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 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 coleman_integrals_on_basis(self, P, Q, algorithm=None):
r"""
Computes the Coleman integrals $\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}$
INPUT:
- P point on self
- Q point on self
- algorithm (optional) = None (uses Frobenius) or teichmuller (uses Teichmuller points)
OUTPUT:
the Coleman integrals $\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}$
EXAMPLES::
sage: K = pAdicField(11, 5)
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.lift_x(2)
sage: Q = C.lift_x(3)
sage: C.coleman_integrals_on_basis(P, Q)
(10*11 + 6*11^3 + 2*11^4 + O(11^5), 11 + 9*11^2 + 7*11^3 + 9*11^4 + O(11^5), 3 + 10*11 + 5*11^2 + 9*11^3 + 4*11^4 + O(11^5), 3 + 11 + 5*11^2 + 4*11^4 + O(11^5))
sage: C.coleman_integrals_on_basis(P, Q, algorithm='teichmuller')
(10*11 + 6*11^3 + 2*11^4 + O(11^5), 11 + 9*11^2 + 7*11^3 + 9*11^4 + O(11^5), 3 + 10*11 + 5*11^2 + 9*11^3 + 4*11^4 + O(11^5), 3 + 11 + 5*11^2 + 4*11^4 + O(11^5))
::
sage: K = pAdicField(11,5)
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.lift_x(11^(-2))
sage: Q = C.lift_x(3*11^(-2))
sage: C.coleman_integrals_on_basis(P, Q)
(3*11^3 + 7*11^4 + 4*11^5 + 7*11^6 + 5*11^7 + O(11^8), 3*11 + 10*11^2 + 8*11^3 + 9*11^4 + 7*11^5 + O(11^6), 4*11^-1 + 2 + 6*11 + 6*11^2 + 7*11^3 + O(11^4), 11^-3 + 6*11^-2 + 2*11^-1 + 2 + O(11^2))
::
sage: R = C(0,1/4)
sage: a = C.coleman_integrals_on_basis(P,R) # long time (7s on sage.math, 2011)
sage: b = C.coleman_integrals_on_basis(R,Q) # long time (9s on sage.math, 2011)
sage: c = C.coleman_integrals_on_basis(P,Q) # long time
sage: a+b == c # long time
True
::
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: T = HK(0,1,0)
sage: Q = HK.lift_x(5^-2)
sage: R = HK.lift_x(4*5^-2)
sage: HK.coleman_integrals_on_basis(S,P)
(2*5^2 + 5^4 + 5^5 + 3*5^6 + 3*5^7 + 2*5^8 + O(5^9), 5 + 2*5^2 + 4*5^3 + 2*5^4 + 3*5^6 + 4*5^7 + 2*5^8 + O(5^9))
sage: HK.coleman_integrals_on_basis(T,P)
(2*5^2 + 5^4 + 5^5 + 3*5^6 + 3*5^7 + 2*5^8 + O(5^9), 5 + 2*5^2 + 4*5^3 + 2*5^4 + 3*5^6 + 4*5^7 + 2*5^8 + O(5^9))
sage: HK.coleman_integrals_on_basis(P,S) == -HK.coleman_integrals_on_basis(S,P)
True
sage: HK.coleman_integrals_on_basis(S,Q)
(4*5 + 4*5^2 + 4*5^3 + O(5^4), 5^-1 + O(5^3))
sage: HK.coleman_integrals_on_basis(Q,R)
(4*5 + 2*5^2 + 2*5^3 + 2*5^4 + 5^5 + 5^6 + 5^7 + 3*5^8 + O(5^9), 2*5^-1 + 4 + 4*5 + 4*5^2 + 4*5^3 + 2*5^4 + 3*5^5 + 2*5^6 + O(5^7))
sage: HK.coleman_integrals_on_basis(S,R) == HK.coleman_integrals_on_basis(S,Q) + HK.coleman_integrals_on_basis(Q,R)
True
sage: HK.coleman_integrals_on_basis(T,T)
(0, 0)
sage: HK.coleman_integrals_on_basis(S,T)
(0, 0)
