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# -*- coding: utf-8 -*- r""" Modular parametrization of elliptic curves over `\QQ`
By the work of Taylor--Wiles et al. it is known that there is a surjective morphism
.. MATH::
\phi_E: X_0(N) \rightarrow E.
from the modular curve `X_0(N)`, where `N` is the conductor of `E`. The map sends the cusp `\infty` to the origin of `E`.
EXAMPLES::
sage: phi = EllipticCurve('11a1').modular_parametrization() sage: phi Modular parameterization from the upper half plane to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field sage: phi(0.5+CDF(I)) (285684.320516... + 7.0...e-11*I : 1.526964169...e8 + 5.6...e-8*I : 1.00000000000000) sage: phi.power_series(prec = 7) (q^-2 + 2*q^-1 + 4 + 5*q + 8*q^2 + q^3 + 7*q^4 + O(q^5), -q^-3 - 3*q^-2 - 7*q^-1 - 13 - 17*q - 26*q^2 - 19*q^3 + O(q^4))
AUTHORS:
- Chris Wuthrich (02/10) - moved from ell_rational_field.py.
""" ###################################################################### # Copyright (C) 2010 William Stein <wstein@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # # This code is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU # General Public License for more details. # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ ###################################################################### from __future__ import absolute_import
from . import heegner
from sage.rings.all import (LaurentSeriesRing, RationalField, ComplexField, QQ) from sage.misc.misc import verbose
class ModularParameterization: r""" This class represents the modular parametrization of an elliptic curve
.. MATH::
\phi_E: X_0(N) \rightarrow E.
Evaluation is done by passing through the lattice representation of `E`.
EXAMPLES::
sage: phi = EllipticCurve('11a1').modular_parametrization() sage: phi Modular parameterization from the upper half plane to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
""" def __init__(self, E): r""" EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_rational_field import ModularParameterization sage: phi = ModularParameterization(EllipticCurve('389a')) sage: phi(CC.0/5) (27.1965586309057 : -144.727322178982 : 1.00000000000000)
sage: phi == loads(dumps(phi)) True """
def curve(self): """ Return the curve associated to this modular parametrization.
EXAMPLES::
sage: E = EllipticCurve('15a') sage: phi = E.modular_parametrization() sage: phi.curve() is E True """
def __repr__(self): """ TESTS::
sage: E = EllipticCurve('37a') sage: phi = E.modular_parametrization() sage: phi Modular parameterization from the upper half plane to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field sage: phi.__repr__() 'Modular parameterization from the upper half plane to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field' """
def __eq__(self, other): r""" Compares two modular parametrizations by simply comparing the elliptic curves.
EXAMPLES::
sage: E = EllipticCurve('37a1') sage: phi = E.modular_parametrization() sage: phi == phi True """ return False
def __ne__(self, other): """ Check whether ``self`` is not equal to ``other``.
EXAMPLES::
sage: E = EllipticCurve('37a1') sage: phi = E.modular_parametrization() sage: phi != phi False """
def __call__(self, z, prec=None): r""" Evaluate ``self`` at a point `z \in X_0(N)` where `z` is given by a representative in the upper half plane.
All computations are done with ``prec`` bits of precision. If ``prec`` is not given, use the precision of `z`.
EXAMPLES::
sage: E = EllipticCurve('37a') sage: phi = E.modular_parametrization() sage: phi((sqrt(7)*I - 17)/74, 53) (...e-16 - ...e-16*I : ...e-16 + ...e-16*I : 1.00000000000000)
Verify that the mapping is invariant under the action of `\Gamma_0(N)` on the upper half plane::
sage: E = EllipticCurve('11a') sage: phi = E.modular_parametrization() sage: tau = CC((1+1j)/5) sage: phi(tau) (-3.92181329652811 - 12.2578555525366*I : 44.9649874434872 + 14.3257120944681*I : 1.00000000000000) sage: phi(tau+1) (-3.92181329652810 - 12.2578555525366*I : 44.9649874434872 + 14.3257120944681*I : 1.00000000000000) sage: phi((6*tau+1) / (11*tau+2)) (-3.9218132965285... - 12.2578555525369*I : 44.964987443489... + 14.325712094467...*I : 1.00000000000000)
We can also apply the modular parametrization to a Heegner point on `X_0(N)`::
sage: H = heegner_points(389,-7,5); H All Heegner points of conductor 5 on X_0(389) associated to QQ[sqrt(-7)] sage: x = H[0]; x Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7 and conductor 5 on X_0(389) sage: E = EllipticCurve('389a'); phi = E.modular_parametrization() sage: phi(x) Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 389 sage: phi(x).quadratic_form() 389*x^2 + 147*x*y + 14*y^2
ALGORITHM:
Integrate the modular form attached to this elliptic curve from `z` to `\infty` to get a point on the lattice representation of `E`, then use the Weierstrass `\wp` function to map it to the curve itself. """ # Map to the CC of CC/PeriodLattice. # Map to E via Weierstrass P
def map_to_complex_numbers(self, z, prec=None): """ Evaluate ``self`` at a point `z \in X_0(N)` where `z` is given by a representative in the upper half plane, returning a point in the complex numbers.
