def local_coordinates_at_nonweierstrass(self, P, prec=20, name='t'):
"""
For a non-Weierstrass point `P = (a,b)` on the hyperelliptic
curve `y^2 = f(x)`, return `(x(t), y(t))` such that `(y(t))^2 = f(x(t))`,
where `t = x - a` is the local parameter.
INPUT:
- ``P = (a, b)`` -- a non-Weierstrass point on self
- ``prec`` -- desired precision of the local coordinates
- ``name`` -- gen of the power series ring (default: ``t``)
OUTPUT:
`(x(t),y(t))` such that `y(t)^2 = f(x(t))` and `t = x - a`
is the local parameter at `P`
EXAMPLES::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: P = H(1,6)
sage: x,y = H.local_coordinates_at_nonweierstrass(P,prec=5)
sage: x
1 + t + O(t^5)
sage: y
6 + t - 7/2*t^2 - 1/2*t^3 - 25/48*t^4 + O(t^5)
sage: Q = H(-2,12)
sage: x,y = H.local_coordinates_at_nonweierstrass(Q,prec=5)
sage: x
-2 + t + O(t^5)
sage: y
12 - 19/2*t - 19/32*t^2 + 61/256*t^3 - 5965/24576*t^4 + O(t^5)
AUTHOR:
- Jennifer Balakrishnan (2007-12)
"""
d = P[1]
if d == 0:
raise TypeError(
"P = %s is a Weierstrass point. Use local_coordinates_at_weierstrass instead!"
% P)
pol = self.hyperelliptic_polynomials()[0]
L = PowerSeriesRing(self.base_ring(), name)
t = L.gen()
L.set_default_prec(prec)
K = PowerSeriesRing(L, 'x')
pol = K(pol)
x = K.gen()
b = P[0]
f = pol(t + b)
for i in range((RR(log(prec) / log(2))).ceil()):
d = (d + f / d) / 2
return t + b + O(t**(prec)), d + O(t**(prec))
def local_coordinates_at_nonweierstrass(self, P, prec=20, name='t'):
"""
For a non-Weierstrass point `P = (a,b)` on the hyperelliptic
curve `y^2 = f(x)`, return `(x(t), y(t))` such that `(y(t))^2 = f(x(t))`,
where `t = x - a` is the local parameter.
INPUT:
- ``P = (a, b)`` -- a non-Weierstrass point on self
- ``prec`` -- desired precision of the local coordinates
- ``name`` -- gen of the power series ring (default: ``t``)
OUTPUT:
`(x(t),y(t))` such that `y(t)^2 = f(x(t))` and `t = x - a`
is the local parameter at `P`
EXAMPLES::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: P = H(1,6)
sage: x,y = H.local_coordinates_at_nonweierstrass(P,prec=5)
sage: x
1 + t + O(t^5)
sage: y
6 + t - 7/2*t^2 - 1/2*t^3 - 25/48*t^4 + O(t^5)
sage: Q = H(-2,12)
sage: x,y = H.local_coordinates_at_nonweierstrass(Q,prec=5)
sage: x
-2 + t + O(t^5)
sage: y
12 - 19/2*t - 19/32*t^2 + 61/256*t^3 - 5965/24576*t^4 + O(t^5)
AUTHOR:
- Jennifer Balakrishnan (2007-12)
"""
d = P[1]
if d == 0:
raise TypeError("P = %s is a Weierstrass point. Use local_coordinates_at_weierstrass instead!"%P)
pol = self.hyperelliptic_polynomials()[0]
L = PowerSeriesRing(self.base_ring(), name)
t = L.gen()
L.set_default_prec(prec)
K = PowerSeriesRing(L, 'x')
pol = K(pol)
x = K.gen()
b = P[0]
f = pol(t+b)
for i in range((RR(log(prec)/log(2))).ceil()):
d = (d + f/d)/2
return t+b+O(t**(prec)), d + O(t**(prec))
def local_coordinates_at_weierstrass(self, P, prec=20, name='t'):
"""
For a finite Weierstrass point on the hyperelliptic
curve y^2 = f(x), returns (x(t), y(t)) such that
(y(t))^2 = f(x(t)), where t = y is the local parameter.
INPUT:
- P a finite Weierstrass point on self
- prec: desired precision of the local coordinates
- name: gen of the power series ring (default: 't')
OUTPUT:
(x(t),y(t)) such that y(t)^2 = f(x(t)) and t = y
is the local parameter at P
EXAMPLES:
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: A = H(4,0)
sage: x,y = H.local_coordinates_at_weierstrass(A,prec =5)
sage: x
4 + 1/360*t^2 - 191/23328000*t^4 + 7579/188956800000*t^6 + O(t^7)
sage: y
t + O(t^7)
sage: B = H(-5,0)
sage: x,y = H.local_coordinates_at_weierstrass(B,prec = 5)
sage: x
-5 + 1/1260*t^2 + 887/2000376000*t^4 + 643759/1587898468800000*t^6 + O(t^7)
sage: y
t + O(t^7)
AUTHOR:
- Jennifer Balakrishnan (2007-12)
"""
if P[1] != 0:
raise TypeError, "P = %s is not a finite Weierstrass point. Use local_coordinates_at_nonweierstrass instead!" % P
pol = self.hyperelliptic_polynomials()[0]
L = PowerSeriesRing(self.base_ring(), name)
t = L.gen()
L.set_default_prec(prec + 2)
K = PowerSeriesRing(L, 'x')
pol = K(pol)
x = K.gen()
b = P[0]
g = pol / (x - b)
c = b + 1 / g(b) * t**2
f = pol - t**2
fprime = f.derivative()
for i in range((RR(log(prec + 2) / log(2))).ceil()):
c = c - f(c) / fprime(c)
return c + O(t**(prec + 2)), t + O(t**(prec + 2))
def local_coordinates_at_weierstrass(self, P, prec=20, name="t"):
"""
For a finite Weierstrass point on the hyperelliptic
curve y^2 = f(x), returns (x(t), y(t)) such that
(y(t))^2 = f(x(t)), where t = y is the local parameter.
