def coleman_integral_S_to_Q(self, w, S, Q):
r"""
Computes the Coleman integral `\int_S^Q w`
**one should be able to feed `S,Q` into coleman_integral,
but currently that segfaults**
INPUT:
- w: a differential
- S: a point with coordinates in an extension of `\QQ_p`
- Q: a non-Weierstrass point defined over `\QQ_p`
OUTPUT:
the Coleman integral `\int_S^Q w`
EXAMPLES::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^3-10*x+9)
sage: K = Qp(5,6)
sage: HK = H.change_ring(K)
sage: J.<a> = K.extension(x^20-5)
sage: HJ = H.change_ring(J)
sage: x,y = HK.monsky_washnitzer_gens()
sage: P = HK(1,0)
sage: Q = HK(0,3)
sage: S = HK.get_boundary_point(HJ,P)
sage: P_to_S = HK.coleman_integral_P_to_S(y.diff(),P,S)
sage: S_to_Q = HJ.coleman_integral_S_to_Q(y.diff(),S,Q)
sage: P_to_S + S_to_Q
3 + O(a^119)
sage: HK.coleman_integral(y.diff(),P,Q)
3 + O(5^6)
AUTHOR:
- Jennifer Balakrishnan
"""
import sage.schemes.hyperelliptic_curves.monsky_washnitzer as monsky_washnitzer
K = self.base_ring()
R = monsky_washnitzer.SpecialHyperellipticQuotientRing(self, K)
MW = monsky_washnitzer.MonskyWashnitzerDifferentialRing(R)
w = MW(w)
f, vec = w.reduce_fast()
g = self.genus()
const = f(Q[0], Q[1]) - f(S[0], S[1])
if vec == vector(2 * g * [0]):
return const
else:
basis_values = self.S_to_Q(S, Q)
dim = len(basis_values)
dot = sum([vec[i] * basis_values[i] for i in range(dim)])
return const + dot
def invariant_differential(self):
"""
Returns $dx/2y$, as an element of the Monsky-Washnitzer cohomology
of self
EXAMPLES::
sage: R.<x> = QQ['x']
sage: C = HyperellipticCurve(x^5 - 4*x + 4)
sage: C.invariant_differential()
1 dx/2y
"""
import sage.schemes.hyperelliptic_curves.monsky_washnitzer as m_w
S = m_w.SpecialHyperellipticQuotientRing(self)
MW = m_w.MonskyWashnitzerDifferentialRing(S)
return MW.invariant_differential()
def coleman_integral(self, w, P, Q, algorithm='None'):
r"""
Returns the Coleman integral `\int_P^Q w`
INPUT:
- w differential (if one of P,Q is Weierstrass, w must be odd)
- P point on self
- Q point on self
- algorithm (optional) = None (uses Frobenius) or teichmuller (uses Teichmuller points)
OUTPUT:
the Coleman integral `\int_P^Q w`
EXAMPLES:
Example of Leprevost from Kiran Kedlaya
The first two should be zero as `(P-Q) = 30(P-Q)` in the Jacobian
and `dx/2y` and `x dx/2y` are holomorphic. ::
sage: K = pAdicField(11, 6)
sage: x = polygen(K)
sage: C = HyperellipticCurve(x^5 + 33/16*x^4 + 3/4*x^3 + 3/8*x^2 - 1/4*x + 1/16)
sage: P = C(-1, 1); P1 = C(-1, -1)
sage: Q = C(0, 1/4); Q1 = C(0, -1/4)
sage: x, y = C.monsky_washnitzer_gens()
sage: w = C.invariant_differential()
sage: w.coleman_integral(P, Q)
O(11^6)
sage: C.coleman_integral(x*w, P, Q)
O(11^6)
sage: C.coleman_integral(x^2*w, P, Q)
7*11 + 6*11^2 + 3*11^3 + 11^4 + 5*11^5 + O(11^6)
::
sage: p = 71; m = 4
sage: K = pAdicField(p, m)
sage: x = polygen(K)
sage: C = HyperellipticCurve(x^5 + 33/16*x^4 + 3/4*x^3 + 3/8*x^2 - 1/4*x + 1/16)
sage: P = C(-1, 1); P1 = C(-1, -1)
sage: Q = C(0, 1/4); Q1 = C(0, -1/4)
sage: x, y = C.monsky_washnitzer_gens()
sage: w = C.invariant_differential()
sage: w.integrate(P, Q), (x*w).integrate(P, Q)
(O(71^4), O(71^4))
sage: R, R1 = C.lift_x(4, all=True)
sage: w.integrate(P, R)
21*71 + 67*71^2 + 27*71^3 + O(71^4)
sage: w.integrate(P, R) + w.integrate(P1, R1)
O(71^4)
A simple example, integrating dx::
sage: R.<x> = QQ['x']
sage: E= HyperellipticCurve(x^3-4*x+4)
sage: K = Qp(5,10)
sage: EK = E.change_ring(K)
sage: P = EK(2, 2)
sage: Q = EK.teichmuller(P)
sage: x, y = EK.monsky_washnitzer_gens()
sage: EK.coleman_integral(x.diff(), P, Q)
5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9 + O(5^10)
sage: Q[0] - P[0]
5 + 2*5^2 + 5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9 + O(5^10)
