Weierstrass \(\wp\)-function for elliptic curves - Elliptic curves

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# Weierstrass \(\wp\)-function for elliptic curves

The Weierstrass \(\wp\) function associated to an elliptic curve over a field \(k\) is a Laurent series of the form

\[\wp(z) = \frac{1}{z^2} + c_2 \cdot z^2 + c_4 \cdot z^4 + \cdots.\]

If the field is contained in \(\mathbb{C}\), then this is the series expansion of the map from \(\mathbb{C}\) to \(E(\mathbb{C})\) whose kernel is the period lattice of \(E\).

Over other fields, like finite fields, this still makes sense as a formal power series with coefficients in \(k\) - at least its first \(p-2\) coefficients where \(p\) is the characteristic of \(k\). It can be defined via the formal group as \(x+c\) in the variable \(z=\log_E(t)\) for a constant \(c\) such that the constant term \(c_0\) in \(\wp(z)\) is zero.

EXAMPLES:

Sage

    sage: E = EllipticCurve([0,1])
    sage: E.weierstrass_p()
    z^-2 - 1/7*z^4 + 1/637*z^10 - 1/84721*z^16 + O(z^20)

Python

    >>> from sage.all import *
    >>> E = EllipticCurve([Integer(0),Integer(1)])
    >>> E.weierstrass_p()
    z^-2 - 1/7*z^4 + 1/637*z^10 - 1/84721*z^16 + O(z^20)

REFERENCES:

- [BMSS2006]

AUTHORS:

- Dan Shumov 04/09: original implementation

- Chris Wuthrich 11/09: major restructuring

- Jeroen Demeyer (2014-03-06): code clean up, fix characteristic bound for quadratic algorithm (see Issue #15855)

sage.schemes.elliptic_curves.ell_wp.compute_wp_fast(k, A, B, m)

  Compute the Weierstrass function of an elliptic curve defined by short Weierstrass model: \(y^2 = x^3 + Ax + B\). It does this with as fast as polynomial of degree \(m\) can be multiplied together in the base ring, i.e. \(O(M(n))\) in the notation of [BMSS2006].

  Let \(p\) be the characteristic of the underlying field: Then we must have either \(p=0\), or \(p > m + 3\).

  INPUT:

  - `k` – the base field of the curve

  - `A` – and

  - `B` – as the coefficients of the short Weierstrass model \(y^2 = x^3 +Ax +B\), and

  - `m` – the precision to which the function is computed to

  OUTPUT: the Weierstrass \(\wp\) function as a Laurent series to precision \(m\)

  ALGORITHM:

  This function uses the algorithm described in section 3.3 of [BMSS2006].

  EXAMPLES:

  Sage

      sage: from sage.schemes.elliptic_curves.ell_wp import compute_wp_fast
      sage: compute_wp_fast(QQ, 1, 8, 7)
      z^-2 - 1/5*z^2 - 8/7*z^4 + 1/75*z^6 + O(z^7)

      sage: k = GF(37)
      sage: compute_wp_fast(k, k(1), k(8), 5)
      z^-2 + 22*z^2 + 20*z^4 + O(z^5)

  Python

      >>> from sage.all import *
      >>> from sage.schemes.elliptic_curves.ell_wp import compute_wp_fast
      >>> compute_wp_fast(QQ, Integer(1), Integer(8), Integer(7))
      z^-2 - 1/5*z^2 - 8/7*z^4 + 1/75*z^6 + O(z^7)

      >>> k = GF(Integer(37))
      >>> compute_wp_fast(k, k(Integer(1)), k(Integer(8)), Integer(5))
      z^-2 + 22*z^2 + 20*z^4 + O(z^5)

sage.schemes.elliptic_curves.ell_wp.compute_wp_pari(E, prec)

  Compute the Weierstrass \(\wp\)-function with the `ellwp` function from PARI.

  EXAMPLES:

  Sage

      sage: E = EllipticCurve([0,1])
      sage: from sage.schemes.elliptic_curves.ell_wp import compute_wp_pari
      sage: compute_wp_pari(E, prec=20)
      z^-2 - 1/7*z^4 + 1/637*z^10 - 1/84721*z^16 + O(z^20)
      sage: compute_wp_pari(E, prec=30)
      z^-2 - 1/7*z^4 + 1/637*z^10 - 1/84721*z^16
      + 3/38548055*z^22 - 4/8364927935*z^28 + O(z^30)

  Python

      >>> from sage.all import *
      >>> E = EllipticCurve([Integer(0),Integer(1)])
      >>> from sage.schemes.elliptic_curves.ell_wp import compute_wp_pari
      >>> compute_wp_pari(E, prec=Integer(20))
      z^-2 - 1/7*z^4 + 1/637*z^10 - 1/84721*z^16 + O(z^20)
      >>> compute_wp_pari(E, prec=Integer(30))
      z^-2 - 1/7*z^4 + 1/637*z^10 - 1/84721*z^16
      + 3/38548055*z^22 - 4/8364927935*z^28 + O(z^30)

sage.schemes.elliptic_curves.ell_wp.compute_wp_quadratic(k, A, B, prec)

  Compute the truncated Weierstrass function of an elliptic curve defined by short Weierstrass model: \(y^2 = x^3 + Ax + B\). Uses an algorithm that is of complexity \(O(prec^2)\).

