Hyperelliptic curves over a \(p\)-adic field¶

class sage.schemes.hyperelliptic_curves.hyperelliptic_padic_field.HyperellipticCurve_padic_field(PP, f, h=None, names=None, genus=None)[source]¶

Bases: HyperellipticCurve_generic, ProjectivePlaneCurve_field

P_to_S(P, S)[source]¶

Given a finite Weierstrass point \(P\) and a point \(S\) in the same disc, compute the Coleman integrals \(\{\int_P^S x^i dx/2y \}_{i=0}^{2g-1}\).

INPUT:

P – finite Weierstrass point
S – point in disc of \(P\)

OUTPUT: Coleman integrals \(\{\int_P^S x^i dx/2y \}_{i=0}^{2g-1}\)

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^3-10*x+9)
sage: K = Qp(5,4)
sage: HK = H.change_ring(K)
sage: P = HK(1,0)
sage: HJ = HK.curve_over_ram_extn(10)
sage: S = HK.get_boundary_point(HJ,P)
sage: HK.P_to_S(P, S)
(2*a + 4*a^3 + 2*a^11 + 4*a^13 + 2*a^17 + 2*a^19 + a^21 + 4*a^23 + a^25 + 2*a^27 + 2*a^29 + 3*a^31 + 4*a^33 + O(a^35), a^-5 + 2*a + 2*a^3 + a^7 + 3*a^11 + a^13 + 3*a^15 + 3*a^17 + 2*a^19 + 4*a^21 + 4*a^23 + 4*a^25 + 2*a^27 + a^29 + a^31 + O(a^33))

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(4))
>>> HK = H.change_ring(K)
>>> P = HK(Integer(1),Integer(0))
>>> HJ = HK.curve_over_ram_extn(Integer(10))
>>> S = HK.get_boundary_point(HJ,P)
>>> HK.P_to_S(P, S)
(2*a + 4*a^3 + 2*a^11 + 4*a^13 + 2*a^17 + 2*a^19 + a^21 + 4*a^23 + a^25 + 2*a^27 + 2*a^29 + 3*a^31 + 4*a^33 + O(a^35), a^-5 + 2*a + 2*a^3 + a^7 + 3*a^11 + a^13 + 3*a^15 + 3*a^17 + 2*a^19 + 4*a^21 + 4*a^23 + 4*a^25 + 2*a^27 + a^29 + a^31 + O(a^33))

AUTHOR:

Jennifer Balakrishnan

S_to_Q(S, Q)[source]¶

Given \(S\) a point on self over an extension field, compute 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 \(\QQ_p\) (with unif. a)
Q – a non-Weierstrass point defined over \(\QQ_p\)

OUTPUT:

The Coleman integrals \(\{\int_S^Q x^i dx/2y \}_{i=0}^{2g-1}\) in terms of \(a\).

EXAMPLES:

Sage

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))

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(6))
>>> HK = H.change_ring(K)
>>> J = K.extension(x**Integer(20)-Integer(5), names=('a',)); (a,) = J._first_ngens(1)
>>> HJ  = H.change_ring(J)
>>> w = HK.invariant_differential()
>>> x,y = HK.monsky_washnitzer_gens()
>>> P = HK(Integer(1),Integer(0))
>>> Q = HK(Integer(0),Integer(3))
>>> S = HK.get_boundary_point(HJ,P)
>>> P_to_S = HK.P_to_S(P,S)
>>> S_to_Q = HJ.S_to_Q(S,Q)
>>> 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))
>>> 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

coleman_integral(w, P, Q, algorithm='None')[source]¶

Return 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 – None (default, 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

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)

Python

>>> from sage.all import *
>>> K = pAdicField(Integer(11), Integer(6))
>>> x = polygen(K)
>>> C = HyperellipticCurve(x**Integer(5) + Integer(33)/Integer(16)*x**Integer(4) + Integer(3)/Integer(4)*x**Integer(3) + Integer(3)/Integer(8)*x**Integer(2) - Integer(1)/Integer(4)*x + Integer(1)/Integer(16))
>>> P = C(-Integer(1), Integer(1)); P1 = C(-Integer(1), -Integer(1))
>>> Q = C(Integer(0), Integer(1)/Integer(4)); Q1 = C(Integer(0), -Integer(1)/Integer(4))
>>> x, y = C.monsky_washnitzer_gens()
>>> w = C.invariant_differential()
>>> w.coleman_integral(P, Q)
O(11^6)
>>> C.coleman_integral(x*w, P, Q)
O(11^6)
>>> C.coleman_integral(x**Integer(2)*w, P, Q)
7*11 + 6*11^2 + 3*11^3 + 11^4 + 5*11^5 + O(11^6)

Sage

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)
50*71 + 3*71^2 + 43*71^3 + O(71^4)
sage: w.integrate(P, R) + w.integrate(P1, R1)
O(71^4)

Python

>>> from sage.all import *
>>> p = Integer(71); m = Integer(4)
>>> K = pAdicField(p, m)
>>> x = polygen(K)
>>> C = HyperellipticCurve(x**Integer(5) + Integer(33)/Integer(16)*x**Integer(4) + Integer(3)/Integer(4)*x**Integer(3) + Integer(3)/Integer(8)*x**Integer(2) - Integer(1)/Integer(4)*x + Integer(1)/Integer(16))
>>> P = C(-Integer(1), Integer(1)); P1 = C(-Integer(1), -Integer(1))
>>> Q = C(Integer(0), Integer(1)/Integer(4)); Q1 = C(Integer(0), -Integer(1)/Integer(4))
>>> x, y = C.monsky_washnitzer_gens()
>>> w = C.invariant_differential()
>>> w.integrate(P, Q), (x*w).integrate(P, Q)
(O(71^4), O(71^4))
>>> R, R1 = C.lift_x(Integer(4), all=True)
>>> w.integrate(P, R)
50*71 + 3*71^2 + 43*71^3 + O(71^4)
>>> w.integrate(P, R) + w.integrate(P1, R1)
O(71^4)

A simple example, integrating dx:

Sage

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)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> E= HyperellipticCurve(x**Integer(3)-Integer(4)*x+Integer(4))
>>> K = Qp(Integer(5),Integer(10))
>>> EK = E.change_ring(K)
>>> P = EK(Integer(2), Integer(2))
>>> Q = EK.teichmuller(P)
>>> x, y = EK.monsky_washnitzer_gens()
>>> 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)
>>> Q[Integer(0)] - P[Integer(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

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)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x*(x-Integer(1))*(x+Integer(9)))
>>> K = Qp(Integer(7),Integer(10))
>>> HK = H.change_ring(K)
>>> import sage.schemes.hyperelliptic_curves.monsky_washnitzer as mw
>>> M_frob, forms = mw.matrix_of_frobenius_hyperelliptic(HK)
>>> w = HK.invariant_differential()
>>> x,y = HK.monsky_washnitzer_gens()
>>> f = forms[Integer(0)]
>>> S = HK(Integer(9),Integer(36))
>>> Q = HK.teichmuller(S)
>>> P = HK(-Integer(1),Integer(4))
>>> b = x*w*w._coeff.parent()(f)
>>> 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

