Canonical heights for elliptic curves over number fields

Also, rigorous lower bounds for the canonical height of non-torsion points, implementing the algorithms in [CS2006] (over \(\QQ\)) and [Tho2010], which also refer to [CPS2006].

AUTHORS:

• Robert Bradshaw (2010): initial version

• John Cremona (2014): added many docstrings and doctests

class sage.schemes.elliptic_curves.height.EllipticCurveCanonicalHeight(E)

Bases: object

Class for computing canonical heights of points on elliptic curves defined over number fields, including rigorous lower bounds for the canonical height of non-torsion points.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.height import EllipticCurveCanonicalHeight
sage: E = EllipticCurve([0,0,0,0,1])
sage: EllipticCurveCanonicalHeight(E)
EllipticCurveCanonicalHeight object associated to
 Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

Normally this object would be created like this:

sage: E.height_function()
EllipticCurveCanonicalHeight object associated to
 Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

B(n, mu)

Return the value \(B_n(\mu)\).

INPUT:

• n (int) – a positive integer

• mu (real) – a positive real number

OUTPUT:

The real value \(B_n(\mu)\) as defined in [Tho2010], section 5.

EXAMPLES:

Example 10.2 from [Tho2010]:

sage: K.<i> = QuadraticField(-1)                                            # needs sage.rings.number_field
sage: E = EllipticCurve([0, 1-i, i, -i, 0])                                 # needs sage.rings.number_field
sage: H = E.height_function()                                               # needs sage.rings.number_field

In [Tho2010] the value is given as 0.772:

sage: RealField(12)( H.B(5, 0.01) )                                         # needs sage.rings.number_field
0.777

DE(n)

Return the value \(D_E(n)\).

INPUT:

• n (int) – a positive integer

OUTPUT:

The value \(D_E(n)\) as defined in [Tho2010], section 4.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, 1+5*i, 3+i])
sage: H = E.height_function()
sage: [H.DE(n) for n in srange(1,6)]
[0, 2*log(5) + 2*log(2), 0, 2*log(13) + 2*log(5) + 4*log(2), 0]

ME()

Return the norm of the ideal \(M_E\).

OUTPUT:

The norm of the ideal \(M_E\) as defined in [Tho2010], section 3.1. This is \(1\) if \(E\) is a global minimal model, and in general measures the non-minimality of \(E\).

EXAMPLES:

sage: K.<i> = QuadraticField(-1)                                            # needs sage.rings.number_field
sage: E = EllipticCurve([0, 0, 0, 1+5*i, 3+i])                              # needs sage.rings.number_field
sage: H = E.height_function()                                               # needs sage.rings.number_field
sage: H.ME()                                                                # needs sage.rings.number_field
1
sage: E = EllipticCurve([0,0,0,0,1])
sage: E.height_function().ME()
1
sage: E = EllipticCurve([0,0,0,0,64])
sage: E.height_function().ME()
4096
sage: E.discriminant()/E.minimal_model().discriminant()
4096

S(xi1, xi2, v)

Return the union of intervals \(S^{(v)}(\xi_1,\xi_2)\).

INPUT:

• xi1, xi2 (real) – real numbers with \(\xi_1\le\xi_2\).

• v (embedding) – a real embedding of the field.

OUTPUT:

The union of intervals \(S^{(v)}(\xi_1,\xi_2)\) defined in [Tho2010] section 6.1.

EXAMPLES:

An example over \(\QQ\):

sage: E = EllipticCurve('389a')
sage: v = QQ.places()[0]
sage: H = E.height_function()
sage: H.S(2, 3, v)
([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105])

An example over a number field:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0,0,0,0,a])
sage: v = K.real_places()[0]
sage: H = E.height_function()
sage: H.S(9, 10, v)
([0.078119444725347..., 0.082342373201640...] U [0.91765762679836..., 0.92188055527465...])

Sn(xi1, xi2, n, v)

Return the union of intervals \(S_n^{(v)}(\xi_1,\xi_2)\).

INPUT:

• xi1, xi2 (real) – real numbers with \(\xi_1\le\xi_2\).

• n (integer) – a positive integer.

• v (embedding) – a real embedding of the field.