AUTHORS:
- Robert Bradshaw (2007-03): non-Weierstrass points
- Jennifer Balakrishnan and Robert Bradshaw (2010-02): Weierstrass points
"""
import sage.schemes.elliptic_curves.monsky_washnitzer as monsky_washnitzer
from sage.misc.profiler import Profiler
prof = Profiler()
prof("setup")
K = self.base_ring()
p = K.prime()
prec = K.precision_cap()
g = self.genus()
dim = 2*g
V = VectorSpace(K, dim)
#if P or Q is Weierstrass, use the Frobenius algorithm
if self.is_weierstrass(P):
if self.is_weierstrass(Q):
return V(0)
else:
PP = None
QQ = Q
TP = None
TQ = self.frobenius(Q)
elif self.is_weierstrass(Q):
PP = P
QQ = None
TQ = None
TP = self.frobenius(P)
elif self.is_same_disc(P,Q):
return self.tiny_integrals_on_basis(P,Q)
elif algorithm == 'teichmuller':
prof("teichmuller")
PP = TP = self.teichmuller(P)
QQ = TQ = self.teichmuller(Q)
evalP, evalQ = TP, TQ
else:
prof("frobPQ")
TP = self.frobenius(P)
TQ = self.frobenius(Q)
PP, QQ = P, Q
prof("tiny integrals")
if TP == None:
P_to_TP = V(0)
else:
if TP!=None:
TPv = (TP[0]**g/TP[1]).valuation()
xTPv = TP[0].valuation()
else:
xTPv = TPv = +Infinity
if TQ!=None:
TQv = (TQ[0]**g/TQ[1]).valuation()
xTQv = TQ[0].valuation()
else:
xTQv = TQv = +Infinity
offset = (2*g-1)*max(TPv, TQv)
if offset == +Infinity:
offset = (2*g-1)*min(TPv,TQv)
if (offset > prec and (xTPv <0 or xTQv <0) and (self.residue_disc(P) == self.change_ring(GF(p))(0,1,0) or self.residue_disc(Q) == self.change_ring(GF(p))(0,1,0))):
newprec = offset + prec
K = pAdicField(p,newprec)
A = PolynomialRing(RationalField(),'x')
f = A(self.hyperelliptic_polynomials()[0])
from sage.schemes.hyperelliptic_curves.constructor import HyperellipticCurve
self = HyperellipticCurve(f).change_ring(K)
xP = P[0]
xPv = xP.valuation()
xPnew = K(sum(xP.list()[i]*p**(xPv + i) for i in range(len(xP.list()))))
PP = P = self.lift_x(xPnew)
TP = self.frobenius(P)
xQ = Q[0]
xQv = xQ.valuation()
xQnew = K(sum(xQ.list()[i]*p**(xQv + i) for i in range(len(xQ.list()))))
QQ = Q = self.lift_x(xQnew)
TQ = self.frobenius(Q)
V = VectorSpace(K,dim)
P_to_TP = V(self.tiny_integrals_on_basis(P, TP))
if TQ == None:
TQ_to_Q = V(0)
else:
TQ_to_Q = V(self.tiny_integrals_on_basis(TQ, Q))
prof("mw calc")
try:
M_frob, forms = self._frob_calc
except AttributeError:
M_frob, forms = self._frob_calc = monsky_washnitzer.matrix_of_frobenius_hyperelliptic(self)
prof("eval f")
R = forms[0].base_ring()
try:
prof("eval f %s"%R)
if PP is None:
L = [-f(R(QQ[0]), R(QQ[1])) for f in forms] ##changed
elif QQ is None:
L = [f(R(PP[0]), R(PP[1])) for f in forms]
else:
L = [f(R(PP[0]), R(PP[1])) - f(R(QQ[0]), R(QQ[1])) for f in forms]
except ValueError:
prof("changing rings")
