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: 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)

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 trac ticket #15855)

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

Computes 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: 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)

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

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

EXAMPLES:

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)

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: 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)

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

Solves 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: 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

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: 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