  Let p be the characteristic of the underlying field. Then we must have either p = 0, or p > prec + 2.

  INPUT:

  - `k` – the field of definition of the curve

  - `A` – and

  - `B` – the coefficients of the elliptic curve

  - `prec` – the precision to which we compute the series

  OUTPUT:

  A Laurent series approximating the Weierstrass \(\wp\)-function to precision `prec`.

  ALGORITHM:

  This function uses the algorithm described in section 3.2 of [BMSS2006].

  EXAMPLES:

  Sage

      sage: E = EllipticCurve([7,0])
      sage: E.weierstrass_p(prec=10, algorithm='quadratic')
      z^-2 - 7/5*z^2 + 49/75*z^6 + O(z^10)

      sage: E = EllipticCurve(GF(103), [1,2])
      sage: E.weierstrass_p(algorithm='quadratic')
      z^-2 + 41*z^2 + 88*z^4 + 11*z^6 + 57*z^8 + 55*z^10 + 73*z^12
       + 11*z^14 + 17*z^16 + 50*z^18 + O(z^20)

      sage: from sage.schemes.elliptic_curves.ell_wp import compute_wp_quadratic
      sage: compute_wp_quadratic(E.base_ring(), E.a4(), E.a6(), prec=10)
      z^-2 + 41*z^2 + 88*z^4 + 11*z^6 + 57*z^8 + O(z^10)

  Python

      >>> from sage.all import *
      >>> E = EllipticCurve([Integer(7),Integer(0)])
      >>> E.weierstrass_p(prec=Integer(10), algorithm='quadratic')
      z^-2 - 7/5*z^2 + 49/75*z^6 + O(z^10)

      >>> E = EllipticCurve(GF(Integer(103)), [Integer(1),Integer(2)])
      >>> E.weierstrass_p(algorithm='quadratic')
      z^-2 + 41*z^2 + 88*z^4 + 11*z^6 + 57*z^8 + 55*z^10 + 73*z^12
       + 11*z^14 + 17*z^16 + 50*z^18 + O(z^20)

      >>> from sage.schemes.elliptic_curves.ell_wp import compute_wp_quadratic
      >>> compute_wp_quadratic(E.base_ring(), E.a4(), E.a6(), prec=Integer(10))
      z^-2 + 41*z^2 + 88*z^4 + 11*z^6 + 57*z^8 + O(z^10)

sage.schemes.elliptic_curves.ell_wp.solve_linear_differential_system(a, b, c, alpha)

  Solve a system of linear differential equations: \(af' + bf = c\) and \(f'(0) = \alpha\) where \(a\), \(b\), and \(c\) are power series in one variable and \(\alpha\) is a constant in the coefficient ring.

  ALGORITHM:

  due to Brent and Kung ‘78.

  EXAMPLES:

  Sage

      sage: from sage.schemes.elliptic_curves.ell_wp import solve_linear_differential_system
      sage: k = GF(17)
      sage: R.<x> = PowerSeriesRing(k)
      sage: a = 1 + x + O(x^7); b = x + O(x^7); c = 1 + x^3 + O(x^7); alpha = k(3)
      sage: f = solve_linear_differential_system(a, b, c, alpha)
      sage: f
      3 + x + 15*x^2 + x^3 + 10*x^5 + 3*x^6 + 13*x^7 + O(x^8)
      sage: a*f.derivative() + b*f - c
      O(x^7)
      sage: f(0) == alpha
      True

  Python

      >>> from sage.all import *
      >>> from sage.schemes.elliptic_curves.ell_wp import solve_linear_differential_system
      >>> k = GF(Integer(17))
      >>> R = PowerSeriesRing(k, names=('x',)); (x,) = R._first_ngens(1)
      >>> a = Integer(1) + x + O(x**Integer(7)); b = x + O(x**Integer(7)); c = Integer(1) + x**Integer(3) + O(x**Integer(7)); alpha = k(Integer(3))
      >>> f = solve_linear_differential_system(a, b, c, alpha)
      >>> f
      3 + x + 15*x^2 + x^3 + 10*x^5 + 3*x^6 + 13*x^7 + O(x^8)
      >>> a*f.derivative() + b*f - c
      O(x^7)
      >>> f(Integer(0)) == alpha
      True

sage.schemes.elliptic_curves.ell_wp.weierstrass_p(E, prec=20, algorithm=None)

  Compute the Weierstrass \(\wp\)-function on an elliptic curve.

  INPUT:

  - `E` – an elliptic curve

  - `prec` – precision

  - `algorithm` – string or `None` (default: `None`); a choice of algorithm among `'pari'`, `'fast'`, `'quadratic'`, or `None` to let this function determine the best algorithm to use

  OUTPUT:

  a Laurent series in one variable \(z\) with coefficients in the base field \(k\) of \(E\).