forms = [f.change_ring(self.base_ring()) for f in forms]
prof("eval f %s"%self.base_ring())
if PP is None:
L = [-f(QQ[0], QQ[1]) for f in forms] ##changed
elif QQ is None:
L = [f(PP[0], PP[1]) for f in forms]
else:
L = [f(PP[0], PP[1]) - f(QQ[0], QQ[1]) for f in forms]
b = V(L)
if PP is None:
b -= TQ_to_Q
elif QQ is None:
b -= P_to_TP
elif algorithm != 'teichmuller':
b -= P_to_TP + TQ_to_Q
prof("lin alg")
M_sys = matrix(K, M_frob).transpose() - 1
TP_to_TQ = M_sys**(-1) * b
prof("done")
# print prof
if algorithm == 'teichmuller':
return P_to_TP + TP_to_TQ + TQ_to_Q
else:
return TP_to_TQ
def S_to_Q(self,S,Q):
r"""
Given $S$ a point on self over an extension field, computes the
Coleman integrals $\{\int_S^Q x^i dx/2y \}_{i=0}^{2g-1}$
**one should be able to feed $S,Q$ into coleman_integral,
but currently that segfaults**
INPUT:
- S: a point with coordinates in an extension of $\Q_p$ (with unif. a)
- Q: a non-Weierstrass point defined over $\Q_p$
OUTPUT:
the Coleman integrals $\{\int_S^Q x^i dx/2y \}_{i=0}^{2g-1}$ in terms of $a$
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: w = HK.invariant_differential()
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.P_to_S(P,S)
sage: S_to_Q = HJ.S_to_Q(S,Q)
sage: P_to_S + S_to_Q
(2*a^40 + a^80 + a^100 + O(a^105), a^20 + 2*a^40 + 4*a^60 + 2*a^80 + O(a^103))
sage: HK.coleman_integrals_on_basis(P,Q)
(2*5^2 + 5^4 + 5^5 + 3*5^6 + O(5^7), 5 + 2*5^2 + 4*5^3 + 2*5^4 + 5^6 + O(5^7))
AUTHOR:
- Jennifer Balakrishnan
"""
FS = self.frobenius(S)
FS = (FS[0],FS[1])
FQ = self.frobenius(Q)
import sage.schemes.elliptic_curves.monsky_washnitzer as monsky_washnitzer
try:
M_frob, forms = self._frob_calc
except AttributeError:
M_frob, forms = self._frob_calc = monsky_washnitzer.matrix_of_frobenius_hyperelliptic(self)
try:
HJ = self._curve_over_ram_extn
K = HJ.base_ring()
except AttributeError:
HJ = S.scheme()
K = self.base_ring()
g = self.genus()
prec2 = K.precision_cap()
p = K.prime()
dim = 2*g
V = VectorSpace(K,dim)
if S == FS:
S_to_FS = V(dim*[0])
else:
P = self(ZZ(FS[0][0]),ZZ(FS[1][0]))
x,y = self.local_coord(P,prec2)
integrals = [(x**i*x.derivative()/(2*y)).integral() for i in range(dim)]
S_to_FS = vector([I.polynomial()(FS[1]) - I.polynomial()(S[1]) for I in integrals])
if HJ(Q[0],Q[1]) == HJ(FQ):
FQ_to_Q = V(dim*[0])
else:
FQ_to_Q = V(self.tiny_integrals_on_basis(FQ, Q))
try:
L = [f(K(S[0]), K(S[1])) - f(K(Q[0]), K(Q[1])) for f in forms]
except ValueError:
forms = [f.change_ring(K) for f in forms]
L = [f(S[0], S[1]) - f(Q[0], Q[1]) for f in forms]
b = V(L)
M_sys = matrix(K, M_frob).transpose() - 1
B = (~M_sys)
v = [B.list()[i].valuation() for i in range(len(B.list()))]
vv= min(v)
B = (p**(-vv)*B).change_ring(K)
