def sigma(self, prec=10):
"""
EXAMPLES::
sage: E = EllipticCurve('14a')
sage: F = E.formal_group()
sage: F.sigma(5)
t + 1/2*t^2 + (1/2*c + 1/3)*t^3 + (3/4*c + 3/4)*t^4 + O(t^5)
"""
a1,a2,a3,a4,a6 = self.curve().ainvs()
k = self.curve().base_ring()
fl = self.log(prec)
R = rings.PolynomialRing(k,'c'); c = R.gen()
F = fl.reverse()
S = rings.LaurentSeriesRing(R,'z')
c = S(c)
z = S.gen()
F = F(z + O(z**prec))
wp = self.x()(F)
e2 = 12*c - a1**2 - 4*a2
g = (1/z**2 - wp + e2/12).power_series()
h = g.integral().integral()
sigma_of_z = z.power_series() * h.exp()
T = rings.PowerSeriesRing(R,'t')
fl = fl(T.gen()+O(T.gen()**prec))
sigma_of_t = sigma_of_z(fl)
return sigma_of_t
def sigma(self, prec=10):
r"""
Return the Weierstrass sigma function as a formal power series
solution to the differential equation
.. MATH::
\frac{d^2 \log \sigma}{dz^2} = - \wp(z)
with initial conditions `\sigma(O)=0` and `\sigma'(O)=1`,
expressed in the variable `t=\log_E(z)` of the formal group.
INPUT:
- ``prec`` - integer (default 10)
OUTPUT: a power series with given precision
Other solutions can be obtained by multiplication with
a function of the form `\exp(c z^2)`.
If the curve has good ordinary reduction at a prime `p`
then there is a canonical choice of `c` that produces
the canonical `p`-adic sigma function.
To obtain that, please use ``E.padic_sigma(p)`` instead.
See :meth:`~sage.schemes.elliptic_curves.ell_rational_field.EllipticCurve_rational_field.padic_sigma`
EXAMPLES::
sage: E = EllipticCurve('14a')
sage: F = E.formal_group()
sage: F.sigma(5)
t + 1/2*t^2 + 1/3*t^3 + 3/4*t^4 + O(t^5)
"""
a1, a2, a3, a4, a6 = self.curve().ainvs()
k = self.curve().base_ring()
fl = self.log(prec)
F = fl.reverse()
S = rings.LaurentSeriesRing(k, 'z')
z = S.gen()
F = F(z + O(z**prec))
wp = self.x()(F) + (a1**2 + 4 * a2) / 12
g = (1 / z**2 - wp).power_series()
h = g.integral().integral()
sigma_of_z = z.power_series() * h.exp()
T = rings.PowerSeriesRing(k, 't')
fl = fl(T.gen() + O(T.gen()**prec))
sigma_of_t = sigma_of_z(fl)
return sigma_of_t
def w(self, prec=20):
r"""
The formal group power series w.
INPUT:
- ``prec`` - integer (default 20)
OUTPUT: a power series with given precision
DETAILS: Return the formal power series
.. MATH::
w(t) = t^3 + a_1 t^4 + (a_2 + a_1^2) t^5 + \cdots
to precision `O(t^{prec})` of Proposition IV.1.1 of
[Silverman AEC1]. This is the formal expansion of
`w = -1/y` about the formal parameter `t = -x/y` at
`\\infty`.
The result is cached, and a cached version is returned if
possible.
.. warning::
The resulting power series will have precision prec, but
its parent PowerSeriesRing will have default precision 20
(or whatever the default default is).
ALGORITHM: Uses Newton's method to solve the elliptic curve
equation at the origin. Complexity is roughly `O(M(n))`
where `n` is the precision and `M(n)` is the time
required to multiply polynomials of length `n` over the
coefficient ring of `E`.
AUTHOR:
- David Harvey (2006-09-09): modified to use Newton's
method instead of a recurrence formula.