OUTPUT:

The union of intervals \(S_n^{(v)}(\xi_1,\xi_2)\) defined in [Tho2010] (Lemma 6.1).

EXAMPLES:

An example over \(\QQ\):

sage: E = EllipticCurve('389a')
sage: v = QQ.places()[0]
sage: H = E.height_function()
sage: H.S(2, 3, v), H.Sn(2, 3, 1, v)
(([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105]),
([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105]))
sage: H.Sn(2, 3, 6, v)
([0.0374187795653158, 0.0457574702661884] U [0.120909196400478, 0.129247887101351] U [0.204085446231982, 0.212424136932855] U [0.287575863067145, 0.295914553768017] U [0.370752112898649, 0.379090803599522] U [0.454242529733812, 0.462581220434684] U [0.537418779565316, 0.545757470266188] U [0.620909196400478, 0.629247887101351] U [0.704085446231982, 0.712424136932855] U [0.787575863067145, 0.795914553768017] U [0.870752112898649, 0.879090803599522] U [0.954242529733812, 0.962581220434684])

An example over a number field:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0,0,0,0,a])
sage: v = K.real_places()[0]
sage: H = E.height_function()
sage: H.S(2, 3, v), H.Sn(2, 3, 1, v)
(([0.142172065860075, 0.172845716928584] U [0.827154283071416, 0.857827934139925]),
 ([0.142172065860075, 0.172845716928584] U [0.827154283071416, 0.857827934139925]))
sage: H.Sn(2, 3, 6, v)
([0.0236953443100124, 0.0288076194880974] U [0.137859047178569, 0.142971322356654] U [0.190362010976679, 0.195474286154764] U [0.304525713845236, 0.309637989023321] U [0.357028677643346, 0.362140952821431] U [0.471192380511903, 0.476304655689988] U [0.523695344310012, 0.528807619488097] U [0.637859047178569, 0.642971322356654] U [0.690362010976679, 0.695474286154764] U [0.804525713845236, 0.809637989023321] U [0.857028677643346, 0.862140952821431] U [0.971192380511903, 0.976304655689988])

alpha(v, tol=0.01)

Return the constant \(\alpha_v\) associated to the embedding v.

INPUT:

• v – an embedding of the base field into \(\RR\) or \(\CC\)

OUTPUT:

The constant \(\alpha_v\). In the notation of [CPS2006] and [Tho2010] (section 3.2), \(\alpha_v^3=\epsilon_v\). The result is cached since it only depends on the curve.

EXAMPLES:

Example 1 from [CPS2006]:

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, 1 + 5*i, 3 + i])
sage: H = E.height_function()
sage: alpha = H.alpha(K.places()[0])
sage: alpha
1.12272013439355

Compare with \(\log(\epsilon_v)=0.344562...\) in [CPS2006]:

sage: 3*alpha.log()                                                         # needs sage.rings.number_field
0.347263296676126

base_field()

Return the base field.

EXAMPLES:

sage: E = EllipticCurve([0,0,0,0,1])
sage: H = E.height_function()
sage: H.base_field()
Rational Field

complex_intersection_is_empty(Bk, v, verbose=False, use_half=True)

Returns True iff an intersection of \(T_n^{(v)}\) sets is empty.

INPUT:

• Bk (list) – a list of reals.

• v (embedding) – a complex embedding of the number field.

• verbose (boolean, default False) – verbosity flag.

• use_half (boolean, default False) – if True, use only half the fundamental region.

OUTPUT:

True or False, according as the intersection of the unions of intervals \(T_n^{(v)}(-b,b)\) for \(b\) in the list Bk (see [Tho2010], section 7) is empty or not. When Bk is the list of \(b=\sqrt{B_n(\mu)}\) for \(n=1,2,3,\dots\) for some \(\mu>0\) this means that all non-torsion points on \(E\) with everywhere good reduction have canonical height strictly greater than \(\mu\), by [Tho2010], Proposition 7.8.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0,0,0,0,a])
sage: v = K.complex_embeddings()[0]
sage: H = E.height_function()