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(5, 1 + 3*5^3 + 2*5^4 + 2*5^5 + 3*5^7)
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)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)+Integer(1))
>>> K = Qp(Integer(5),Integer(8))
>>> HK = H.change_ring(K)
>>> w = HK.invariant_differential()
>>> P = HK(Integer(0),Integer(1))
>>> Q = HK(Integer(5), Integer(1) + Integer(3)*Integer(5)**Integer(3) + Integer(2)*Integer(5)**Integer(4) + Integer(2)*Integer(5)**Integer(5) + Integer(3)*Integer(5)**Integer(7))
>>> x,y = HK.monsky_washnitzer_gens()
>>> (Integer(2)*y*w).coleman_integral(P,Q)
5 + O(5^9)
>>> xloc,yloc,zloc = HK.local_analytic_interpolation(P,Q)
>>> I2 = (xloc.derivative()/(Integer(2)*yloc)).integral()
>>> I2.polynomial()(Integer(1)) - I2(Integer(0))
3*5 + 2*5^2 + 2*5^3 + 5^4 + 4*5^6 + 5^7 + O(5^9)
>>> 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

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)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(8))
>>> HK = H.change_ring(K)
>>> S = HK(Integer(1),Integer(0))
>>> P = HK(Integer(0),Integer(3))
>>> negP = HK(Integer(0),-Integer(3))
>>> T = HK(Integer(0),Integer(1),Integer(0))
>>> w = HK.invariant_differential()
>>> x,y = HK.monsky_washnitzer_gens()
>>> HK.coleman_integral(w*x**Integer(3),S,T)
0
>>> HK.coleman_integral(w*x**Integer(3),T,S)
0
>>> 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)
>>> 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)
>>> HK.coleman_integral(w*x**Integer(3),T,P)
5^2 + 2*5^3 + 3*5^6 + 3*5^7 + O(5^8)
>>> HK.coleman_integral(w*x**Integer(3),S,P)
5^2 + 2*5^3 + 3*5^6 + 3*5^7 + O(5^8)
>>> 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)
>>> 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)

coleman_integral_P_to_S(w, P, S)[source]¶

Given a finite Weierstrass point \(P\) and a point \(S\) in the same disc, compute the Coleman integral \(\int_P^S w\).

INPUT:

w – differential
P – Weierstrass point
S – point in the same disc of \(P\) (S is defined over an extension of \(\QQ_p\); coordinates of S are given in terms of uniformizer \(a\))

OUTPUT: Coleman integral \(\int_P^S w\) in terms of \(a\)

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^3-10*x+9)
sage: K = Qp(5,4)
sage: HK = H.change_ring(K)
sage: P = HK(1,0)
sage: J.<a> = K.extension(x^10-5)
sage: HJ  = H.change_ring(J)
sage: S = HK.get_boundary_point(HJ,P)
sage: x,y = HK.monsky_washnitzer_gens()
sage: S[0]-P[0] == HK.coleman_integral_P_to_S(x.diff(),P,S)
True
sage: HK.coleman_integral_P_to_S(HK.invariant_differential(),P,S) == HK.P_to_S(P,S)[0]
True

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(4))
>>> HK = H.change_ring(K)
>>> P = HK(Integer(1),Integer(0))
>>> J = K.extension(x**Integer(10)-Integer(5), names=('a',)); (a,) = J._first_ngens(1)
>>> HJ  = H.change_ring(J)
>>> S = HK.get_boundary_point(HJ,P)
>>> x,y = HK.monsky_washnitzer_gens()
>>> S[Integer(0)]-P[Integer(0)] == HK.coleman_integral_P_to_S(x.diff(),P,S)
True
>>> HK.coleman_integral_P_to_S(HK.invariant_differential(),P,S) == HK.P_to_S(P,S)[Integer(0)]
True

AUTHOR:

Jennifer Balakrishnan

coleman_integral_S_to_Q(w, S, Q)[source]¶

Compute 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

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)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(6))
>>> HK = H.change_ring(K)
>>> J = K.extension(x**Integer(20)-Integer(5), names=('a',)); (a,) = J._first_ngens(1)
>>> HJ  = H.change_ring(J)
>>> x,y = HK.monsky_washnitzer_gens()
>>> P = HK(Integer(1),Integer(0))
>>> Q = HK(Integer(0),Integer(3))
>>> S = HK.get_boundary_point(HJ,P)
>>> P_to_S = HK.coleman_integral_P_to_S(y.diff(),P,S)
>>> S_to_Q = HJ.coleman_integral_S_to_Q(y.diff(),S,Q)
>>> P_to_S  + S_to_Q
3 + O(a^119)
>>> HK.coleman_integral(y.diff(),P,Q)
3 + O(5^6)

AUTHOR:

Jennifer Balakrishnan

coleman_integral_from_weierstrass_via_boundary(w, P, Q, d)[source]¶

Compute the Coleman integral \(\int_P^Q w\) via a boundary point in the disc of \(P\), defined over a degree \(d\) extension

INPUT:

w – a differential
P – a Weierstrass point
Q – a non-Weierstrass point
d – degree of extension where coordinates of boundary point lie

OUTPUT:

the Coleman integral \(\int_P^Q w\), written in terms of the uniformizer \(a\) of the degree \(d\) extension

EXAMPLES:

Sage

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: P = HK(1,0)
sage: Q = HK(0,3)
sage: x,y = HK.monsky_washnitzer_gens()
sage: HK.coleman_integral_from_weierstrass_via_boundary(y.diff(),P,Q,20)
3 + O(a^119)
sage: HK.coleman_integral(y.diff(),P,Q)
3 + O(5^6)
sage: w = HK.invariant_differential()
sage: HK.coleman_integral_from_weierstrass_via_boundary(w,P,Q,20)
2*a^40 + a^80 + a^100 + O(a^105)
sage: HK.coleman_integral(w,P,Q)
2*5^2 + 5^4 + 5^5 + 3*5^6 + O(5^7)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(6))
>>> HK = H.change_ring(K)
>>> P = HK(Integer(1),Integer(0))
>>> Q = HK(Integer(0),Integer(3))
>>> x,y = HK.monsky_washnitzer_gens()
>>> HK.coleman_integral_from_weierstrass_via_boundary(y.diff(),P,Q,Integer(20))
3 + O(a^119)
>>> HK.coleman_integral(y.diff(),P,Q)
3 + O(5^6)
>>> w = HK.invariant_differential()
>>> HK.coleman_integral_from_weierstrass_via_boundary(w,P,Q,Integer(20))
2*a^40 + a^80 + a^100 + O(a^105)
>>> HK.coleman_integral(w,P,Q)
2*5^2 + 5^4 + 5^5 + 3*5^6 + O(5^7)

AUTHOR:

Jennifer Balakrishnan

coleman_integrals_on_basis(P, Q, algorithm=None)[source]¶

Compute the Coleman integrals \(\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}\).

INPUT:

P – point on self
Q – point on self
algorithm – None (default, uses Frobenius) or teichmuller (uses Teichmuller points)

OUTPUT: the Coleman integrals \(\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}\)

EXAMPLES:

Sage

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)
(9*11^2 + 7*11^3 + 5*11^4 + O(11^5), 11 + 3*11^2 + 7*11^3 + 11^4 + O(11^5), 10*11 + 11^2 + 5*11^3 + 5*11^4 + O(11^5), 3 + 9*11^2 + 6*11^3 + 11^4 + O(11^5))
sage: C.coleman_integrals_on_basis(P, Q, algorithm='teichmuller')
(9*11^2 + 7*11^3 + 5*11^4 + O(11^5), 11 + 3*11^2 + 7*11^3 + 11^4 + O(11^5), 10*11 + 11^2 + 5*11^3 + 5*11^4 + O(11^5), 3 + 9*11^2 + 6*11^3 + 11^4 + O(11^5))