B = p**(vv)*B
return B*(b-S_to_FS-FQ_to_Q)
def coleman_integrals_on_basis(self, P, Q, algorithm=None):
r"""
Computes the Coleman integrals $\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}$
INPUT:
- P point on self
- Q point on self
- algorithm (optional) = None (uses Frobenius) or teichmuller (uses Teichmuller points)
OUTPUT:
the Coleman integrals $\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}$
EXAMPLES::
sage: K = pAdicField(11, 5)
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.lift_x(2)
sage: Q = C.lift_x(3)
sage: C.coleman_integrals_on_basis(P, Q)
(10*11 + 6*11^3 + 2*11^4 + O(11^5), 11 + 9*11^2 + 7*11^3 + 9*11^4 + O(11^5), 3 + 10*11 + 5*11^2 + 9*11^3 + 4*11^4 + O(11^5), 3 + 11 + 5*11^2 + 4*11^4 + O(11^5))
sage: C.coleman_integrals_on_basis(P, Q, algorithm='teichmuller')
(10*11 + 6*11^3 + 2*11^4 + O(11^5), 11 + 9*11^2 + 7*11^3 + 9*11^4 + O(11^5), 3 + 10*11 + 5*11^2 + 9*11^3 + 4*11^4 + O(11^5), 3 + 11 + 5*11^2 + 4*11^4 + O(11^5))
::
sage: K = pAdicField(11,5)
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.lift_x(11^(-2))
sage: Q = C.lift_x(3*11^(-2))
sage: C.coleman_integrals_on_basis(P, Q)
(3*11^3 + 7*11^4 + 4*11^5 + 7*11^6 + 5*11^7 + O(11^8), 3*11 + 10*11^2 + 8*11^3 + 9*11^4 + 7*11^5 + O(11^6), 4*11^-1 + 2 + 6*11 + 6*11^2 + 7*11^3 + O(11^4), 11^-3 + 6*11^-2 + 2*11^-1 + 2 + O(11^2))
::
sage: R = C(0,1/4)
sage: a = C.coleman_integrals_on_basis(P,R) # long time (7s on sage.math, 2011)
sage: b = C.coleman_integrals_on_basis(R,Q) # long time (9s on sage.math, 2011)
sage: c = C.coleman_integrals_on_basis(P,Q) # long time
sage: a+b == c # long time
True
::
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: T = HK(0,1,0)
sage: Q = HK.lift_x(5^-2)
sage: R = HK.lift_x(4*5^-2)
sage: HK.coleman_integrals_on_basis(S,P)
(2*5^2 + 5^4 + 5^5 + 3*5^6 + 3*5^7 + 2*5^8 + O(5^9), 5 + 2*5^2 + 4*5^3 + 2*5^4 + 3*5^6 + 4*5^7 + 2*5^8 + O(5^9))
sage: HK.coleman_integrals_on_basis(T,P)
(2*5^2 + 5^4 + 5^5 + 3*5^6 + 3*5^7 + 2*5^8 + O(5^9), 5 + 2*5^2 + 4*5^3 + 2*5^4 + 3*5^6 + 4*5^7 + 2*5^8 + O(5^9))
sage: HK.coleman_integrals_on_basis(P,S) == -HK.coleman_integrals_on_basis(S,P)
True
sage: HK.coleman_integrals_on_basis(S,Q)
(4*5 + 4*5^2 + 4*5^3 + O(5^4), 5^-1 + O(5^3))
sage: HK.coleman_integrals_on_basis(Q,R)
(4*5 + 2*5^2 + 2*5^3 + 2*5^4 + 5^5 + 5^6 + 5^7 + 3*5^8 + O(5^9), 2*5^-1 + 4 + 4*5 + 4*5^2 + 4*5^3 + 2*5^4 + 3*5^5 + 2*5^6 + O(5^7))
sage: HK.coleman_integrals_on_basis(S,R) == HK.coleman_integrals_on_basis(S,Q) + HK.coleman_integrals_on_basis(Q,R)
True
sage: HK.coleman_integrals_on_basis(T,T)
(0, 0)
sage: HK.coleman_integrals_on_basis(S,T)
(0, 0)
AUTHORS:
- Robert Bradshaw (2007-03): non-Weierstrass points