The following two lines prove that the heights of non-torsion points on \(E\) with everywhere good reduction have canonical height strictly greater than 0.02, but fail to prove the same for 0.03. For the first proof, using only \(n=1,2,3\) is not sufficient:

sage: H.complex_intersection_is_empty([H.B(n,0.02) for n in [1,2,3]], v)  # long time, needs sage.rings.number_field
False
sage: H.complex_intersection_is_empty([H.B(n,0.02) for n in [1,2,3,4]], v)  # needs sage.rings.number_field
True
sage: H.complex_intersection_is_empty([H.B(n,0.03) for n in [1,2,3,4]], v)  # long time, needs sage.rings.number_field
False

Using \(n\le6\) enables us to prove the lower bound 0.03. Note that it takes longer when the result is False than when it is True:

sage: H.complex_intersection_is_empty([H.B(n,0.03) for n in [1..6]], v)     # needs sage.rings.number_field
True

curve()

Return the elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([0,0,0,0,1])
sage: H = E.height_function()
sage: H.curve()
Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

e_p(p)

Return the exponent of the group over the residue field at p.

INPUT:

• p – a prime ideal of \(K\) (or a prime number if \(K=\QQ\)).

OUTPUT:

A positive integer \(e_p\), the exponent of the group of nonsingular points on the reduction of the elliptic curve modulo \(p\). The result is cached.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, 1 + 5*i, 3 + i])
sage: H = E.height_function()
sage: H.e_p(K.prime_above(2))
2
sage: H.e_p(K.prime_above(3))
10
sage: H.e_p(K.prime_above(5))
9
sage: E.conductor().norm().factor()
2^10 * 20921
sage: p1, p2 = K.primes_above(20921)
sage: E.local_data(p1)
Local data at Fractional ideal (-40*i + 139):
Reduction type: bad split multiplicative
...
sage: H.e_p(p1)
20920
sage: E.local_data(p2)
Local data at Fractional ideal (40*i + 139):
Reduction type: good
...
sage: H.e_p(p2)
20815

fk_intervals(v=None, N=20, domain=Complex Interval Field with 53 bits of precision)

Return a function approximating the Weierstrass function, with error.

INPUT:

• v (embedding) – an embedding of the number field. If None (default) use the real embedding if the field is \(\QQ\) and raise an error for other fields.

• N (int) – The number of terms to use in the \(q\)-expansion of \(\wp\).

• domain (complex field) – the model of \(\CC\) to use, for example CDF of CIF (default).

OUTPUT:

A pair of functions fk, err which can be evaluated at complex numbers \(z\) (in the correct domain) to give an approximation to \(\wp(z)\) and an upper bound on the error, respectively. The Weierstrass function returned is with respect to the normalised lattice \([1,\tau]\) associated to the given embedding.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: L = E.period_lattice()
sage: w1, w2 = L.normalised_basis()
sage: z = CDF(0.3, 0.4)

Compare the value give by the standard elliptic exponential (scaled since fk is with respect to the normalised lattice):

sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2 ** 2
-1.82543539306049 - 2.49336319992847*I

to the value given by this function, and see the error:

sage: fk, err = E.height_function().fk_intervals(N=10)
sage: fk(CIF(z))
-1.82543539306049? - 2.49336319992847?*I
sage: err(CIF(z))
2.71750621458744e-31

The same, but in the domain CDF instead of CIF:

sage: fk, err = E.height_function().fk_intervals(N=10, domain=CDF)
sage: fk(z)
-1.8254353930604... - 2.493363199928...*I

min(tol, n_max, verbose=False)

Returns a lower bound for all points of infinite order.

INPUT:

• tol – tolerance in output (see below).

• n_max – how many multiples to use in iteration.

• verbose (boolean, default False) – verbosity flag.

OUTPUT:

A positive real \(\mu\) for which it has been established rigorously that every point of infinite order on the elliptic curve (defined over its ground field) has canonical height greater than \(\mu\), and such that it is not possible (at least without increasing n_max) to prove the same for \(\mu\cdot\text{tol}\).