Python

>>> from sage.all import *
>>> K = pAdicField(Integer(11), Integer(5))
>>> x = polygen(K)
>>> C = HyperellipticCurve(x**Integer(5) + Integer(33)/Integer(16)*x**Integer(4) + Integer(3)/Integer(4)*x**Integer(3) + Integer(3)/Integer(8)*x**Integer(2) - Integer(1)/Integer(4)*x + Integer(1)/Integer(16))
>>> P = C.lift_x(Integer(2))
>>> Q = C.lift_x(Integer(3))
>>> C.coleman_integrals_on_basis(P, Q)
(9*11^2 + 7*11^3 + 5*11^4 + O(11^5), 11 + 3*11^2 + 7*11^3 + 11^4 + O(11^5), 10*11 + 11^2 + 5*11^3 + 5*11^4 + O(11^5), 3 + 9*11^2 + 6*11^3 + 11^4 + O(11^5))
>>> C.coleman_integrals_on_basis(P, Q, algorithm='teichmuller')
(9*11^2 + 7*11^3 + 5*11^4 + O(11^5), 11 + 3*11^2 + 7*11^3 + 11^4 + O(11^5), 10*11 + 11^2 + 5*11^3 + 5*11^4 + O(11^5), 3 + 9*11^2 + 6*11^3 + 11^4 + O(11^5))

Sage

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(11^-2, 10*11^-5 + 10*11^-4 + 10*11^-3 + 2*11^-2 + 10*11^-1)
sage: Q = C(3*11^-2, 11^-5 + 11^-3 + 10*11^-2 + 7*11^-1)
sage: C.coleman_integrals_on_basis(P, Q)
(6*11^3 + 3*11^4 + 8*11^5 + 4*11^6 + 9*11^7 + O(11^8), 11 + 10*11^2 + 8*11^3 + 7*11^4 + 5*11^5 + O(11^6), 6*11^-1 + 2 + 6*11 + 3*11^3 + O(11^4), 9*11^-3 + 9*11^-2 + 9*11^-1 + 2*11 + O(11^2))

Python

>>> from sage.all import *
>>> K = pAdicField(Integer(11),Integer(5))
>>> x = polygen(K)
>>> C = HyperellipticCurve(x**Integer(5) + Integer(33)/Integer(16)*x**Integer(4) + Integer(3)/Integer(4)*x**Integer(3) + Integer(3)/Integer(8)*x**Integer(2) - Integer(1)/Integer(4)*x + Integer(1)/Integer(16))
>>> P = C(Integer(11)**-Integer(2), Integer(10)*Integer(11)**-Integer(5) + Integer(10)*Integer(11)**-Integer(4) + Integer(10)*Integer(11)**-Integer(3) + Integer(2)*Integer(11)**-Integer(2) + Integer(10)*Integer(11)**-Integer(1))
>>> Q = C(Integer(3)*Integer(11)**-Integer(2), Integer(11)**-Integer(5) + Integer(11)**-Integer(3) + Integer(10)*Integer(11)**-Integer(2) + Integer(7)*Integer(11)**-Integer(1))
>>> C.coleman_integrals_on_basis(P, Q)
(6*11^3 + 3*11^4 + 8*11^5 + 4*11^6 + 9*11^7 + O(11^8), 11 + 10*11^2 + 8*11^3 + 7*11^4 + 5*11^5 + O(11^6), 6*11^-1 + 2 + 6*11 + 3*11^3 + O(11^4), 9*11^-3 + 9*11^-2 + 9*11^-1 + 2*11 + O(11^2))

Sage

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

Python

>>> from sage.all import *
>>> R = C(Integer(0),Integer(1)/Integer(4))
>>> a = C.coleman_integrals_on_basis(P,R)  # long time (7s on sage.math, 2011)
>>> b = C.coleman_integrals_on_basis(R,Q)  # long time (9s on sage.math, 2011)
>>> c = C.coleman_integrals_on_basis(P,Q)  # long time
>>> a+b == c  # long time
True

Sage

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)
(5 + O(5^4), 4*5^-1 + 4 + 4*5 + 4*5^2 + O(5^3))
sage: HK.coleman_integrals_on_basis(Q,R)
(5 + 2*5^2 + 2*5^3 + 2*5^4 + 3*5^5 + 3*5^6 + 3*5^7 + 5^8 + O(5^9), 3*5^-1 + 2*5^4 + 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)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(8))
>>> HK = H.change_ring(K)
>>> S = HK(Integer(1),Integer(0))
>>> P = HK(Integer(0),Integer(3))
>>> T = HK(Integer(0),Integer(1),Integer(0))
>>> Q = HK.lift_x(Integer(5)**-Integer(2))
>>> R = HK.lift_x(Integer(4)*Integer(5)**-Integer(2))
>>> 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))
>>> 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))
>>> HK.coleman_integrals_on_basis(P,S) == -HK.coleman_integrals_on_basis(S,P)
True
>>> HK.coleman_integrals_on_basis(S,Q)
(5 + O(5^4), 4*5^-1 + 4 + 4*5 + 4*5^2 + O(5^3))
>>> HK.coleman_integrals_on_basis(Q,R)
(5 + 2*5^2 + 2*5^3 + 2*5^4 + 3*5^5 + 3*5^6 + 3*5^7 + 5^8 + O(5^9), 3*5^-1 + 2*5^4 + 5^5 + 2*5^6 + O(5^7))
>>> HK.coleman_integrals_on_basis(S,R) == HK.coleman_integrals_on_basis(S,Q) + HK.coleman_integrals_on_basis(Q,R)
True
>>> HK.coleman_integrals_on_basis(T,T)
(0, 0)
>>> 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

coleman_integrals_on_basis_hyperelliptic(P, Q, algorithm=None)[source]¶

Compute the Coleman integrals \(\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}\).

INPUT:

P – point on self
Q – point on self
algorithm – None (default, uses Frobenius) or teichmuller (uses Teichmuller points)

OUTPUT: the Coleman integrals \(\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}\)

EXAMPLES:

Sage

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)
(9*11^2 + 7*11^3 + 5*11^4 + O(11^5), 11 + 3*11^2 + 7*11^3 + 11^4 + O(11^5), 10*11 + 11^2 + 5*11^3 + 5*11^4 + O(11^5), 3 + 9*11^2 + 6*11^3 + 11^4 + O(11^5))
sage: C.coleman_integrals_on_basis(P, Q, algorithm='teichmuller')
(9*11^2 + 7*11^3 + 5*11^4 + O(11^5), 11 + 3*11^2 + 7*11^3 + 11^4 + O(11^5), 10*11 + 11^2 + 5*11^3 + 5*11^4 + O(11^5), 3 + 9*11^2 + 6*11^3 + 11^4 + O(11^5))

Python

>>> from sage.all import *
>>> K = pAdicField(Integer(11), Integer(5))
>>> x = polygen(K)
>>> C = HyperellipticCurve(x**Integer(5) + Integer(33)/Integer(16)*x**Integer(4) + Integer(3)/Integer(4)*x**Integer(3) + Integer(3)/Integer(8)*x**Integer(2) - Integer(1)/Integer(4)*x + Integer(1)/Integer(16))
>>> P = C.lift_x(Integer(2))
>>> Q = C.lift_x(Integer(3))
>>> C.coleman_integrals_on_basis(P, Q)
(9*11^2 + 7*11^3 + 5*11^4 + O(11^5), 11 + 3*11^2 + 7*11^3 + 11^4 + O(11^5), 10*11 + 11^2 + 5*11^3 + 5*11^4 + O(11^5), 3 + 9*11^2 + 6*11^3 + 11^4 + O(11^5))
>>> C.coleman_integrals_on_basis(P, Q, algorithm='teichmuller')
(9*11^2 + 7*11^3 + 5*11^4 + O(11^5), 11 + 3*11^2 + 7*11^3 + 11^4 + O(11^5), 10*11 + 11^2 + 5*11^3 + 5*11^4 + O(11^5), 3 + 9*11^2 + 6*11^3 + 11^4 + O(11^5))