EXAMPLES::
sage: e = EllipticCurve([0, 0, 1, -1, 0])
sage: e.formal_group().w(10)
t^3 + t^6 - t^7 + 2*t^9 + O(t^10)
Check that caching works::
sage: e = EllipticCurve([3, 2, -4, -2, 5])
sage: e.formal_group().w(20)
t^3 + 3*t^4 + 11*t^5 + 35*t^6 + 101*t^7 + 237*t^8 + 312*t^9 - 949*t^10 - 10389*t^11 - 57087*t^12 - 244092*t^13 - 865333*t^14 - 2455206*t^15 - 4366196*t^16 + 6136610*t^17 + 109938783*t^18 + 688672497*t^19 + O(t^20)
sage: e.formal_group().w(7)
t^3 + 3*t^4 + 11*t^5 + 35*t^6 + O(t^7)
sage: e.formal_group().w(35)
t^3 + 3*t^4 + 11*t^5 + 35*t^6 + 101*t^7 + 237*t^8 + 312*t^9 - 949*t^10 - 10389*t^11 - 57087*t^12 - 244092*t^13 - 865333*t^14 - 2455206*t^15 - 4366196*t^16 + 6136610*t^17 + 109938783*t^18 + 688672497*t^19 + 3219525807*t^20 + 12337076504*t^21 + 38106669615*t^22 + 79452618700*t^23 - 33430470002*t^24 - 1522228110356*t^25 - 10561222329021*t^26 - 52449326572178*t^27 - 211701726058446*t^28 - 693522772940043*t^29 - 1613471639599050*t^30 - 421817906421378*t^31 + 23651687753515182*t^32 + 181817896829144595*t^33 + 950887648021211163*t^34 + O(t^35)
"""
prec = max(prec, 0)
k = self.curve().base_ring()
try:
# Try cached version
w = self.__w
cached_prec = w.prec()
R = w.parent()
except AttributeError:
# No cached version available
R = rings.PowerSeriesRing(k, "t")
w = R([k(0), k(0), k(0), k(1)], 4)
cached_prec = 4
self.__w = w
if prec < cached_prec:
return R(w, prec)
# We use the following iteration, which doubles the precision
# at each step:
#
# z^3 - a_3 w^2 - a_4 z w^2 - 2 a_6 w^3
# w' = -----------------------------------------------------
# 1 - a_1 z - a_2 z^2 - 2 a_3 w - 2 a_4 z w - 3 a_6 w^2
a1, a2, a3, a4, a6 = self.curve().ainvs()
current_prec = cached_prec
w = w.truncate() # work with polynomials instead of power series
numerator_const = w.parent()([0, 0, 0, 1]) # z^3
denominator_const = w.parent()([1, -a1, -a2]) # 1 - a_1 z - a_2 z^2
last_prec = 0
for next_prec in misc.newton_method_sizes(prec):
if next_prec > current_prec:
if w.degree() - 1 > last_prec:
# Here it's best to throw away some precision to get us
# in sync with the sizes recommended by
# newton_method_sizes(). This is especially counter-
# intuitive when we throw away almost half of our
# cached data!
# todo: this might not actually be true, depending on
# the overhead of truncate(), which is currently very
# high e.g. for NTL based polynomials (but this is
# slated to be fixed...)
w = w.truncate(last_prec)
w_squared = w.square()
w_cubed = (w_squared * w).truncate(next_prec)
numerator = numerator_const \
- a3 * w_squared \
- a4 * w_squared.shift(1) \
- (2*a6) * w_cubed
denominator = denominator_const \
- (2*a3) * w \
- (2*a4) * w.shift(1) \
- (3*a6) * w_squared
# todo: this is quite inefficient, because it gets
# converted to a power series, then the power series
# inversion works with polynomials again, and then
# it gets converted *back* to a power series, and
# then we convert it to a polynomial again! That's four
# useless conversions!!!
inverse = ~R(denominator, prec=next_prec)
inverse = inverse.truncate(next_prec)
w = (numerator * inverse).truncate(next_prec)
last_prec = next_prec
# convert back to power series
w = R(w, prec)
self.__w = (prec, w)
return self.__w[1]
def mult_by_n(self, n, prec=10):
"""
The formal 'multiplication by n' endomorphism `[n]`.
INPUT:
- ``prec`` - integer (default 10)
OUTPUT: a power series with given precision
DETAILS: Return the formal power series
.. MATH::
[n](t) = n t + \cdots
to precision `O(t^{prec})` of Proposition 2.3 of [Silverman
AEC1].
.. warning::
The resulting power series will have precision prec, but
its parent PowerSeriesRing will have default precision 20
(or whatever the default default is).
AUTHORS:
- Nick Alexander: minor fixes, docstring
- David Harvey (2007-03): faster algorithm for char 0 field
case
- Hamish Ivey-Law (2009-06): double-and-add algorithm for
non char 0 field case.
- Tom Boothby (2009-06): slight improvement to double-and-add
- Francis Clarke (2012-08): adjustments and simplifications using group_law
code as modified to yield a two-variable power series.