All computations are done with ``prec`` bits of precision. If ``prec`` is not given, use the precision of `z`. Use self(z) to compute the image of z on the Weierstrass equation of the curve.
EXAMPLES::
sage: E = EllipticCurve('37a'); phi = E.modular_parametrization() sage: tau = (sqrt(7)*I - 17)/74 sage: z = phi.map_to_complex_numbers(tau); z 0.929592715285395 - 1.22569469099340*I sage: E.elliptic_exponential(z) (...e-16 - ...e-16*I : ...e-16 + ...e-16*I : 1.00000000000000) sage: phi(tau) (...e-16 - ...e-16*I : ...e-16 + ...e-16*I : 1.00000000000000) """ raise NotImplementedError raise ValueError("Point must be in the upper half plane") # TODO: for very small imaginary part, maybe try to transform under # \Gamma_0(N) to a better representative? # nterms'th term is less than 2**-(prec+10) (c.f. eclib code) # Use Horner's rule to sum the integral of the form
def power_series(self, prec=20): r""" Return the power series of this modular parametrization.
The curve must be a a minimal model. The prec parameter determines the number of significant terms. This means that X will be given up to O(q^(prec-2)) and Y will be given up to O(q^(prec-3)).
OUTPUT: A list of two Laurent series ``[X(x),Y(x)]`` of degrees -2, -3 respectively, which satisfy the equation of the elliptic curve. There are modular functions on `\Gamma_0(N)` where `N` is the conductor.
The series should satisfy the differential equation
.. MATH::
\frac{\mathrm{d}X}{2Y + a_1 X + a_3} = \frac{f(q)\, \mathrm{d}q}{q}
where `f` is ``self.curve().q_expansion()``.
EXAMPLES::
sage: E = EllipticCurve('389a1') sage: phi = E.modular_parametrization() sage: X,Y = phi.power_series(prec=10) sage: X q^-2 + 2*q^-1 + 4 + 7*q + 13*q^2 + 18*q^3 + 31*q^4 + 49*q^5 + 74*q^6 + 111*q^7 + O(q^8) sage: Y -q^-3 - 3*q^-2 - 8*q^-1 - 17 - 33*q - 61*q^2 - 110*q^3 - 186*q^4 - 320*q^5 - 528*q^6 + O(q^7) sage: X,Y = phi.power_series() sage: X q^-2 + 2*q^-1 + 4 + 7*q + 13*q^2 + 18*q^3 + 31*q^4 + 49*q^5 + 74*q^6 + 111*q^7 + 173*q^8 + 251*q^9 + 379*q^10 + 560*q^11 + 824*q^12 + 1199*q^13 + 1773*q^14 + 2548*q^15 + 3722*q^16 + 5374*q^17 + O(q^18) sage: Y -q^-3 - 3*q^-2 - 8*q^-1 - 17 - 33*q - 61*q^2 - 110*q^3 - 186*q^4 - 320*q^5 - 528*q^6 - 861*q^7 - 1383*q^8 - 2218*q^9 - 3472*q^10 - 5451*q^11 - 8447*q^12 - 13020*q^13 - 19923*q^14 - 30403*q^15 - 46003*q^16 + O(q^17)
The following should give 0, but only approximately::
sage: q = X.parent().gen() sage: E.defining_polynomial()(X,Y,1) + O(q^11) == 0 True
Note that below we have to change variable from `x` to `q`::
sage: a1,_,a3,_,_ = E.a_invariants() sage: f = E.q_expansion(17) sage: q = f.parent().gen() sage: f/q == (X.derivative()/(2*Y+a1*X+a3)) True """ raise NotImplementedError("only implemented for minimal curves") |