m = int(n / n2)
lamb = int(math.floor(math.sqrt(prec)))
adjusted_prec = prec + 2 * arith.valuation(n, p) # this is M'
R = rings.Integers(p ** (adjusted_prec + 2*lamb))
sigma = self.padic_sigma_truncated(p, N=adjusted_prec, E2=E2, lamb=lamb)
# K is the field for the final result
K = Qp(p, prec=adjusted_prec-1)
E = self
def height(P, check=True):
if P.is_finite_order():
return K(0)
if check:
assert P.curve() == E, "the point P must lie on the curve " \
"from which the height function was created"
Q = n2 * P
alpha, beta, d = _multiply_point(E, R, Q, m * p**lamb)
assert beta.lift() % p != 0, "beta should be a unit!"
assert d.lift() % p == 0, "d should not be a unit!"
t = -d * alpha / beta
total = R(1)
t_power = t
for k in range(2, sigma.prec()):
total = total + t_power * sigma[k].lift()
t_power = t_power * t
total = (-alpha / beta) * total
L = Qp(p, prec=adjusted_prec + 2*lamb)
total = L(total.lift(), adjusted_prec + 2*lamb)
# changed sign to make it correct for p-adic bsd
answer = -total.log() * 2 / (n * p**lamb)**2
if check:
assert answer.precision_absolute() >= prec, "we should have got an " \
"answer with precision at least prec, but we didn't."
return K(answer)
# (man... I love python's local function definitions...)
return height
def padic_sigma(self, p, N=20, E2=None, check=False, check_hypotheses=True):
r"""
Computes the p-adic sigma function with respect to the standard
invariant differential `dx/(2y + a_1 x + a_3)`, as
defined by Mazur and Tate, as a power series in the usual
uniformiser `t` at the origin.
The equation of the curve must be minimal at `p`.
INPUT:
- ``p`` - prime = 5 for which the curve has good
ordinary reduction
- ``N`` - integer = 1, indicates precision of result;
see OUTPUT section for description
- ``E2`` - precomputed value of E2. If not supplied,
this function will call padic_E2 to compute it. The value supplied
must be correct mod `p^{N-2}`.
- ``check`` - boolean, whether to perform a
consistency check (i.e. verify that the computed sigma satisfies
the defining
- ``differential equation`` - note that this does NOT
guarantee correctness of all the returned digits, but it comes
pretty close :-))
- ``check_hypotheses`` - boolean, whether to check
that this is a curve for which the p-adic sigma function makes
sense
OUTPUT: A power series `t + \cdots` with coefficients in
`\ZZ_p`.
The output series will be truncated at `O(t^{N+1})`, and
the coefficient of `t^n` for `n \geq 1` will be
correct to precision `O(p^{N-n+1})`.
In practice this means the following. If `t_0 = p^k u`,
where `u` is a `p`-adic unit with at least
`N` digits of precision, and `k \geq 1`, then the
returned series may be used to compute `\sigma(t_0)`
correctly modulo `p^{N+k}` (i.e. with `N` correct
`p`-adic digits).
ALGORITHM: Described in "Efficient Computation of p-adic Heights"
(David Harvey), which is basically an optimised version of the
algorithm from "p-adic Heights and Log Convergence" (Mazur, Stein,