Sage

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(11^-2, 10*11^-5 + 10*11^-4 + 10*11^-3 + 2*11^-2 + 10*11^-1)
sage: Q = C(3*11^-2, 11^-5 + 11^-3 + 10*11^-2 + 7*11^-1)
sage: C.coleman_integrals_on_basis(P, Q)
(6*11^3 + 3*11^4 + 8*11^5 + 4*11^6 + 9*11^7 + O(11^8), 11 + 10*11^2 + 8*11^3 + 7*11^4 + 5*11^5 + O(11^6), 6*11^-1 + 2 + 6*11 + 3*11^3 + O(11^4), 9*11^-3 + 9*11^-2 + 9*11^-1 + 2*11 + O(11^2))

Python

>>> from sage.all import *
>>> K = pAdicField(Integer(11),Integer(5))
>>> x = polygen(K)
>>> C = HyperellipticCurve(x**Integer(5) + Integer(33)/Integer(16)*x**Integer(4) + Integer(3)/Integer(4)*x**Integer(3) + Integer(3)/Integer(8)*x**Integer(2) - Integer(1)/Integer(4)*x + Integer(1)/Integer(16))
>>> P = C(Integer(11)**-Integer(2), Integer(10)*Integer(11)**-Integer(5) + Integer(10)*Integer(11)**-Integer(4) + Integer(10)*Integer(11)**-Integer(3) + Integer(2)*Integer(11)**-Integer(2) + Integer(10)*Integer(11)**-Integer(1))
>>> Q = C(Integer(3)*Integer(11)**-Integer(2), Integer(11)**-Integer(5) + Integer(11)**-Integer(3) + Integer(10)*Integer(11)**-Integer(2) + Integer(7)*Integer(11)**-Integer(1))
>>> C.coleman_integrals_on_basis(P, Q)
(6*11^3 + 3*11^4 + 8*11^5 + 4*11^6 + 9*11^7 + O(11^8), 11 + 10*11^2 + 8*11^3 + 7*11^4 + 5*11^5 + O(11^6), 6*11^-1 + 2 + 6*11 + 3*11^3 + O(11^4), 9*11^-3 + 9*11^-2 + 9*11^-1 + 2*11 + O(11^2))

Sage

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

Python

>>> from sage.all import *
>>> R = C(Integer(0),Integer(1)/Integer(4))
>>> a = C.coleman_integrals_on_basis(P,R)  # long time (7s on sage.math, 2011)
>>> b = C.coleman_integrals_on_basis(R,Q)  # long time (9s on sage.math, 2011)
>>> c = C.coleman_integrals_on_basis(P,Q)  # long time
>>> a+b == c  # long time
True

Sage

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)
(5 + O(5^4), 4*5^-1 + 4 + 4*5 + 4*5^2 + O(5^3))
sage: HK.coleman_integrals_on_basis(Q,R)
(5 + 2*5^2 + 2*5^3 + 2*5^4 + 3*5^5 + 3*5^6 + 3*5^7 + 5^8 + O(5^9), 3*5^-1 + 2*5^4 + 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)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(8))
>>> HK = H.change_ring(K)
>>> S = HK(Integer(1),Integer(0))
>>> P = HK(Integer(0),Integer(3))
>>> T = HK(Integer(0),Integer(1),Integer(0))
>>> Q = HK.lift_x(Integer(5)**-Integer(2))
>>> R = HK.lift_x(Integer(4)*Integer(5)**-Integer(2))
>>> 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))
>>> 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))
>>> HK.coleman_integrals_on_basis(P,S) == -HK.coleman_integrals_on_basis(S,P)
True
>>> HK.coleman_integrals_on_basis(S,Q)
(5 + O(5^4), 4*5^-1 + 4 + 4*5 + 4*5^2 + O(5^3))
>>> HK.coleman_integrals_on_basis(Q,R)
(5 + 2*5^2 + 2*5^3 + 2*5^4 + 3*5^5 + 3*5^6 + 3*5^7 + 5^8 + O(5^9), 3*5^-1 + 2*5^4 + 5^5 + 2*5^6 + O(5^7))
>>> HK.coleman_integrals_on_basis(S,R) == HK.coleman_integrals_on_basis(S,Q) + HK.coleman_integrals_on_basis(Q,R)
True
>>> HK.coleman_integrals_on_basis(T,T)
(0, 0)
>>> 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

curve_over_ram_extn(deg)[source]¶

Return self over \(\QQ_p(p^(1/deg))\).

INPUT:

deg – the degree of the ramified extension

OUTPUT: self over \(\QQ_p(p^(1/deg))\)

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: K = Qp(11,5)
sage: HK = H.change_ring(K)
sage: HL = HK.curve_over_ram_extn(2)
sage: HL
Hyperelliptic Curve over 11-adic Eisenstein Extension Field in a defined by x^2 - 11 defined by (1 + O(a^10))*y^2 = (1 + O(a^10))*x^5 + (10 + 8*a^2 + 10*a^4 + 10*a^6 + 10*a^8 + O(a^10))*x^3 + (7 + a^2 + O(a^10))*x^2 + (7 + 3*a^2 + O(a^10))*x

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(5)-Integer(23)*x**Integer(3)+Integer(18)*x**Integer(2)+Integer(40)*x)
>>> K = Qp(Integer(11),Integer(5))
>>> HK = H.change_ring(K)
>>> HL = HK.curve_over_ram_extn(Integer(2))
>>> HL
Hyperelliptic Curve over 11-adic Eisenstein Extension Field in a defined by x^2 - 11 defined by (1 + O(a^10))*y^2 = (1 + O(a^10))*x^5 + (10 + 8*a^2 + 10*a^4 + 10*a^6 + 10*a^8 + O(a^10))*x^3 + (7 + a^2 + O(a^10))*x^2 + (7 + 3*a^2 + O(a^10))*x

AUTHOR:

Jennifer Balakrishnan

find_char_zero_weier_point(Q)[source]¶

Given \(Q\) a point on self in a Weierstrass disc, finds the center of the Weierstrass disc (if defined over self.base_ring()).

EXAMPLES:

Sage

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: P = HK.lift_x(1 + 2*5^2)
sage: Q = HK.lift_x(5^-2)
sage: S = HK(1,0)
sage: T = HK(0,1,0)
sage: HK.find_char_zero_weier_point(P)
(1 + O(5^8) : 0 : 1 + O(5^8))
sage: HK.find_char_zero_weier_point(Q)
(0 : 1 + O(5^8) : 0)
sage: HK.find_char_zero_weier_point(S)
(1 + O(5^8) : 0 : 1 + O(5^8))
sage: HK.find_char_zero_weier_point(T)
(0 : 1 + O(5^8) : 0)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(8))
>>> HK = H.change_ring(K)
>>> P = HK.lift_x(Integer(1) + Integer(2)*Integer(5)**Integer(2))
>>> Q = HK.lift_x(Integer(5)**-Integer(2))
>>> S = HK(Integer(1),Integer(0))
>>> T = HK(Integer(0),Integer(1),Integer(0))
>>> HK.find_char_zero_weier_point(P)
(1 + O(5^8) : 0 : 1 + O(5^8))
>>> HK.find_char_zero_weier_point(Q)
(0 : 1 + O(5^8) : 0)
>>> HK.find_char_zero_weier_point(S)
(1 + O(5^8) : 0 : 1 + O(5^8))
>>> HK.find_char_zero_weier_point(T)
(0 : 1 + O(5^8) : 0)

AUTHOR:

Jennifer Balakrishnan

frobenius(P=None)[source]¶

Return the \(p\)-th power lift of Frobenius of \(P\).