EXAMPLES::
sage: e = EllipticCurve([1, 2, 3, 4, 6])
sage: e.formal_group().mult_by_n(0, 5)
O(t^5)
sage: e.formal_group().mult_by_n(1, 5)
t + O(t^5)
We verify an identity of low degree::
sage: none = e.formal_group().mult_by_n(-1, 5)
sage: two = e.formal_group().mult_by_n(2, 5)
sage: ntwo = e.formal_group().mult_by_n(-2, 5)
sage: ntwo - none(two)
O(t^5)
sage: ntwo - two(none)
O(t^5)
It's quite fast::
sage: E = EllipticCurve("37a"); F = E.formal_group()
sage: F.mult_by_n(100, 20)
100*t - 49999950*t^4 + 3999999960*t^5 + 14285614285800*t^7 - 2999989920000150*t^8 + 133333325333333400*t^9 - 3571378571674999800*t^10 + 1402585362624965454000*t^11 - 146666057066712847999500*t^12 + 5336978000014213190385000*t^13 - 519472790950932256570002000*t^14 + 93851927683683567270392002800*t^15 - 6673787211563812368630730325175*t^16 + 320129060335050875009191524993000*t^17 - 45670288869783478472872833214986000*t^18 + 5302464956134111125466184947310391600*t^19 + O(t^20)
TESTS::
sage: F = EllipticCurve(GF(17), [1, 1]).formal_group()
sage: F.mult_by_n(10, 50) # long time (13s on sage.math, 2011)
10*t + 5*t^5 + 7*t^7 + 13*t^9 + t^11 + 16*t^13 + 13*t^15 + 9*t^17 + 16*t^19 + 15*t^23 + 15*t^25 + 2*t^27 + 10*t^29 + 8*t^31 + 15*t^33 + 6*t^35 + 7*t^37 + 9*t^39 + 10*t^41 + 5*t^43 + 4*t^45 + 6*t^47 + 13*t^49 + O(t^50)
sage: F = EllipticCurve(GF(101), [1, 1]).formal_group()
sage: F.mult_by_n(100, 20)
100*t + O(t^20)
sage: P.<a1, a2, a3, a4, a6> = PolynomialRing(ZZ, 5)
sage: E = EllipticCurve(list(P.gens()))
sage: E.formal().mult_by_n(2,prec=5)
2*t - a1*t^2 - 2*a2*t^3 + (a1*a2 - 7*a3)*t^4 + O(t^5)
sage: E = EllipticCurve(QQ, [1,2,3,4,6])
sage: E.formal().mult_by_n(2,prec=5)
2*t - t^2 - 4*t^3 - 19*t^4 + O(t^5)
"""
if self.curve().base_ring().is_field() and self.curve().base_ring().characteristic() == 0 and n != 0:
# The following algorithm only works over a field of
# characteristic zero. I don't know whether something similar
# can be done over a general ring. It would be nice if it did,
# since it's much faster than using the formal group law.
# -- dmharvey
# Create a "formal point" on the original curve E.
# Our answer only needs prec-1 coefficients (since lowest term
# is t^1), and x(t) = t^(-2) + ... and y(t) = t^(-3) + ...,
# so we only need x(t) mod t^(prec-3) and y(t) mod t^(prec-4)
x = self.x(prec-3)
y = self.y(prec-4)
R = x.parent() # the Laurent series ring over the base ring
X = self.curve().change_ring(R)
P = X(x, y)
# and multiply it by n, using the group law on E
Q = n*P
# express it in terms of the formal parameter
return -Q[0] / Q[1]
# Now the general case, not necessarily over a field.
R = rings.PowerSeriesRing(self.curve().base_ring(), "t")
t = R.gen()
if n == 1:
return t.add_bigoh(prec)
if n == 0:
return R(0).add_bigoh(prec)
if n == -1:
return R(self.inverse(prec))
if n < 0:
return self.inverse(prec)(self.mult_by_n(-n, prec))
F = self.group_law(prec)
result = t
if n < 4:
for m in range(n - 1):
result = F(result, t)
return result
# Double and add is faster than the naive method when n >= 4.
g = t
if n & 1:
result = g
else:
result = 0
n = n >> 1
while n > 0:
g = F(g, g)
if n & 1:
result = F(result, g)
n = n >> 1
return result
def group_law(self, prec=10):
r"""
The formal group law.
INPUT:
- ``prec`` - integer (default 10)
OUTPUT: a power series with given precision in R[['t1','t2']], where
the curve is defined over R.
