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)

Bases: sage.schemes.hyperelliptic_curves.hyperelliptic_generic.HyperellipticCurve_generic, sage.schemes.curves.projective_curve.ProjectivePlaneCurve_field

P_to_S(P, S)

Given a finite Weierstrass point \(P\) and a point \(S\) in the same disc, computes 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: 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))

AUTHOR:

• Jennifer Balakrishnan

S_to_Q(S, Q)

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 \(\QQ_p\) (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: 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

coleman_integral(w, P, Q, algorithm='None')

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)

coleman_integral_P_to_S(w, P, S)

Given a finite Weierstrass point \(P\) and a point \(S\) in the same disc, computes 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: 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

AUTHOR:

• Jennifer Balakrishnan

coleman_integral_S_to_Q(w, S, Q)

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: 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

coleman_integral_from_weierstrass_via_boundary(w, P, Q, d)

Computes 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: 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)

AUTHOR:

• Jennifer Balakrishnan

coleman_integrals_on_basis(P, Q, algorithm=None)

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

coleman_integrals_on_basis_hyperelliptic(P, Q, algorithm=None)

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

curve_over_ram_extn(deg)

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: 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

AUTHOR:

• Jennifer Balakrishnan

find_char_zero_weier_point(Q)

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: 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)

AUTHOR:

• Jennifer Balakrishnan

frobenius(P=None)

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

EXAMPLES:

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) : 5 + 9*11 + 2*11^2 + 2*11^3 + 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) : 5 + 9*11 + 6*11^2 + 11^3 + 6*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) : 5 + 9*11 + 6*11^2 + 11^3 + 6*11^4 + O(11^5) : 1 + O(11^5))

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))

AUTHORS:

• Robert Bradshaw and Jennifer Balakrishnan (2010-02)

get_boundary_point(curve_over_extn, P)

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: 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))

AUTHOR:

• Jennifer Balakrishnan

is_in_weierstrass_disc(P)

Checks if \(P\) is in a Weierstrass disc

EXAMPLES:

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) : 2*5 + 4*5^3 + 3*5^4 + 5^5 + 3*5^6 + O(5^7) : 1 + O(5^8))
sage: HK.is_in_weierstrass_disc(T)
True

AUTHOR:

• Jennifer Balakrishnan (2010-02)

is_same_disc(P, Q)

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

EXAMPLES:

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

is_weierstrass(P)

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

EXAMPLES:

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) : 2*5 + 4*5^3 + 3*5^4 + 5^5 + 3*5^6 + O(5^7) : 1 + O(5^8))
sage: HK.is_weierstrass(T)
False

AUTHOR:

• Jennifer Balakrishnan (2010-02)

local_analytic_interpolation(P, Q)

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 and Q points on self in the same residue disc

OUTPUT:

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

1. \(X(0) = P\) and \(X(1) = Q\) if \(P, Q\) are not in the infinite disc

2. \(X(P[0]^g/P[1]) = P\) and \(X(Q[0]^g/Q[1]) = Q\) if \(P, Q\) are in the infinite disc

EXAMPLES:

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

A non-Weierstrass disc:

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)

A finite Weierstrass disc:

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)

The infinite disc:

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

An error if points are not in the same disc:

sage: x,y,z = HK.local_analytic_interpolation(P,HK(1,0))
Traceback (most recent call last):
...
ValueError: (5^-2 + O(5^6) : 5^-3 + 4*5^2 + 5^3 + 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)

Takes 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: 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)

AUTHOR:

• Jennifer Balakrishnan

residue_disc(P)

Gives the residue disc of \(P\)

EXAMPLES:

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)

AUTHOR:

• Jennifer Balakrishnan

teichmuller(P)

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: 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)

tiny_integrals(F, P, Q)

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 a list of functions \(f_i\)

• P a point on self

• Q a point on self (in the same residue disc as P)

OUTPUT:

The integrals \(\int_P^Q f_i dx/2y\)

EXAMPLES:

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

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)
(3*11^3 + 7*11^4 + 4*11^5 + 7*11^6 + 5*11^7 + O(11^8))

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

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

tiny_integrals_on_basis(P, Q)

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 a point on self

• Q a 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: 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))

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)
(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))

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

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

weierstrass_points()

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: 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))]