EXAMPLES:

Example 1 from [CS2006] (where the same lower bound of 0.1126 was given):

sage: E = EllipticCurve([1, 0, 1, 421152067, 105484554028056]) # 60490d1
sage: E.height_function().min(.0001, 5)
0.0011263287309893311

Example 10.1 from [Tho2010] (where a lower bound of 0.18 was given):

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, 91 - 26*i, -144 - 323*i])
sage: H = E.height_function()
sage: H.min(0.1, 4)                 # long time
0.1621049443313762

Example 10.2 from [Tho2010]:

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 1 - i, i, -i, 0])
sage: H = E.height_function()
sage: H.min(0.01, 5)                # long time
0.020153685521979152

In this example the point \(P=(0,0)\) has height 0.023 so our lower bound is quite good:

sage: P = E((0,0))                                                          # needs sage.rings.number_field
sage: P.height()                                                            # needs sage.rings.number_field
0.0230242154471211

Example 10.3 from [Tho2010] (where the same bound of 0.0625 is given):

sage: # needs sage.rings.number_field
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0, 0, 0, -3*a - a^2, a^2])
sage: H = E.height_function()
sage: H.min(0.1, 5)                 # long time
0.0625

More examples over \(\QQ\):

sage: E = EllipticCurve('37a')
sage: h = E.height_function()
sage: h.min(.01, 5)
0.03987318057488725
sage: E.gen(0).height()
0.0511114082399688

After base change the lower bound can decrease:

sage: K.<a> = QuadraticField(-5)                                            # needs sage.rings.number_field
sage: E.change_ring(K).height_function().min(0.5, 10)       # long time, needs sage.rings.number_field
0.04419417382415922

sage: E = EllipticCurve('389a')
sage: h = E.height_function()
sage: h.min(0.1, 5)
0.05731275270029196
sage: [P.height() for P in E.gens()]
[0.686667083305587, 0.327000773651605]

min_gr(tol, n_max, verbose=False)

Returns a lower bound for points of infinite order with good reduction.

INPUT:

• tol – tolerance in output (see below).

• n_max – how many multiples to use in iteration.

• verbose (boolean, default False) – verbosity flag.

OUTPUT:

A positive real \(\mu\) for which it has been established rigorously that every point of infinite order on the elliptic curve (defined over its ground field), which has good reduction at all primes, has canonical height greater than \(\mu\), and such that it is not possible (at least without increasing n_max) to prove the same for \(\mu\cdot\text{tol}\).

EXAMPLES:

Example 1 from [CS2006] (where a lower bound of 1.9865 was given):

sage: E = EllipticCurve([1, 0, 1, 421152067, 105484554028056]) # 60490d1
sage: E.height_function().min_gr(.0001, 5)
1.98684388146518

Example 10.1 from [Tho2010] (where a lower bound of 0.18 was given):

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, 91 - 26*i, -144 - 323*i])
sage: H = E.height_function()
sage: H.min_gr(0.1, 4)              # long time
0.1621049443313762

Example 10.2 from [Tho2010]:

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 1 - i, i, -i, 0])
sage: H = E.height_function()
sage: H.min_gr(0.01, 5)             # long time
0.020153685521979152

In this example the point \(P=(0,0)\) has height 0.023 so our lower bound is quite good:

sage: P = E((0,0))                                                          # needs sage.rings.number_field
sage: P.has_good_reduction()                                                # needs sage.rings.number_field
True
sage: P.height()                                                            # needs sage.rings.number_field
0.0230242154471211

Example 10.3 from [Tho2010] (where the same bound of 0.25 is given):

sage: # needs sage.rings.number_field
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0, 0, 0, -3*a - a^2, a^2])
sage: H = E.height_function()
sage: H.min_gr(0.1, 5)              # long time
0.25

psi(xi, v)

Return the normalised elliptic log of a point with this x-coordinate.

INPUT:

• xi (real) – the real x-coordinate of a point on the curve in the connected component with respect to a real embedding.

• v (embedding) – a real embedding of the number field.

OUTPUT:

A real number in the interval [0.5,1] giving the elliptic logarithm of a point on \(E\) with \(x\)-coordinate xi, on the connected component with respect to the embedding \(v\), scaled by the real period.