DETAILS: Return the formal power series
.. MATH::
F(t_1, t_2) = t_1 + t_2 - a_1 t_1 t_2 - \cdots
to precision `O(t1,t2)^{prec}` of page 115 of [Silverman AEC1].
The result is cached, and a cached version is returned if possible.
AUTHORS:
- Nick Alexander: minor fixes, docstring
- Francis Clarke (2012-08): modified to use two-variable power series ring
EXAMPLES::
sage: e = EllipticCurve([1, 2])
sage: e.formal_group().group_law(6)
t1 + t2 - 2*t1^4*t2 - 4*t1^3*t2^2 - 4*t1^2*t2^3 - 2*t1*t2^4 + O(t1, t2)^6
sage: e = EllipticCurve('14a1')
sage: ehat = e.formal()
sage: ehat.group_law(3)
t1 + t2 - t1*t2 + O(t1, t2)^3
sage: ehat.group_law(5)
t1 + t2 - t1*t2 - 2*t1^3*t2 - 3*t1^2*t2^2 - 2*t1*t2^3 + O(t1, t2)^5
sage: e = EllipticCurve(GF(7), [3, 4])
sage: ehat = e.formal()
sage: ehat.group_law(3)
t1 + t2 + O(t1, t2)^3
sage: F = ehat.group_law(7); F
t1 + t2 + t1^4*t2 + 2*t1^3*t2^2 + 2*t1^2*t2^3 + t1*t2^4 + O(t1, t2)^7
TESTS::
sage: R.<x,y,z> = GF(7)[[]]
sage: F(x, ehat.inverse()(x))
0 + O(x, y, z)^7
sage: F(x, y) == F(y, x)
True
sage: F(x, F(y, z)) == F(F(x, y), z)
True
Let's ensure caching with changed precision is working::
sage: e.formal_group().group_law(4)
t1 + t2 + O(t1, t2)^4
Test for :trac:`9646`::
sage: P.<a1, a2, a3, a4, a6> = PolynomialRing(ZZ, 5)
sage: E = EllipticCurve(list(P.gens()))
sage: F = E.formal().group_law(prec=5)
sage: t1, t2 = F.parent().gens()
sage: F(t1, 0)
t1 + O(t1, t2)^5
sage: F(0, t2)
t2 + O(t1, t2)^5
sage: F.coefficients()[t1*t2^2]
-a2
"""
prec = max(prec,0)
if prec <= 0:
raise ValueError("The precision must be positive.")
R = rings.PowerSeriesRing(self.curve().base_ring(), 2, 't1,t2')
t1, t2 = R.gens()
if prec == 1:
return R(0)
elif prec == 2:
return t1 + t2 - self.curve().a1()*t1*t2
try:
pr, F = self.__group_law
if prec <= pr:
return F.add_bigoh(prec)
except AttributeError:
pass
w = self.w(prec+1)
lam = sum([w[n]*sum(t2**m * t1**(n-m-1) for m in range(n)) for n in range(3, prec+1)])
lam = lam.add_bigoh(prec)
nu = w(t1) - lam*t1
a1, a2, a3, a4, a6 = self.curve().ainvs()
lam2 = lam*lam
lam3 = lam2*lam
# note that the following formula differs from the one in Silverman page 119.
# See trac ticket 9646 for the explanation and justification.
t3 = -t1 - t2 - \
(a1*lam + a3*lam2 + a2*nu + 2*a4*lam*nu + 3*a6*lam2*nu)/ \
(1 + a2*lam + a4*lam2 + a6*lam3)
inv = self.inverse(prec)
F = inv(t3).add_bigoh(prec)
self.__group_law = (prec, F)
return F
def group_law(self, prec=10):
r"""
The formal group law.
INPUT:
- ``prec`` - integer (default 10)
OUTPUT: a power series with given precision in
ZZ[[ ZZ[['t1']],'t2']]
DETAILS: Return the formal power series
.. math::
F(t_1, t_2) = t_1 + t_2 - a_1 t_1 t_2 - \cdots
to precision `O(t^{prec})` of page 115 of [Silverman AEC1].
The result is cached, and a cached version is returned if
possible.
.. warning::
The resulting power series will have precision prec, but
its parent PowerSeriesRing will have default precision 20
(or whatever the default default is).
