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, 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))
>>> 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, 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))
>>> 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 (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)
>>> 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: 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)
>>> 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: 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)
>>> 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: 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)
>>> 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: 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)
>>> 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: 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)
>>> 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, 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
>>> 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: 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)
>>> 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]#
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)
>>> 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]#
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) (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))
>>> 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: 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))
>>> 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: 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
>>> 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: 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)
>>> 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]#
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) (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))
>>> 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: 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))
>>> 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: 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
>>> 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: 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)
>>> 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: 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
>>> 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: 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)
>>> 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]#
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) : 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))
>>> 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: 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))
>>> 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: 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))
>>> 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]#
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) : 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
>>> 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]#
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
>>> 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]#
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) : 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
>>> 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 and 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: R.<x> = QQ['x'] sage: H = HyperellipticCurve(x^3-10*x+9) sage: K = Qp(5,8) sage: HK = H.change_ring(K)
>>> 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: 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)
>>> 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: 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)
>>> 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: 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
>>> 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: 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
>>> 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]#
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 extensionx0
– seeds the Newton iterationprec
– 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)
>>> 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]#
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)
>>> 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: 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)
>>> 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 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
>>> 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: 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))
>>> 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: 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
>>> 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 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))
>>> 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: 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))
>>> 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: 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
>>> 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: 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))]
>>> 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))]