EXAMPLES:

Sage

sage: K = Qp(11, 5)
sage: R.<x> = K[]
sage: E = HyperellipticCurve(x^5 - 21*x - 20)
sage: P = E.lift_x(2)
sage: E.frobenius(P)
(2 + 10*11 + 5*11^2 + 11^3 + O(11^5) : 6 + 11 + 8*11^2 + 8*11^3 + 10*11^4 + O(11^5) : 1 + O(11^5))
sage: Q = E.teichmuller(P); Q
(2 + 10*11 + 4*11^2 + 9*11^3 + 11^4 + O(11^5) : 6 + 11 + 4*11^2 + 9*11^3 + 4*11^4 + O(11^5) : 1 + O(11^5))
sage: E.frobenius(Q)
(2 + 10*11 + 4*11^2 + 9*11^3 + 11^4 + O(11^5) : 6 + 11 + 4*11^2 + 9*11^3 + 4*11^4 + O(11^5) : 1 + O(11^5))

Python

>>> from sage.all import *
>>> K = Qp(Integer(11), Integer(5))
>>> R = K['x']; (x,) = R._first_ngens(1)
>>> E = HyperellipticCurve(x**Integer(5) - Integer(21)*x - Integer(20))
>>> P = E.lift_x(Integer(2))
>>> E.frobenius(P)
(2 + 10*11 + 5*11^2 + 11^3 + O(11^5) : 6 + 11 + 8*11^2 + 8*11^3 + 10*11^4 + O(11^5) : 1 + O(11^5))
>>> Q = E.teichmuller(P); Q
(2 + 10*11 + 4*11^2 + 9*11^3 + 11^4 + O(11^5) : 6 + 11 + 4*11^2 + 9*11^3 + 4*11^4 + O(11^5) : 1 + O(11^5))
>>> E.frobenius(Q)
(2 + 10*11 + 4*11^2 + 9*11^3 + 11^4 + O(11^5) : 6 + 11 + 4*11^2 + 9*11^3 + 4*11^4 + O(11^5) : 1 + O(11^5))

Sage

sage: R.<x> = QQ[]
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: Q = H(0,0)
sage: u,v = H.local_coord(Q,prec=100)
sage: K = Qp(11,5)
sage: L.<a> = K.extension(x^20-11)
sage: HL = H.change_ring(L)
sage: S = HL(u(a),v(a))
sage: HL.frobenius(S)
(8*a^22 + 10*a^42 + 4*a^44 + 2*a^46 + 9*a^48 + 8*a^50 + a^52 + 7*a^54 +
7*a^56 + 5*a^58 + 9*a^62 + 5*a^64 + a^66 + 6*a^68 + a^70 + 6*a^74 +
2*a^76 + 2*a^78 + 4*a^82 + 5*a^84 + 2*a^86 + 7*a^88 + a^90 + 6*a^92 +
a^96 + 5*a^98 + 2*a^102 + 2*a^106 + 6*a^108 + 8*a^110 + 3*a^112 +
a^114 + 8*a^116 + 10*a^118 + 3*a^120 + O(a^122) :
a^11 + 7*a^33 + 7*a^35 + 4*a^37 + 6*a^39 + 9*a^41 + 8*a^43 + 8*a^45 +
a^47 + 7*a^51 + 4*a^53 + 5*a^55 + a^57 + 7*a^59 + 5*a^61 + 9*a^63 +
4*a^65 + 10*a^69 + 3*a^71 + 2*a^73 + 9*a^75 + 10*a^77 + 6*a^79 +
10*a^81 + 7*a^85 + a^87 + 4*a^89 + 8*a^91 + a^93 + 8*a^95 + 2*a^97 +
7*a^99 + a^101 + 3*a^103 + 6*a^105 + 7*a^107 + 4*a^109 + O(a^111) :
1 + O(a^100))

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(5)-Integer(23)*x**Integer(3)+Integer(18)*x**Integer(2)+Integer(40)*x)
>>> Q = H(Integer(0),Integer(0))
>>> u,v = H.local_coord(Q,prec=Integer(100))
>>> K = Qp(Integer(11),Integer(5))
>>> L = K.extension(x**Integer(20)-Integer(11), names=('a',)); (a,) = L._first_ngens(1)
>>> HL = H.change_ring(L)
>>> S = HL(u(a),v(a))
>>> HL.frobenius(S)
(8*a^22 + 10*a^42 + 4*a^44 + 2*a^46 + 9*a^48 + 8*a^50 + a^52 + 7*a^54 +
7*a^56 + 5*a^58 + 9*a^62 + 5*a^64 + a^66 + 6*a^68 + a^70 + 6*a^74 +
2*a^76 + 2*a^78 + 4*a^82 + 5*a^84 + 2*a^86 + 7*a^88 + a^90 + 6*a^92 +
a^96 + 5*a^98 + 2*a^102 + 2*a^106 + 6*a^108 + 8*a^110 + 3*a^112 +
a^114 + 8*a^116 + 10*a^118 + 3*a^120 + O(a^122) :
a^11 + 7*a^33 + 7*a^35 + 4*a^37 + 6*a^39 + 9*a^41 + 8*a^43 + 8*a^45 +
a^47 + 7*a^51 + 4*a^53 + 5*a^55 + a^57 + 7*a^59 + 5*a^61 + 9*a^63 +
4*a^65 + 10*a^69 + 3*a^71 + 2*a^73 + 9*a^75 + 10*a^77 + 6*a^79 +
10*a^81 + 7*a^85 + a^87 + 4*a^89 + 8*a^91 + a^93 + 8*a^95 + 2*a^97 +
7*a^99 + a^101 + 3*a^103 + 6*a^105 + 7*a^107 + 4*a^109 + O(a^111) :
1 + O(a^100))

AUTHORS:

Robert Bradshaw and Jennifer Balakrishnan (2010-02)

get_boundary_point(curve_over_extn, P)[source]¶

Given self over an extension field, find a point in the disc of \(P\) near the boundary.

INPUT:

curve_over_extn – self over a totally ramified extension
P – Weierstrass point

OUTPUT: a point in the disc of \(P\) near the boundary

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^3-10*x+9)
sage: K = Qp(3,6)
sage: HK = H.change_ring(K)
sage: P = HK(1,0)
sage: J.<a> = K.extension(x^30-3)
sage: HJ  = H.change_ring(J)
sage: S = HK.get_boundary_point(HJ,P)
sage: S
(1 + 2*a^2 + 2*a^6 + 2*a^18 + a^32 + a^34 + a^36 + 2*a^38 + 2*a^40 + a^42 + 2*a^44 + a^48 + 2*a^50 + 2*a^52 + a^54 + a^56 + 2*a^60 + 2*a^62 + a^70 + 2*a^72 + a^76 + 2*a^78 + a^82 + a^88 + a^96 + 2*a^98 + 2*a^102 + a^104 + 2*a^106 + a^108 + 2*a^110 + a^112 + 2*a^116 + a^126 + 2*a^130 + 2*a^132 + a^144 + 2*a^148 + 2*a^150 + a^152 + 2*a^154 + a^162 + a^164 + a^166 + a^168 + a^170 + a^176 + a^178 + O(a^180) : a + O(a^180) : 1 + O(a^180))

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(3),Integer(6))
>>> HK = H.change_ring(K)
>>> P = HK(Integer(1),Integer(0))
>>> J = K.extension(x**Integer(30)-Integer(3), names=('a',)); (a,) = J._first_ngens(1)
>>> HJ  = H.change_ring(J)
>>> S = HK.get_boundary_point(HJ,P)
>>> S
(1 + 2*a^2 + 2*a^6 + 2*a^18 + a^32 + a^34 + a^36 + 2*a^38 + 2*a^40 + a^42 + 2*a^44 + a^48 + 2*a^50 + 2*a^52 + a^54 + a^56 + 2*a^60 + 2*a^62 + a^70 + 2*a^72 + a^76 + 2*a^78 + a^82 + a^88 + a^96 + 2*a^98 + 2*a^102 + a^104 + 2*a^106 + a^108 + 2*a^110 + a^112 + 2*a^116 + a^126 + 2*a^130 + 2*a^132 + a^144 + 2*a^148 + 2*a^150 + a^152 + 2*a^154 + a^162 + a^164 + a^166 + a^168 + a^170 + a^176 + a^178 + O(a^180) : a + O(a^180) : 1 + O(a^180))

AUTHOR:

Jennifer Balakrishnan

is_in_weierstrass_disc(P)[source]¶

Check if \(P\) is in a Weierstrass disc.