EXAMPLES:

An example over \(\QQ\):

sage: E = EllipticCurve('389a')
sage: v = QQ.places()[0]
sage: L = E.period_lattice(v)
sage: P = E.lift_x(10/9)
sage: L(P)
0.958696500380439
sage: L(P) / L.real_period()
0.384985810227885
sage: H = E.height_function()
sage: H.psi(10/9, v)
0.615014189772115

An example over a number field:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0,0,0,0,a])
sage: P = E.lift_x(1/3*a^2 + a + 5/3)
sage: v = K.real_places()[0]
sage: L = E.period_lattice(v)
sage: L(P)
3.51086196882538
sage: L(P) / L.real_period()
0.867385122699931
sage: xP = v(P.x())
sage: H = E.height_function()
sage: H.psi(xP, v)
0.867385122699931
sage: H.psi(1.23, v)
0.785854718241495

real_intersection_is_empty(Bk, v)

Returns True iff an intersection of \(S_n^{(v)}\) sets is empty.

INPUT:

• Bk (list) – a list of reals.

• v (embedding) – a real embedding of the number field.

OUTPUT:

True or False, according as the intersection of the unions of intervals \(S_n^{(v)}(-b,b)\) for \(b\) in the list Bk is empty or not. When Bk is the list of \(b=B_n(\mu)\) for \(n=1,2,3,\dots\) for some \(\mu>0\) this means that all non-torsion points on \(E\) with everywhere good reduction have canonical height strictly greater than \(\mu\), by [Tho2010], Proposition 6.2.

EXAMPLES:

An example over \(\QQ\):

sage: E = EllipticCurve('389a')
sage: v = QQ.places()[0]
sage: H = E.height_function()

The following two lines prove that the heights of non-torsion points on \(E\) with everywhere good reduction have canonical height strictly greater than 0.2, but fail to prove the same for 0.3:

sage: H.real_intersection_is_empty([H.B(n,0.2) for n in srange(1,10)], v)
True
sage: H.real_intersection_is_empty([H.B(n,0.3) for n in srange(1,10)], v)
False

An example over a number field:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0,0,0,0,a])
sage: v = K.real_places()[0]
sage: H = E.height_function()

The following two lines prove that the heights of non-torsion points on \(E\) with everywhere good reduction have canonical height strictly greater than 0.07, but fail to prove the same for 0.08:

sage: H.real_intersection_is_empty([H.B(n,0.07) for n in srange(1,5)], v)  # long time, needs sage.rings.number_field
True
sage: H.real_intersection_is_empty([H.B(n,0.08) for n in srange(1,5)], v)   # needs sage.rings.number_field
False

tau(v)

Return the normalised upper half-plane parameter \(\tau\) for the period lattice with respect to the embedding \(v\).

INPUT:

• v (embedding) – a real or complex embedding of the number field.

OUTPUT:

(Complex) \(\tau = \omega_1/\omega_2\) in the fundamental region of the upper half-plane.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: H = E.height_function()
sage: H.tau(QQ.places()[0])
1.22112736076463*I

test_mu(mu, N, verbose=True)

Return True if we can prove that \(\mu\) is a lower bound.

INPUT:

• mu (real) – a positive real number

• N (integer) – upper bound on the multiples to be used.

• verbose (boolean, default True) – verbosity flag.

OUTPUT:

True or False, according to whether we succeed in proving that \(\mu\) is a lower bound for the canonical heights of points of infinite order with everywhere good reduction.

Note

A True result is rigorous; False only means that the attempt failed: trying again with larger \(N\) may yield True.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2)                                          # needs sage.rings.number_field
sage: E = EllipticCurve([0,0,0,0,a])                                        # needs sage.rings.number_field
sage: H = E.height_function()                                               # needs sage.rings.number_field

This curve does have a point of good reduction whose canonical point is approximately 1.68:

sage: P = E.gens(lim3=5)[0]; P                                              # needs sage.rings.number_field
(1/3*a^2 + a + 5/3 : -2*a^2 - 4/3*a - 5/3 : 1)
sage: P.height()                                                            # needs sage.rings.number_field
1.68038085233673
sage: P.has_good_reduction()                                                # needs sage.rings.number_field
True