INPUT:
- P a finite Weierstrass point on self
- prec: desired precision of the local coordinates
- name: gen of the power series ring (default: 't')
OUTPUT:
(x(t),y(t)) such that y(t)^2 = f(x(t)) and t = y
is the local parameter at P
EXAMPLES:
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: A = H(4,0)
sage: x,y = H.local_coordinates_at_weierstrass(A,prec =5)
sage: x
4 + 1/360*t^2 - 191/23328000*t^4 + 7579/188956800000*t^6 + O(t^7)
sage: y
t + O(t^7)
sage: B = H(-5,0)
sage: x,y = H.local_coordinates_at_weierstrass(B,prec = 5)
sage: x
-5 + 1/1260*t^2 + 887/2000376000*t^4 + 643759/1587898468800000*t^6 + O(t^7)
sage: y
t + O(t^7)
AUTHOR:
- Jennifer Balakrishnan (2007-12)
"""
if P[1] != 0:
raise TypeError, "P = %s is not a finite Weierstrass point. Use local_coordinates_at_nonweierstrass instead!" % P
pol = self.hyperelliptic_polynomials()[0]
L = PowerSeriesRing(self.base_ring(), name)
t = L.gen()
L.set_default_prec(prec + 2)
K = PowerSeriesRing(L, "x")
pol = K(pol)
x = K.gen()
b = P[0]
g = pol / (x - b)
c = b + 1 / g(b) * t ** 2
f = pol - t ** 2
fprime = f.derivative()
for i in range((RR(log(prec + 2) / log(2))).ceil()):
c = c - f(c) / fprime(c)
return c + O(t ** (prec + 2)), t + O(t ** (prec + 2))
def double_integral_zero_infty(Phi, tau1, tau2):
p = Phi.parent().prime()
K = tau1.parent()
R = PolynomialRing(K, 'x')
x = R.gen()
R1 = PowerSeriesRing(K, 'r1')
r1 = R1.gen()
try:
R1.set_default_prec(Phi.precision_absolute())
except AttributeError:
R1.set_default_prec(Phi.precision_relative())
level = Phi._map._manin.level()
E0inf = [M2Z([0, -1, level, 0])]
E0Zp = [M2Z([p, a, 0, 1]) for a in range(p)]
predicted_evals = num_evals(tau1, tau2)
a, b, c, d = find_center(p, level, tau1, tau2).list()
h = M2Z([a, b, c, d])
E = [h * e0 for e0 in E0Zp + E0inf]
resadd = 0
resmul = 1
total_evals = 0
percentage = QQ(0)
ii = 0
f = (x - tau2) / (x - tau1)
while len(E) > 0:
ii += 1
increment = QQ((100 - percentage) / len(E))
verbose(
'remaining %s percent (and done %s of %s evaluations)' %
(RealField(10)(100 - percentage), total_evals, predicted_evals))
newE = []
for e in E:
a, b, c, d = e.list()
assert ZZ(c) % level == 0
try:
y0 = f((a * r1 + b) / (c * r1 + d))
val = y0(y0.parent().base_ring()(0))
if all([xx.valuation(p) > 0 for xx in (y0 / val - 1).list()]):
if total_evals % 100 == 0:
Phi._map._codomain.clear_cache()
pol = val.log(p_branch=0) + (
(y0.derivative() / y0).integral())
V = [0] * pol.valuation() + pol.shift(
-pol.valuation()).list()
try:
phimap = Phi._map(M2Z([b, d, a, c]))
except OverflowError:
print(a, b, c, d)
raise OverflowError, 'Matrix too large?'
# mu_e0 = ZZ(phimap.moment(0).rational_reconstruction())
mu_e0 = ZZ(Phi._liftee._map(M2Z([b, d, a, c])).moment(0))
mu_e = [mu_e0] + [
phimap.moment(o).lift() for o in range(1, len(V))
]
resadd += sum(starmap(mul, izip(V, mu_e)))
resmul *= val**mu_e0
percentage += increment
total_evals += 1
else:
newE.extend([e * e0 for e0 in E0Zp])
except ZeroDivisionError:
#raise RuntimeError,'Probably not enough working precision...'
newE.extend([e * e0 for e0 in E0Zp])
E = newE
verbose('total evaluations = %s' % total_evals)
val = resmul.valuation()
return p**val * K.teichmuller(p**(-val) * resmul) * resadd.exp()