EXAMPLES:

Sage

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: P = HK(0,3)
sage: HK.is_in_weierstrass_disc(P)
False
sage: Q = HK(0,1,0)
sage: HK.is_in_weierstrass_disc(Q)
True
sage: S = HK(1,0)
sage: HK.is_in_weierstrass_disc(S)
True
sage: T = HK.lift_x(1+3*5^2); T
(1 + 3*5^2 + O(5^8) : 3*5 + 4*5^2 + 5^4 + 3*5^5 + 5^6 + O(5^7) : 1 + O(5^8))
sage: HK.is_in_weierstrass_disc(T)
True

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(8))
>>> HK = H.change_ring(K)
>>> P = HK(Integer(0),Integer(3))
>>> HK.is_in_weierstrass_disc(P)
False
>>> Q = HK(Integer(0),Integer(1),Integer(0))
>>> HK.is_in_weierstrass_disc(Q)
True
>>> S = HK(Integer(1),Integer(0))
>>> HK.is_in_weierstrass_disc(S)
True
>>> T = HK.lift_x(Integer(1)+Integer(3)*Integer(5)**Integer(2)); T
(1 + 3*5^2 + O(5^8) : 3*5 + 4*5^2 + 5^4 + 3*5^5 + 5^6 + O(5^7) : 1 + O(5^8))
>>> HK.is_in_weierstrass_disc(T)
True

AUTHOR:

Jennifer Balakrishnan (2010-02)

is_same_disc(P, Q)[source]¶

Check if \(P,Q\) are in same residue disc.

EXAMPLES:

Sage

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: P = HK.lift_x(1 + 2*5^2)
sage: Q = HK.lift_x(5^-2)
sage: S = HK(1,0)
sage: HK.is_same_disc(P,Q)
False
sage: HK.is_same_disc(P,S)
True
sage: HK.is_same_disc(Q,S)
False

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(8))
>>> HK = H.change_ring(K)
>>> P = HK.lift_x(Integer(1) + Integer(2)*Integer(5)**Integer(2))
>>> Q = HK.lift_x(Integer(5)**-Integer(2))
>>> S = HK(Integer(1),Integer(0))
>>> HK.is_same_disc(P,Q)
False
>>> HK.is_same_disc(P,S)
True
>>> HK.is_same_disc(Q,S)
False

is_weierstrass(P)[source]¶

Check if \(P\) is a Weierstrass point (i.e., fixed by the hyperelliptic involution).

EXAMPLES:

Sage

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: P = HK(0,3)
sage: HK.is_weierstrass(P)
False
sage: Q = HK(0,1,0)
sage: HK.is_weierstrass(Q)
True
sage: S = HK(1,0)
sage: HK.is_weierstrass(S)
True
sage: T = HK.lift_x(1+3*5^2); T
(1 + 3*5^2 + O(5^8) : 3*5 + 4*5^2 + 5^4 + 3*5^5 + 5^6 + O(5^7) : 1 + O(5^8))
sage: HK.is_weierstrass(T)
False

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(8))
>>> HK = H.change_ring(K)
>>> P = HK(Integer(0),Integer(3))
>>> HK.is_weierstrass(P)
False
>>> Q = HK(Integer(0),Integer(1),Integer(0))
>>> HK.is_weierstrass(Q)
True
>>> S = HK(Integer(1),Integer(0))
>>> HK.is_weierstrass(S)
True
>>> T = HK.lift_x(Integer(1)+Integer(3)*Integer(5)**Integer(2)); T
(1 + 3*5^2 + O(5^8) : 3*5 + 4*5^2 + 5^4 + 3*5^5 + 5^6 + O(5^7) : 1 + O(5^8))
>>> HK.is_weierstrass(T)
False

AUTHOR:

Jennifer Balakrishnan (2010-02)

local_analytic_interpolation(P, Q)[source]¶

For points \(P\), \(Q\) in the same residue disc, this constructs an interpolation from \(P\) to \(Q\) (in homogeneous coordinates) in a power series in the local parameter \(t\), with precision equal to the \(p\)-adic precision of the underlying ring.

INPUT:

P, Q – points on self in the same residue disc

OUTPUT:

Returns a point \(X(t) = ( x(t) : y(t) : z(t) )\) such that:

\(X(0) = P\) and \(X(1) = Q\) if \(P, Q\) are not in the infinite disc
\(X(P[0]^g/P[1]) = P\) and \(X(Q[0]^g/Q[1]) = Q\) if \(P, Q\) are in the infinite disc

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^3-10*x+9)
sage: K = Qp(5,8)
sage: HK = H.change_ring(K)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(8))
>>> HK = H.change_ring(K)

A non-Weierstrass disc:

Sage

sage: P = HK(0,3)
sage: Q = HK(5, 3 + 3*5^2 + 2*5^3 + 3*5^4 + 2*5^5 + 2*5^6 + 3*5^7 + O(5^8))
sage: x,y,z, = HK.local_analytic_interpolation(P,Q)
sage: x(0) == P[0], x(1) == Q[0], y(0) == P[1], y.polynomial()(1) == Q[1]
(True, True, True, True)

Python

>>> from sage.all import *
>>> P = HK(Integer(0),Integer(3))
>>> Q = HK(Integer(5), Integer(3) + Integer(3)*Integer(5)**Integer(2) + Integer(2)*Integer(5)**Integer(3) + Integer(3)*Integer(5)**Integer(4) + Integer(2)*Integer(5)**Integer(5) + Integer(2)*Integer(5)**Integer(6) + Integer(3)*Integer(5)**Integer(7) + O(Integer(5)**Integer(8)))
>>> x,y,z, = HK.local_analytic_interpolation(P,Q)
>>> x(Integer(0)) == P[Integer(0)], x(Integer(1)) == Q[Integer(0)], y(Integer(0)) == P[Integer(1)], y.polynomial()(Integer(1)) == Q[Integer(1)]
(True, True, True, True)

A finite Weierstrass disc:

Sage

sage: P = HK.lift_x(1 + 2*5^2)
sage: Q = HK.lift_x(1 + 3*5^2)
sage: x,y,z = HK.local_analytic_interpolation(P,Q)
sage: x(0) == P[0], x.polynomial()(1) == Q[0], y(0) == P[1], y(1) == Q[1]
(True, True, True, True)

Python

>>> from sage.all import *
>>> P = HK.lift_x(Integer(1) + Integer(2)*Integer(5)**Integer(2))
>>> Q = HK.lift_x(Integer(1) + Integer(3)*Integer(5)**Integer(2))
>>> x,y,z = HK.local_analytic_interpolation(P,Q)
>>> x(Integer(0)) == P[Integer(0)], x.polynomial()(Integer(1)) == Q[Integer(0)], y(Integer(0)) == P[Integer(1)], y(Integer(1)) == Q[Integer(1)]
(True, True, True, True)