Using \(N=5\) we can prove that 0.1 is a lower bound (in fact we only need \(N=2\)), but not that 0.2 is:

sage: H.test_mu(0.1, 5)                                                     # needs sage.rings.number_field
B_1(0.100000000000000) = 1.51580969677387
B_2(0.100000000000000) = 0.932072561526720
True
sage: H.test_mu(0.2, 5)                                                     # needs sage.rings.number_field
B_1(0.200000000000000) = 2.04612906979932
B_2(0.200000000000000) = 3.09458988474327
B_3(0.200000000000000) = 27.6251108409484
B_4(0.200000000000000) = 1036.24722370223
B_5(0.200000000000000) = 3.67090854562318e6
False

Since 0.1 is a lower bound we can deduce that the point \(P\) is either primitive or divisible by either 2 or 3. In fact it is primitive:

sage: (P.height()/0.1).sqrt()                                               # needs sage.rings.number_field
4.09924487233530
sage: P.division_points(2)                                                  # needs sage.rings.number_field
[]
sage: P.division_points(3)                                                  # needs sage.rings.number_field
[]

wp_c(v)

Return a bound for the Weierstrass \(\wp\)-function.

INPUT:

• v (embedding) – a real or complex embedding of the number field.

OUTPUT:

(Real) \(c>0\) such that

\[|\wp(z) - z^-2| \le \frac{c^2|z|^2}{1-c|z|^2}\]

whenever \(c|z|^2<1\). Given the recurrence relations for the Laurent series expansion of \(\wp\), it is easy to see that there is such a constant \(c\). [Reference?]

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: H = E.height_function()
sage: H.wp_c(QQ.places()[0])
2.68744508779950

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, 1 + 5*i, 3 + i])
sage: H = E.height_function()
sage: H.wp_c(K.places()[0])
2.66213425640096

wp_intervals(v=None, N=20, abs_only=False)

Return a function approximating the Weierstrass function.

INPUT:

• v (embedding) – an embedding of the number field. If None (default) use the real embedding if the field is \(\QQ\) and raise an error for other fields.

• N (int, default 20) – The number of terms to use in the \(q\)-expansion of \(\wp\).

• abs_only (boolean, default False) – flag to determine whether (if True) the error adjustment should use the absolute value or (if False) the real and imaginary parts.

OUTPUT:

A function wp which can be evaluated at complex numbers \(z\) to give an approximation to \(\wp(z)\). The Weierstrass function returned is with respect to the normalised lattice \([1,\tau]\) associated to the given embedding. For \(z\) which are not near a lattice point the function fk is used, otherwise a better approximation is used.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: wp = E.height_function().wp_intervals()
sage: z = CDF(0.3, 0.4)
sage: wp(CIF(z))
-1.82543539306049? - 2.4933631999285?*I

sage: L = E.period_lattice()
sage: w1, w2 = L.normalised_basis()
sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2^2
-1.82543539306049 - 2.49336319992847*I

sage: z = CDF(0.3, 0.1)
sage: wp(CIF(z))
8.5918243572165? - 5.4751982004351?*I
sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2^2
8.59182435721650 - 5.47519820043503*I

wp_on_grid(v, N, half=False)

Return an array of the values of \(\wp\) on an \(N\times N\) grid.

INPUT:

• v (embedding) – an embedding of the number field.

• N (int) – The number of terms to use in the \(q\)-expansion of \(\wp\).

• half (boolean, default False) – if True, use an array of size \(N\times N/2\) instead of \(N\times N\).

OUTPUT:

An array of size either \(N\times N/2\) or \(N\times N\) whose \((i,j)\) entry is the value of the Weierstrass \(\wp\)-function at \((i+.5)/N + (j+.5)*\tau/N\), a grid of points in the fundamental region for the lattice \([1,\tau]\).

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: H = E.height_function()
sage: v = QQ.places()[0]

The array of values on the grid shows symmetry, since \(\wp\) is even:

sage: H.wp_on_grid(v,4)
array([[25.43920182,  5.28760943,  5.28760943, 25.43920182],
       [ 6.05099485,  1.83757786,  1.83757786,  6.05099485],
       [ 6.05099485,  1.83757786,  1.83757786,  6.05099485],
       [25.43920182,  5.28760943,  5.28760943, 25.43920182]])

The array of values on the half-grid:

sage: H.wp_on_grid(v,4,True)
array([[25.43920182,  5.28760943],
       [ 6.05099485,  1.83757786],
       [ 6.05099485,  1.83757786],
       [25.43920182,  5.28760943]])

class sage.schemes.elliptic_curves.height.UnionOfIntervals(endpoints)

Bases: object

A class representing a finite union of closed intervals in \(\RR\) which can be scaled, shifted, intersected, etc.