Yet another example::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x*(x-1)*(x+9))
sage: K = Qp(7,10)
sage: HK = H.change_ring(K)
sage: import sage.schemes.hyperelliptic_curves.monsky_washnitzer as mw
sage: M_frob, forms = mw.matrix_of_frobenius_hyperelliptic(HK)
sage: w = HK.invariant_differential()
sage: x,y = HK.monsky_washnitzer_gens()
sage: f = forms[0]
sage: S = HK(9,36)
sage: Q = HK.teichmuller(S)
sage: P = HK(-1,4)
sage: b = x*w*w._coeff.parent()(f)
sage: HK.coleman_integral(b,P,Q)
7 + 7^2 + 4*7^3 + 5*7^4 + 3*7^5 + 7^6 + 5*7^7 + 3*7^8 + 4*7^9 + 4*7^10 + O(7^11)
::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^3+1)
sage: K = Qp(5,8)
sage: HK = H.change_ring(K)
sage: w = HK.invariant_differential()
sage: P = HK(0,1)
sage: Q = HK.lift_x(5)
sage: x,y = HK.monsky_washnitzer_gens()
sage: (2*y*w).coleman_integral(P,Q)
5 + O(5^9)
sage: xloc,yloc,zloc = HK.local_analytic_interpolation(P,Q)
sage: I2 = (xloc.derivative()/(2*yloc)).integral()
sage: I2.polynomial()(1) - I2(0)
3*5 + 2*5^2 + 2*5^3 + 5^4 + 4*5^6 + 5^7 + O(5^9)
sage: HK.coleman_integral(w,P,Q)
3*5 + 2*5^2 + 2*5^3 + 5^4 + 4*5^6 + 5^7 + O(5^9)
Integrals involving Weierstrass points::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^3-10*x+9)
sage: K = Qp(5,8)
sage: HK = H.change_ring(K)
sage: S = HK(1,0)
sage: P = HK(0,3)
sage: negP = HK(0,-3)
sage: T = HK(0,1,0)
sage: w = HK.invariant_differential()
sage: x,y = HK.monsky_washnitzer_gens()
sage: HK.coleman_integral(w*x^3,S,T)
0
sage: HK.coleman_integral(w*x^3,T,S)
0
sage: HK.coleman_integral(w,S,P)
2*5^2 + 5^4 + 5^5 + 3*5^6 + 3*5^7 + 2*5^8 + O(5^9)
sage: HK.coleman_integral(w,T,P)
2*5^2 + 5^4 + 5^5 + 3*5^6 + 3*5^7 + 2*5^8 + O(5^9)
sage: HK.coleman_integral(w*x^3,T,P)
5^2 + 2*5^3 + 3*5^6 + 3*5^7 + O(5^8)
sage: HK.coleman_integral(w*x^3,S,P)
5^2 + 2*5^3 + 3*5^6 + 3*5^7 + O(5^8)
sage: HK.coleman_integral(w, P, negP, algorithm='teichmuller')
5^2 + 4*5^3 + 2*5^4 + 2*5^5 + 3*5^6 + 2*5^7 + 4*5^8 + O(5^9)
sage: HK.coleman_integral(w, P, negP)
5^2 + 4*5^3 + 2*5^4 + 2*5^5 + 3*5^6 + 2*5^7 + 4*5^8 + O(5^9)
AUTHORS:
- Robert Bradshaw (2007-03)
- Kiran Kedlaya (2008-05)
- Jennifer Balakrishnan (2010-02)
"""
# TODO: implement Jacobians and show the relationship directly
import sage.schemes.hyperelliptic_curves.monsky_washnitzer as monsky_washnitzer
K = self.base_ring()
prec = K.precision_cap()
S = monsky_washnitzer.SpecialHyperellipticQuotientRing(self, K)
MW = monsky_washnitzer.MonskyWashnitzerDifferentialRing(S)
w = MW(w)
f, vec = w.reduce_fast()
basis_values = self.coleman_integrals_on_basis(P, Q, algorithm)
dim = len(basis_values)
x, y = self.local_coordinates_at_infinity(2 * prec)
if self.is_weierstrass(P):
if self.is_weierstrass(Q):
return 0
elif f == 0:
return sum([vec[i] * basis_values[i] for i in range(dim)])
elif w._coeff(x, -y) * x.derivative() / (-2 * y) + w._coeff(
x, y) * x.derivative() / (2 * y) == 0:
return self.coleman_integral(w, self(Q[0], -Q[1]),
self(Q[0], Q[1]), algorithm) / 2
else:
raise ValueError(
"The differential is not odd: use coleman_integral_from_weierstrass_via_boundary"
)
elif self.is_weierstrass(Q):
if f == 0:
return sum([vec[i] * basis_values[i] for i in range(dim)])
elif w._coeff(x, -y) * x.derivative() / (-2 * y) + w._coeff(
x, y) * x.derivative() / (2 * y) == 0:
return -self.coleman_integral(w, self(P[0], -P[1]),
self(P[0], P[1]), algorithm) / 2
else:
raise ValueError(
"The differential is not odd: use coleman_integral_from_weierstrass_via_boundary"
)
else:
return f(Q[0], Q[1]) - f(P[0], P[1]) + sum(
[vec[i] * basis_values[i]
for i in range(dim)]) # this is just a dot product...
def monsky_washnitzer_gens(self):
import sage.schemes.hyperelliptic_curves.monsky_washnitzer as monsky_washnitzer
S = monsky_washnitzer.SpecialHyperellipticQuotientRing(self)
return S.gens()