The infinite disc:

Sage

sage: P = HK.lift_x(5^-2)
sage: Q = HK.lift_x(4*5^-2)
sage: x,y,z = HK.local_analytic_interpolation(P,Q)
sage: x = x/z
sage: y = y/z
sage: x(P[0]/P[1]) == P[0]
True
sage: x(Q[0]/Q[1]) == Q[0]
True
sage: y(P[0]/P[1]) == P[1]
True
sage: y(Q[0]/Q[1]) == Q[1]
True

Python

>>> from sage.all import *
>>> P = HK.lift_x(Integer(5)**-Integer(2))
>>> Q = HK.lift_x(Integer(4)*Integer(5)**-Integer(2))
>>> x,y,z = HK.local_analytic_interpolation(P,Q)
>>> x = x/z
>>> y = y/z
>>> x(P[Integer(0)]/P[Integer(1)]) == P[Integer(0)]
True
>>> x(Q[Integer(0)]/Q[Integer(1)]) == Q[Integer(0)]
True
>>> y(P[Integer(0)]/P[Integer(1)]) == P[Integer(1)]
True
>>> y(Q[Integer(0)]/Q[Integer(1)]) == Q[Integer(1)]
True

An error if points are not in the same disc:

Sage

sage: x,y,z = HK.local_analytic_interpolation(P,HK(1,0))
Traceback (most recent call last):
...
ValueError: (5^-2 + O(5^6) : 4*5^-3 + 4*5^-2 + 4*5^-1 + 4 + 4*5 + 3*5^3 + 5^4 + O(5^5) : 1 + O(5^8)) and (1 + O(5^8) : 0 : 1 + O(5^8)) are not in the same residue disc

Python

>>> from sage.all import *
>>> x,y,z = HK.local_analytic_interpolation(P,HK(Integer(1),Integer(0)))
Traceback (most recent call last):
...
ValueError: (5^-2 + O(5^6) : 4*5^-3 + 4*5^-2 + 4*5^-1 + 4 + 4*5 + 3*5^3 + 5^4 + O(5^5) : 1 + O(5^8)) and (1 + O(5^8) : 0 : 1 + O(5^8)) are not in the same residue disc

AUTHORS:

Robert Bradshaw (2007-03)
Jennifer Balakrishnan (2010-02)

newton_sqrt(f, x0, prec)[source]¶

Take the square root of the power series \(f\) by Newton’s method.

NOTE:

this function should eventually be moved to \(p\)-adic power series ring

INPUT:

f – power series with coefficients in \(\QQ_p\) or an extension
x0 – seeds the Newton iteration
prec – precision

OUTPUT: the square root of \(f\)

EXAMPLES:

Sage

sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: Q = H(0,0)
sage: u,v = H.local_coord(Q,prec=100)
sage: K = Qp(11,5)
sage: HK = H.change_ring(K)
sage: L.<a> = K.extension(x^20-11)
sage: HL = H.change_ring(L)
sage: S = HL(u(a),v(a))
sage: f = H.hyperelliptic_polynomials()[0]
sage: y = HK.newton_sqrt( f(u(a)^11), a^11,5)
sage: y^2 - f(u(a)^11)
O(a^122)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(5)-Integer(23)*x**Integer(3)+Integer(18)*x**Integer(2)+Integer(40)*x)
>>> Q = H(Integer(0),Integer(0))
>>> u,v = H.local_coord(Q,prec=Integer(100))
>>> K = Qp(Integer(11),Integer(5))
>>> HK = H.change_ring(K)
>>> L = K.extension(x**Integer(20)-Integer(11), names=('a',)); (a,) = L._first_ngens(1)
>>> HL = H.change_ring(L)
>>> S = HL(u(a),v(a))
>>> f = H.hyperelliptic_polynomials()[Integer(0)]
>>> y = HK.newton_sqrt( f(u(a)**Integer(11)), a**Integer(11),Integer(5))
>>> y**Integer(2) - f(u(a)**Integer(11))
O(a^122)

AUTHOR:

Jennifer Balakrishnan

residue_disc(P)[source]¶

Give the residue disc of \(P\).

EXAMPLES:

Sage

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: P = HK.lift_x(1 + 2*5^2)
sage: HK.residue_disc(P)
(1 : 0 : 1)
sage: Q = HK(0,3)
sage: HK.residue_disc(Q)
(0 : 3 : 1)
sage: S = HK.lift_x(5^-2)
sage: HK.residue_disc(S)
(0 : 1 : 0)
sage: T = HK(0,1,0)
sage: HK.residue_disc(T)
(0 : 1 : 0)

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> H = HyperellipticCurve(x**Integer(3)-Integer(10)*x+Integer(9))
>>> K = Qp(Integer(5),Integer(8))
>>> HK = H.change_ring(K)
>>> P = HK.lift_x(Integer(1) + Integer(2)*Integer(5)**Integer(2))
>>> HK.residue_disc(P)
(1 : 0 : 1)
>>> Q = HK(Integer(0),Integer(3))
>>> HK.residue_disc(Q)
(0 : 3 : 1)
>>> S = HK.lift_x(Integer(5)**-Integer(2))
>>> HK.residue_disc(S)
(0 : 1 : 0)
>>> T = HK(Integer(0),Integer(1),Integer(0))
>>> HK.residue_disc(T)
(0 : 1 : 0)

AUTHOR:

Jennifer Balakrishnan

teichmuller(P)[source]¶

Find a Teichm:uller point in the same residue class of \(P\).

Because this lift of frobenius acts as \(x \mapsto x^p\), take the Teichmuller lift of \(x\) and then find a matching \(y\) from that.

EXAMPLES:

Sage

sage: K = pAdicField(7, 5)
sage: E = EllipticCurve(K, [-31/3, -2501/108]) # 11a
sage: P = E(K(14/3), K(11/2))
sage: E.frobenius(P) == P
False
sage: TP = E.teichmuller(P); TP
(0 : 2 + 3*7 + 3*7^2 + 3*7^4 + O(7^5) : 1 + O(7^5))
sage: E.frobenius(TP) == TP
True
sage: (TP[0] - P[0]).valuation() > 0, (TP[1] - P[1]).valuation() > 0
(True, True)

Python

>>> from sage.all import *
>>> K = pAdicField(Integer(7), Integer(5))
>>> E = EllipticCurve(K, [-Integer(31)/Integer(3), -Integer(2501)/Integer(108)]) # 11a
>>> P = E(K(Integer(14)/Integer(3)), K(Integer(11)/Integer(2)))
>>> E.frobenius(P) == P
False
>>> TP = E.teichmuller(P); TP
(0 : 2 + 3*7 + 3*7^2 + 3*7^4 + O(7^5) : 1 + O(7^5))
>>> E.frobenius(TP) == TP
True
>>> (TP[Integer(0)] - P[Integer(0)]).valuation() > Integer(0), (TP[Integer(1)] - P[Integer(1)]).valuation() > Integer(0)
(True, True)

tiny_integrals(F, P, Q)[source]¶

Evaluate the integrals of \(f_i dx/2y\) from \(P\) to \(Q\) for each \(f_i\) in \(F\) by formally integrating a power series in a local parameter \(t\)

\(P\) and \(Q\) MUST be in the same residue disc for this result to make sense.