  EXAMPLES:

  Sage

      sage: E = EllipticCurve('11a1')
      sage: E.weierstrass_p(prec=10)
      z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + 77531/41580*z^8 + O(z^10)
      sage: E.weierstrass_p(prec=8)
      z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + O(z^8)
      sage: Esh = E.short_weierstrass_model()
      sage: Esh.weierstrass_p(prec=8)
      z^-2 + 13392/5*z^2 + 1080432/7*z^4 + 59781888/25*z^6 + O(z^8)

      sage: E.weierstrass_p(prec=8, algorithm='pari')
      z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + O(z^8)
      sage: E.weierstrass_p(prec=8, algorithm='quadratic')
      z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + O(z^8)

      sage: k = GF(11)
      sage: E = EllipticCurve(k, [1,1])
      sage: E.weierstrass_p(prec=6, algorithm='fast')
      z^-2 + 2*z^2 + 3*z^4 + O(z^6)
      sage: E.weierstrass_p(prec=7, algorithm='fast')
      Traceback (most recent call last):
      ...
      ValueError: for computing the Weierstrass p-function via the fast algorithm,
      the characteristic (11) of the underlying field must be greater than prec + 4 = 11
      sage: E.weierstrass_p(prec=8)
      z^-2 + 2*z^2 + 3*z^4 + 5*z^6 + O(z^8)
      sage: E.weierstrass_p(prec=8, algorithm='quadratic')
      z^-2 + 2*z^2 + 3*z^4 + 5*z^6 + O(z^8)
      sage: E.weierstrass_p(prec=8, algorithm='pari')
      z^-2 + 2*z^2 + 3*z^4 + 5*z^6 + O(z^8)
      sage: E.weierstrass_p(prec=9)
      Traceback (most recent call last):
      ...
      NotImplementedError: currently no algorithms for computing the Weierstrass
      p-function for that characteristic / precision pair is implemented.
      Lower the precision below char(k) - 2
      sage: E.weierstrass_p(prec=9, algorithm='quadratic')
      Traceback (most recent call last):
      ...
      ValueError: for computing the Weierstrass p-function via the quadratic
      algorithm, the characteristic (11) of the underlying field must be greater
      than prec + 2 = 11
      sage: E.weierstrass_p(prec=9, algorithm='pari')
      Traceback (most recent call last):
      ...
      ValueError: for computing the Weierstrass p-function via pari, the
      characteristic (11) of the underlying field must be greater than prec + 2 = 11

  Python

      >>> from sage.all import *
      >>> E = EllipticCurve('11a1')
      >>> E.weierstrass_p(prec=Integer(10))
      z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + 77531/41580*z^8 + O(z^10)
      >>> E.weierstrass_p(prec=Integer(8))
      z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + O(z^8)
      >>> Esh = E.short_weierstrass_model()
      >>> Esh.weierstrass_p(prec=Integer(8))
      z^-2 + 13392/5*z^2 + 1080432/7*z^4 + 59781888/25*z^6 + O(z^8)

      >>> E.weierstrass_p(prec=Integer(8), algorithm='pari')
      z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + O(z^8)
      >>> E.weierstrass_p(prec=Integer(8), algorithm='quadratic')
      z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + O(z^8)

      >>> k = GF(Integer(11))
      >>> E = EllipticCurve(k, [Integer(1),Integer(1)])
      >>> E.weierstrass_p(prec=Integer(6), algorithm='fast')
      z^-2 + 2*z^2 + 3*z^4 + O(z^6)
      >>> E.weierstrass_p(prec=Integer(7), algorithm='fast')
      Traceback (most recent call last):
      ...
      ValueError: for computing the Weierstrass p-function via the fast algorithm,
      the characteristic (11) of the underlying field must be greater than prec + 4 = 11
      >>> E.weierstrass_p(prec=Integer(8))
      z^-2 + 2*z^2 + 3*z^4 + 5*z^6 + O(z^8)
      >>> E.weierstrass_p(prec=Integer(8), algorithm='quadratic')
      z^-2 + 2*z^2 + 3*z^4 + 5*z^6 + O(z^8)
      >>> E.weierstrass_p(prec=Integer(8), algorithm='pari')
      z^-2 + 2*z^2 + 3*z^4 + 5*z^6 + O(z^8)
      >>> E.weierstrass_p(prec=Integer(9))
      Traceback (most recent call last):
      ...
      NotImplementedError: currently no algorithms for computing the Weierstrass
      p-function for that characteristic / precision pair is implemented.
      Lower the precision below char(k) - 2
      >>> E.weierstrass_p(prec=Integer(9), algorithm='quadratic')
      Traceback (most recent call last):
      ...
      ValueError: for computing the Weierstrass p-function via the quadratic
      algorithm, the characteristic (11) of the underlying field must be greater
      than prec + 2 = 11
      >>> E.weierstrass_p(prec=Integer(9), algorithm='pari')
      Traceback (most recent call last):
      ...
      ValueError: for computing the Weierstrass p-function via pari, the
      characteristic (11) of the underlying field must be greater than prec + 2 = 11