AUTHORS:
- Nick Alexander: minor fixes, docstring
EXAMPLES::
sage: e = EllipticCurve([1, 2])
sage: e.formal_group().group_law(5)
t1 + O(t1^5) + (1 - 2*t1^4 + O(t1^5))*t2 + (-4*t1^3 + O(t1^5))*t2^2 + (-4*t1^2 - 30*t1^4 + O(t1^5))*t2^3 + (-2*t1 - 30*t1^3 + O(t1^5))*t2^4 + O(t2^5)
sage: e = EllipticCurve('14a1')
sage: ehat = e.formal()
sage: ehat.group_law(3)
t1 + O(t1^3) + (1 - t1 - 4*t1^2 + O(t1^3))*t2 + (-4*t1 + O(t1^3))*t2^2 + O(t2^3)
sage: ehat.group_law(5)
t1 + O(t1^5) + (1 - t1 - 2*t1^3 - 32*t1^4 + O(t1^5))*t2 + (-3*t1^2 - 59*t1^3 - 120*t1^4 + O(t1^5))*t2^2 + (-2*t1 - 59*t1^2 - 141*t1^3 - 347*t1^4 + O(t1^5))*t2^3 + (-32*t1 - 120*t1^2 - 347*t1^3 - 951*t1^4 + O(t1^5))*t2^4 + O(t2^5)
sage: e = EllipticCurve(GF(7),[3,4])
sage: ehat = e.formal()
sage: ehat.group_law(3)
t1 + O(t1^3) + (1 + O(t1^3))*t2 + O(t2^3)
sage: F = ehat.group_law(7); F
t1 + O(t1^7) + (1 + t1^4 + 2*t1^6 + O(t1^7))*t2 + (2*t1^3 + 6*t1^5 + O(t1^7))*t2^2 + (2*t1^2 + 3*t1^4 + 2*t1^6 + O(t1^7))*t2^3 + (t1 + 3*t1^3 + 4*t1^5 + O(t1^7))*t2^4 + (6*t1^2 + 4*t1^4 + O(t1^7))*t2^5 + (2*t1 + 2*t1^3 + O(t1^7))*t2^6 + O(t2^7)
TESTS::
sage: i = e.formal_group().inverse(7)
sage: Fx = F.base_extend(F.base_ring().base_extend(i.parent()))
sage: Fx (i.parent().gen()) (i)
O(t^7)
Let's ensure caching with changed precision is working::
sage: e.formal_group().group_law(4)
t1 + O(t1^4) + (1 + O(t1^4))*t2 + (2*t1^3 + O(t1^4))*t2^2 + (2*t1^2 + O(t1^4))*t2^3 + O(t2^4)
Test for trac ticket 9646::
sage: P.<a1, a2, a3, a4, a6> = PolynomialRing(ZZ, 5)
sage: E = EllipticCurve(list(P.gens()))
sage: F = E.formal().group_law(prec = 4)
sage: t2 = F.parent().gen()
sage: t1 = F.parent().base_ring().gen()
sage: F(t1, 0)
t1
sage: F(0, t2)
t2
sage: F[2][1]
-a2
"""
prec = max(prec, 0)
if prec <= 0:
raise ValueError, "The precision must be positive."
R1 = rings.PowerSeriesRing(self.curve().base_ring(), "t1")
R2 = rings.PowerSeriesRing(R1, "t2")
t1 = R1.gen().add_bigoh(prec)
t2 = R2.gen().add_bigoh(prec)
if prec == 1:
return R2(O(t2))
elif prec == 2:
return R2(t1 + t2 - self.curve().a1() * t1 * t2)
fix_prec = lambda F, final_prec: R2([c + O(t1**final_prec)
for c in F]) + O(t2**final_prec)
try:
pr, F = self.__group_law
except AttributeError:
pr = -1
if prec <= pr:
# we have to 'fix up' coefficient precisions
return fix_prec(F, prec)
w = self.w(prec + 1)
tsum = lambda n: sum([t2**m * t1**(n - m - 1) for m in range(n)])
lam = sum([tsum(n) * w[n] for n in range(3, prec + 1)])
w1 = R1(w, prec + 1)
nu = w1 - lam * t1 + O(t1**prec)
a1, a2, a3, a4, a6 = self.curve().ainvs()
lam2 = lam * lam
lam3 = lam2 * lam
# note that the following formula differs from the one in Silverman page 119. See trac ticket 9646 for the explanation and justification.
t3 = -t1 - t2 - \
(a1*lam + a3*lam2 + a2*nu + 2*a4*lam*nu + 3*a6*lam2*nu)/ \
(1 + a2*lam + a4*lam2 + a6*lam3)
inv = self.inverse(prec)
# we have to 'fix up' precision issues
F = fix_prec(inv(t3), prec)
self.__group_law = (prec, F)
return F