The intervals are represented as an ordered list of their endpoints, which may include \(-\infty\) and \(+\infty\).

EXAMPLES:

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: R = UnionOfIntervals([1, 2, 3, infinity]); R
([1, 2] U [3, +Infinity])
sage: R + 5
([6, 7] U [8, +Infinity])
sage: ~R
([-Infinity, 1] U [2, 3])
sage: ~R | (10*R + 100)
([-Infinity, 1] U [2, 3] U [110, 120] U [130, +Infinity])

Todo

Unify UnionOfIntervals with the class RealSet introduced by github issue #13125; see github issue #16063.

finite_endpoints()

Returns the finite endpoints of this union of intervals.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: UnionOfIntervals([0, 1]).finite_endpoints()
[0, 1]
sage: UnionOfIntervals([-infinity, 0, 1, infinity]).finite_endpoints()
[0, 1]

classmethod intersection(L)

Return the intersection of a list of UnionOfIntervals.

INPUT:

• L (list) – a list of UnionOfIntervals instances

OUTPUT:

A new UnionOfIntervals instance representing the intersection of the UnionOfIntervals in the list.

Note

This is a class method.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: A = UnionOfIntervals([1,3,5,7]); A
([1, 3] U [5, 7])
sage: B = A + 1; B
([2, 4] U [6, 8])
sage: A.intersection([A,B])
([2, 3] U [6, 7])

intervals()

Returns the intervals in self, as a list of 2-tuples.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: UnionOfIntervals(list(range(10))).intervals()
[(0, 1), (2, 3), (4, 5), (6, 7), (8, 9)]
sage: UnionOfIntervals([-infinity, pi, 17, infinity]).intervals()           # needs sage.symbolic
[(-Infinity, pi), (17, +Infinity)]

is_empty()

Returns whether self is empty.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: UnionOfIntervals([3, 4]).is_empty()
False
sage: all = UnionOfIntervals([-infinity, infinity])
sage: all.is_empty()
False
sage: (~all).is_empty()
True
sage: A = UnionOfIntervals([0, 1]) & UnionOfIntervals([2, 3])
sage: A.is_empty()
True

static join(L, condition)

Utility function to form the union or intersection of a list of UnionOfIntervals.

INPUT:

• L (list) – a list of UnionOfIntervals instances

• condition (function) – either any or all, or some other boolean function of a list of boolean values.

OUTPUT:

A new UnionOfIntervals instance representing the subset of ‘RR’ equal to those reals in any/all/condition of the UnionOfIntervals in the list.

Note

This is a static method for the class.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: A = UnionOfIntervals([1,3,5,7]); A
([1, 3] U [5, 7])
sage: B = A + 1; B
([2, 4] U [6, 8])
sage: A.join([A,B], any) # union
([1, 4] U [5, 8])
sage: A.join([A,B], all) # intersection
([2, 3] U [6, 7])
sage: A.join([A,B], sum) # symmetric difference
([1, 2] U [3, 4] U [5, 6] U [7, 8])

classmethod union(L)

Return the union of a list of UnionOfIntervals.

INPUT:

• L (list) – a list of UnionOfIntervals instances

OUTPUT:

A new UnionOfIntervals instance representing the union of the UnionOfIntervals in the list.

Note

This is a class method.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: A = UnionOfIntervals([1,3,5,7]); A
([1, 3] U [5, 7])
sage: B = A + 1; B
([2, 4] U [6, 8])
sage: A.union([A,B])
([1, 4] U [5, 8])

sage.schemes.elliptic_curves.height.eps(err, is_real)

Return a Real or Complex interval centered on 0 with radius err.