INPUT:

F – list of functions \(f_i\)
P – point on self
Q – point on self (in the same residue disc as \(P\))

OUTPUT: the integrals \(\int_P^Q f_i dx/2y\)

EXAMPLES:

Sage

sage: K = pAdicField(17, 5)
sage: E = EllipticCurve(K, [-31/3, -2501/108]) # 11a
sage: P = E(K(14/3), K(11/2))
sage: TP = E.teichmuller(P);
sage: x,y = E.monsky_washnitzer_gens()
sage: E.tiny_integrals([1,x],P, TP) == E.tiny_integrals_on_basis(P,TP)
True

Python

>>> from sage.all import *
>>> K = pAdicField(Integer(17), Integer(5))
>>> E = EllipticCurve(K, [-Integer(31)/Integer(3), -Integer(2501)/Integer(108)]) # 11a
>>> P = E(K(Integer(14)/Integer(3)), K(Integer(11)/Integer(2)))
>>> TP = E.teichmuller(P);
>>> x,y = E.monsky_washnitzer_gens()
>>> E.tiny_integrals([Integer(1),x],P, TP) == E.tiny_integrals_on_basis(P,TP)
True

Sage

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.tiny_integrals([1],P,Q)
(5*11^3 + 7*11^4 + 2*11^5 + 6*11^6 + 11^7 + O(11^8))

Python

>>> from sage.all import *
>>> K = pAdicField(Integer(11), Integer(5))
>>> x = polygen(K)
>>> C = HyperellipticCurve(x**Integer(5) + Integer(33)/Integer(16)*x**Integer(4) + Integer(3)/Integer(4)*x**Integer(3) + Integer(3)/Integer(8)*x**Integer(2) - Integer(1)/Integer(4)*x + Integer(1)/Integer(16))
>>> P = C.lift_x(Integer(11)**(-Integer(2)))
>>> Q = C.lift_x(Integer(3)*Integer(11)**(-Integer(2)))
>>> C.tiny_integrals([Integer(1)],P,Q)
(5*11^3 + 7*11^4 + 2*11^5 + 6*11^6 + 11^7 + O(11^8))

Note that this fails if the points are not in the same residue disc:

Sage

sage: S = C(0,1/4)
sage: C.tiny_integrals([1,x,x^2,x^3],P,S)
Traceback (most recent call last):
...
ValueError: (11^-2 + O(11^3) : 11^-5 + 8*11^-2 + O(11^0) : 1 + O(11^5)) and (0 : 3 + 8*11 + 2*11^2 + 8*11^3 + 2*11^4 + O(11^5) : 1 + O(11^5)) are not in the same residue disc

Python

>>> from sage.all import *
>>> S = C(Integer(0),Integer(1)/Integer(4))
>>> C.tiny_integrals([Integer(1),x,x**Integer(2),x**Integer(3)],P,S)
Traceback (most recent call last):
...
ValueError: (11^-2 + O(11^3) : 11^-5 + 8*11^-2 + O(11^0) : 1 + O(11^5)) and (0 : 3 + 8*11 + 2*11^2 + 8*11^3 + 2*11^4 + O(11^5) : 1 + O(11^5)) are not in the same residue disc

tiny_integrals_on_basis(P, Q)[source]¶

Evaluate the integrals \(\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}\) by formally integrating a power series in a local parameter \(t\). \(P\) and \(Q\) MUST be in the same residue disc for this result to make sense.

INPUT:

P – point on self
Q – point on self (in the same residue disc as \(P\))

OUTPUT: the integrals \(\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}\)

EXAMPLES:

Sage

sage: K = pAdicField(17, 5)
sage: E = EllipticCurve(K, [-31/3, -2501/108]) # 11a
sage: P = E(K(14/3), K(11/2))
sage: TP = E.teichmuller(P);
sage: E.tiny_integrals_on_basis(P, TP)
(17 + 14*17^2 + 17^3 + 8*17^4 + O(17^5), 16*17 + 5*17^2 + 8*17^3 + 14*17^4 + O(17^5))

Python

>>> from sage.all import *
>>> K = pAdicField(Integer(17), Integer(5))
>>> E = EllipticCurve(K, [-Integer(31)/Integer(3), -Integer(2501)/Integer(108)]) # 11a
>>> P = E(K(Integer(14)/Integer(3)), K(Integer(11)/Integer(2)))
>>> TP = E.teichmuller(P);
>>> E.tiny_integrals_on_basis(P, TP)
(17 + 14*17^2 + 17^3 + 8*17^4 + O(17^5), 16*17 + 5*17^2 + 8*17^3 + 14*17^4 + O(17^5))

Sage

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.tiny_integrals_on_basis(P,Q)
(5*11^3 + 7*11^4 + 2*11^5 + 6*11^6 + 11^7 + O(11^8), 10*11 + 2*11^3 + 3*11^4 + 5*11^5 + O(11^6), 5*11^-1 + 8 + 4*11 + 10*11^2 + 7*11^3 + O(11^4), 2*11^-3 + 11^-2 + 11^-1 + 10 + 8*11 + O(11^2))

Python

>>> from sage.all import *
>>> K = pAdicField(Integer(11), Integer(5))
>>> x = polygen(K)
>>> C = HyperellipticCurve(x**Integer(5) + Integer(33)/Integer(16)*x**Integer(4) + Integer(3)/Integer(4)*x**Integer(3) + Integer(3)/Integer(8)*x**Integer(2) - Integer(1)/Integer(4)*x + Integer(1)/Integer(16))
>>> P = C.lift_x(Integer(11)**(-Integer(2)))
>>> Q = C.lift_x(Integer(3)*Integer(11)**(-Integer(2)))
>>> C.tiny_integrals_on_basis(P,Q)
(5*11^3 + 7*11^4 + 2*11^5 + 6*11^6 + 11^7 + O(11^8), 10*11 + 2*11^3 + 3*11^4 + 5*11^5 + O(11^6), 5*11^-1 + 8 + 4*11 + 10*11^2 + 7*11^3 + O(11^4), 2*11^-3 + 11^-2 + 11^-1 + 10 + 8*11 + O(11^2))

Note that this fails if the points are not in the same residue disc:

Sage

sage: S = C(0,1/4)
sage: C.tiny_integrals_on_basis(P,S)
Traceback (most recent call last):
...
ValueError: (11^-2 + O(11^3) : 11^-5 + 8*11^-2 + O(11^0) : 1 + O(11^5)) and (0 : 3 + 8*11 + 2*11^2 + 8*11^3 + 2*11^4 + O(11^5) : 1 + O(11^5)) are not in the same residue disc

Python

>>> from sage.all import *
>>> S = C(Integer(0),Integer(1)/Integer(4))
>>> C.tiny_integrals_on_basis(P,S)
Traceback (most recent call last):
...
ValueError: (11^-2 + O(11^3) : 11^-5 + 8*11^-2 + O(11^0) : 1 + O(11^5)) and (0 : 3 + 8*11 + 2*11^2 + 8*11^3 + 2*11^4 + O(11^5) : 1 + O(11^5)) are not in the same residue disc

weierstrass_points()[source]¶

Return the Weierstrass points of self defined over self.base_ring(), that is, the point at infinity and those points in the support of the divisor of \(y\).

EXAMPLES:

Sage

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: C.weierstrass_points()
[(0 : 1 + O(11^5) : 0), (7 + 10*11 + 4*11^3 + O(11^5) : 0 : 1 + O(11^5))]

Python

>>> from sage.all import *
>>> K = pAdicField(Integer(11), Integer(5))
>>> x = polygen(K)
>>> C = HyperellipticCurve(x**Integer(5) + Integer(33)/Integer(16)*x**Integer(4) + Integer(3)/Integer(4)*x**Integer(3) + Integer(3)/Integer(8)*x**Integer(2) - Integer(1)/Integer(4)*x + Integer(1)/Integer(16))
>>> C.weierstrass_points()
[(0 : 1 + O(11^5) : 0), (7 + 10*11 + 4*11^3 + O(11^5) : 0 : 1 + O(11^5))]