- Jennifer Balakrishnan and Robert Bradshaw (2010-02): Weierstrass points
"""
import sage.schemes.elliptic_curves.monsky_washnitzer as monsky_washnitzer
from sage.misc.profiler import Profiler
prof = Profiler()
prof("setup")
K = self.base_ring()
p = K.prime()
prec = K.precision_cap()
g = self.genus()
dim = 2*g
V = VectorSpace(K, dim)
#if P or Q is Weierstrass, use the Frobenius algorithm
if self.is_weierstrass(P):
if self.is_weierstrass(Q):
return V(0)
else:
PP = None
QQ = Q
TP = None
TQ = self.frobenius(Q)
elif self.is_weierstrass(Q):
PP = P
QQ = None
TQ = None
TP = self.frobenius(P)
elif self.is_same_disc(P,Q):
return self.tiny_integrals_on_basis(P,Q)
elif algorithm == 'teichmuller':
prof("teichmuller")
PP = TP = self.teichmuller(P)
QQ = TQ = self.teichmuller(Q)
evalP, evalQ = TP, TQ
else:
prof("frobPQ")
TP = self.frobenius(P)
TQ = self.frobenius(Q)
PP, QQ = P, Q
prof("tiny integrals")
if TP == None:
P_to_TP = V(0)
else:
if TP!=None:
TPv = (TP[0]**g/TP[1]).valuation()
xTPv = TP[0].valuation()
else:
xTPv = TPv = +Infinity
if TQ!=None:
TQv = (TQ[0]**g/TQ[1]).valuation()
xTQv = TQ[0].valuation()
else:
xTQv = TQv = +Infinity
offset = (2*g-1)*max(TPv, TQv)
if offset == +Infinity:
offset = (2*g-1)*min(TPv,TQv)
if (offset > prec and (xTPv <0 or xTQv <0) and (self.residue_disc(P) == self.change_ring(GF(p))(0,1,0) or self.residue_disc(Q) == self.change_ring(GF(p))(0,1,0))):
newprec = offset + prec
K = pAdicField(p,newprec)
A = PolynomialRing(RationalField(),'x')
f = A(self.hyperelliptic_polynomials()[0])
from sage.schemes.hyperelliptic_curves.constructor import HyperellipticCurve
self = HyperellipticCurve(f).change_ring(K)
xP = P[0]
xPv = xP.valuation()
xPnew = K(sum(xP.list()[i]*p**(xPv + i) for i in range(len(xP.list()))))
PP = P = self.lift_x(xPnew)
TP = self.frobenius(P)
xQ = Q[0]
xQv = xQ.valuation()
xQnew = K(sum(xQ.list()[i]*p**(xQv + i) for i in range(len(xQ.list()))))
QQ = Q = self.lift_x(xQnew)
TQ = self.frobenius(Q)
V = VectorSpace(K,dim)
P_to_TP = V(self.tiny_integrals_on_basis(P, TP))
if TQ == None:
TQ_to_Q = V(0)
else:
TQ_to_Q = V(self.tiny_integrals_on_basis(TQ, Q))
prof("mw calc")
try:
M_frob, forms = self._frob_calc
except AttributeError:
M_frob, forms = self._frob_calc = monsky_washnitzer.matrix_of_frobenius_hyperelliptic(self)
prof("eval f")
R = forms[0].base_ring()
try:
prof("eval f %s"%R)
if PP is None:
L = [-f(R(QQ[0]), R(QQ[1])) for f in forms] ##changed
elif QQ is None:
L = [f(R(PP[0]), R(PP[1])) for f in forms]
else:
L = [f(R(PP[0]), R(PP[1])) - f(R(QQ[0]), R(QQ[1])) for f in forms]
except ValueError:
prof("changing rings")
forms = [f.change_ring(self.base_ring()) for f in forms]