INPUT:

• err (real) – a positive real number, the radius of the interval

• is_real (boolean) – if True, returns a real interval in RIF, else a complex interval in CIF

OUTPUT:

An element of RIF or CIF (as specified), centered on 0, with given radius.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.height import eps
sage: eps(0.01, True)
0.0?
sage: eps(0.01, False)
0.0? + 0.0?*I

sage.schemes.elliptic_curves.height.inf_max_abs(f, g, D)

Returns \(\inf_D(\max(|f|, |g|))\).

INPUT:

• f, g (polynomials) – real univariate polynomials

• D (UnionOfIntervals) – a subset of \(\RR\)

OUTPUT:

A real number approximating the value of \(\inf_D(\max(|f|, |g|))\).

ALGORITHM:

The extreme values must occur at an endpoint of a subinterval of \(D\) or at a point where one of \(f\), \(f'\), \(g\), \(g'\), \(f\pm g\) is zero.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.height import inf_max_abs, UnionOfIntervals
sage: x = polygen(RR)
sage: f = (x-10)^4 + 1
sage: g = 2*x^3 + 100
sage: inf_max_abs(f, g, UnionOfIntervals([1,2,3,4,5,6]))
425.638201706391
sage: r0 = (f - g).roots()[0][0]
sage: r0
5.46053402234697
sage: max(abs(f(r0)), abs(g(r0)))
425.638201706391

sage.schemes.elliptic_curves.height.min_on_disk(f, tol, max_iter=10000)

Returns the minimum of a real-valued complex function on a square.

INPUT:

• f – a function from CIF to RIF

• tol (real) – a positive real number

• max_iter (integer, default 10000) – a positive integer bounding the number of iterations to be used

OUTPUT:

A 2-tuple \((s,t)\), where \(t=f(s)\) and \(s\) is a CIF element contained in the disk \(|z|\le1\), at which \(f\) takes its minimum value.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.height import min_on_disk
sage: f = lambda x: (x^2 + 100).abs()
sage: s, t = min_on_disk(f, 0.0001)
sage: s, f(s), t
(0.01? + 1.00?*I, 99.01?, 99.0000000000000)

sage.schemes.elliptic_curves.height.nonneg_region(f)

Returns the UnionOfIntervals representing the region where f is non-negative.

INPUT:

• f (polynomial) – a univariate polynomial over \(\RR\).

OUTPUT:

A UnionOfIntervals representing the set \(\{x \in\RR mid f(x) \ge 0\}\).

EXAMPLES:

sage: from sage.schemes.elliptic_curves.height import nonneg_region
sage: x = polygen(RR)
sage: nonneg_region(x^2 - 1)
([-Infinity, -1.00000000000000] U [1.00000000000000, +Infinity])
sage: nonneg_region(1 - x^2)
([-1.00000000000000, 1.00000000000000])
sage: nonneg_region(1 - x^3)
([-Infinity, 1.00000000000000])
sage: nonneg_region(x^3 - 1)
([1.00000000000000, +Infinity])
sage: nonneg_region((x-1)*(x-2))
([-Infinity, 1.00000000000000] U [2.00000000000000, +Infinity])
sage: nonneg_region(-(x-1)*(x-2))
([1.00000000000000, 2.00000000000000])
sage: nonneg_region((x-1)*(x-2)*(x-3))
([1.00000000000000, 2.00000000000000] U [3.00000000000000, +Infinity])
sage: nonneg_region(-(x-1)*(x-2)*(x-3))
([-Infinity, 1.00000000000000] U [2.00000000000000, 3.00000000000000])
sage: nonneg_region(x^4 + 1)
([-Infinity, +Infinity])
sage: nonneg_region(-x^4 - 1)
()

sage.schemes.elliptic_curves.height.rat_term_CIF(z, try_strict=True)

Compute the value of \(u/(1-u)^2\) in CIF, where \(u=\exp(2\pi i z)\).

INPUT:

• z (complex) – a CIF element

• try_strict (bool) – flag

EXAMPLES:

sage: from sage.schemes.elliptic_curves.height import rat_term_CIF
sage: z = CIF(0.5,0.2)
sage: rat_term_CIF(z)
-0.172467461182437? + 0.?e-16*I
sage: rat_term_CIF(z, False)
-0.172467461182437? + 0.?e-16*I