prof("eval f %s"%self.base_ring())
if PP is None:
L = [-f(QQ[0], QQ[1]) for f in forms] ##changed
elif QQ is None:
L = [f(PP[0], PP[1]) for f in forms]
else:
L = [f(PP[0], PP[1]) - f(QQ[0], QQ[1]) for f in forms]
b = V(L)
if PP is None:
b -= TQ_to_Q
elif QQ is None:
b -= P_to_TP
elif algorithm != 'teichmuller':
b -= P_to_TP + TQ_to_Q
prof("lin alg")
M_sys = matrix(K, M_frob).transpose() - 1
TP_to_TQ = M_sys**(-1) * b
prof("done")
# print prof
if algorithm == 'teichmuller':
return P_to_TP + TP_to_TQ + TQ_to_Q
else:
return TP_to_TQ
def S_to_Q(self,S,Q):
r"""
Given $S$ a point on self over an extension field, computes the
Coleman integrals $\{\int_S^Q x^i dx/2y \}_{i=0}^{2g-1}$
**one should be able to feed $S,Q$ into coleman_integral,
but currently that segfaults**
INPUT:
- S: a point with coordinates in an extension of $\Q_p$ (with unif. a)
- Q: a non-Weierstrass point defined over $\Q_p$
OUTPUT:
the Coleman integrals $\{\int_S^Q x^i dx/2y \}_{i=0}^{2g-1}$ in terms of $a$
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: w = HK.invariant_differential()
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.P_to_S(P,S)
sage: S_to_Q = HJ.S_to_Q(S,Q)
sage: P_to_S + S_to_Q
(2*a^40 + a^80 + a^100 + O(a^105), a^20 + 2*a^40 + 4*a^60 + 2*a^80 + O(a^103))
sage: HK.coleman_integrals_on_basis(P,Q)
(2*5^2 + 5^4 + 5^5 + 3*5^6 + O(5^7), 5 + 2*5^2 + 4*5^3 + 2*5^4 + 5^6 + O(5^7))
AUTHOR:
- Jennifer Balakrishnan
"""
FS = self.frobenius(S)
FS = (FS[0],FS[1])
FQ = self.frobenius(Q)
import sage.schemes.elliptic_curves.monsky_washnitzer as monsky_washnitzer
try:
M_frob, forms = self._frob_calc
except AttributeError:
M_frob, forms = self._frob_calc = monsky_washnitzer.matrix_of_frobenius_hyperelliptic(self)
try:
HJ = self._curve_over_ram_extn
K = HJ.base_ring()
except AttributeError:
HJ = S.scheme()
K = self.base_ring()
g = self.genus()
prec2 = K.precision_cap()
p = K.prime()
dim = 2*g
V = VectorSpace(K,dim)
if S == FS:
S_to_FS = V(dim*[0])
else:
P = self(ZZ(FS[0][0]),ZZ(FS[1][0]))
x,y = self.local_coord(P,prec2)
integrals = [(x**i*x.derivative()/(2*y)).integral() for i in range(dim)]
S_to_FS = vector([I.polynomial()(FS[1]) - I.polynomial()(S[1]) for I in integrals])
if HJ(Q[0],Q[1]) == HJ(FQ):
FQ_to_Q = V(dim*[0])
else:
FQ_to_Q = V(self.tiny_integrals_on_basis(FQ, Q))
try:
L = [f(K(S[0]), K(S[1])) - f(K(Q[0]), K(Q[1])) for f in forms]
except ValueError:
forms = [f.change_ring(K) for f in forms]
L = [f(S[0], S[1]) - f(Q[0], Q[1]) for f in forms]
b = V(L)
M_sys = matrix(K, M_frob).transpose() - 1
B = (~M_sys)
v = [B.list()[i].valuation() for i in range(len(B.list()))]
vv= min(v)
B = (p**(-vv)*B).change_ring(K)
B = p**(vv)*B
return B*(b-S_to_FS-FQ_to_Q)
