Conversions of parameters of Weierstrass \(\wp\)

Wikipedia [1463], Wikipedia [1464], MathWorld [1066], MathWorld [1064], MathWorld [292], MathWorld [291].

The Weierstrass elliptic functions may be defined on a general lattice \(\Lambda = \{m 2\omega_1 + n 2\omega_2 : m, n \in \mathbb{Z} \}\) with half-periods \(\omega_1, \omega_2 \in \mathbb{C}\).

In XlCalcNet, two interfaces are used:

The first interface (described in this section) uses \(\omega = \omega_1\) and \(\tau = \omega_2 / \omega_1\) as parameters; these have a natural geometrical interpretation, specifying respectively the size and shape of the “fundamental parallelogram”. In Flint there is a default choice of \(\omega=1\); therefore \(\omega\) is omitted from the function parameters in Flint, but \(\omega\) is a required parameter in XlCalcNet.

In some contexts the lattice invariants \(g_2\) and \(g_3\) are the natural parameters because they correspond directly with physical quantities; usually \(g_2, g_3 \in \mathbb{R}\) in such cases (and also in XlCalcNet). This interface is described in the next section.

To keep these two interfaces apart with regard to mathematical notation, we write \(\wp(z| \omega, \tau)\) for the first interface and \(\wp_g(z; g_2, g_3)\) for the second, and likewise for the other Weierstrass functions.

To evaluate the functions on a general lattice, we can use the the following homogeneity relations (for \(t \ne 0\)). Note that the period ratio \(\tau\) is preserved:

\[\wp'(tz| t \omega, \tau) = t^{-3} \wp'(z| \omega, \tau)\]

\[\wp(tz| t \omega, \tau) = t^{-2} \wp(z| \omega, \tau)\]

\[\zeta(tz| t \omega, \tau) = t^{-1} \zeta(z| \omega, \tau)\]

\[\sigma(tz| t \omega, \tau) = t \sigma(z| \omega, \tau)\]

\[g_2(t \omega, \tau) = t^{-4} g_2(\omega, \tau)\]

\[g_3(t \omega, \tau) = t^{-6} g_3(\omega, \tau)\]

\[e_i(t \omega, \tau) = t^{-2} e_i(\omega, \tau), \quad i = 1, 2, 3\]

\[\Delta(t \omega, \tau) = t^{-12} \Delta(\omega, \tau)\]

Elliptic lattice invariants \(g_2, g_3\) from lattice roots \(e_1, e_2, e_3\)

ctxflint.elliptic_invariants_from_roots(e1, e2)

Computes the lattice invariants \(g_2, g_3\). The Weierstrass elliptic function satisfies the differential equation \([\wp'(z, \tau)]^2 = 4 [\wp(z,\tau)]^3 - g_2 \wp(z,\tau) - g_3\). Up to constant factors, the lattice invariants are the first two Eisenstein series.

See also: MathWorld [292], MathWorld [291], Flint [837].

\[g_{2}=2({e_{1}}^{2}+{e_{2}}^{2}+{e_{3}}^{2})=-4(e_{2}e_{3}+e_{3}e_{1}+e_{1}e_{2}),\]

\[g_{3}=4e_{1}e_{2}e_{3}=\tfrac{4}{3}({e_{1}}^{3}+{e_{2}}^{3}+{e_{3}}^{3}).\]

An example with purely imaginary input, producing real output:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; tau = '0.0 + 0.7j'
>>> dx2, dx3 = dec.elliptic_invariants(tau); mx2, mx3 = mpm.elliptic_invariants(tau);
>>> gx2, gx3  = gmp.elliptic_invariants(tau)
>>> fx2, fx3 = fpm.elliptic_invariants(tau); ax2, ax3 = apm.elliptic_invariants(tau)
>>> print("lattice invariant g2")
>>> mpm.show([dx2, mx2, gx2, fx2, ax2], aligned=True)
>>> print("lattice invariant g3")
>>> mpm.show([dx3, mx3, gx3, fx3, ax3], aligned=True)

lattice invariant g2
dec: 5.573660602292979232289778942418842671744E+2
mpm: 5.573660602292979232289778942418842671737e+2
gmp: 5.573660602292979232289778942418842671749E+02
fpm: 5.57366060229298E+02
apm: 5.573660602292979232289778942418842671735e+2 (7.909e-38%)

lattice invariant g3
dec: -2.266333763304145743477283974834873543570E+3
mpm: -2.266333763304145743477283974834873543563e+3
gmp: -2.266333763304145743477283974834873543571E+03
fpm: -2.26633376330415E+03
apm: -2.266333763304145743477283974834873543560e+3 (-1.184e-36%)

Elliptic lattice invariants \(g_2, g_3\) from \((\omega, \tau)\)

ctxflint.g2g3_from_tau(omega, tau)

Computes the lattice invariants \(g_2, g_3\) in terms of half-period \(\omega_1\) and elliptic period ratio \(\tau\).

We have \(g_2(t \omega, \tau) = t^{-4} g_2(\omega, \tau)\) and \(g_3(t \omega, \tau) = t^{-6} g_3(\omega, \tau)\).

See also: MathWorld [292], MathWorld [291], Flint [837].

See also: https://en.wikipedia.org/wiki/J-invariant#Expressions_in_terms_of_theta_functions

See also: https://dlmf.nist.gov/23.3#i

We obtain the elliptic lattice invariants from the Eisenstein series \(G_4(\tau)\) and \(G_6(\tau)\):

\[g_2(\tau) = 60G_4(\tau) = 60\sum_{(m,n) \neq (0,0)} \left(m + n\tau\right)^{-4}\]

\[g_3(\tau) = 140G_6(\tau) = 140\sum_{(m,n) \neq (0,0)} \left(m + n\tau\right)^{-6}\]

Define \(q = e^{\pi i \tau}\)

Let \(a = \theta_{2}(q)\), \(b = \theta_{3}(q)\), \(c = \theta_{4}(q)\), where \(a^4 - b^4 + c^4 = 0\). Then

\(g_2(\tau) = \tfrac{2}{3}\pi^4 \left(a^8 + b^8 + c^8\right)\), and

\(g_3(\tau) = \tfrac{4}{27}\pi^6 \sqrt{\frac{\left(a^8+b^8+c^8\right)^3-54\left(abc\right)^8}{2}}\).

Elliptic lattice roots \(e_1, e_2, e_3\) from (\(\omega, \tau\))

ctxflint.elliptic_roots_from_tau(omega, tau)

Computes the lattice roots \(e_1, e_2, e_3\), in terms of half-period \(\omega_1\) and elliptic period ratio \(\tau\). They are the roots of the polynomial \(4z^3 - g_2 z - g_3\).

We have \(e_i(t \omega, \tau) = t^{-2} e_i(\omega, \tau), \quad i = 1, 2, 3\).

See also Wikipedia [1464], Flint [837].

See also: https://dlmf.nist.gov/23.3#i

With \(\tau\) as input, the roots \(e_{1}\), \(e_{2}\), and \(e_{3}\) can be expressed in terms of the Jacobi theta functions \(\theta _{i}(0;q)\) . Let \(q = e^{2 i \pi \tau}\) denote the nome. Then

\[a=\theta _{2}(0;q); \quad b=\theta _{3}(0;q); \quad c=\theta _{4}(0;q), \quad \text{with } a^4 - b^4 + c^4 = 0, \quad \text{and}\]

\[e_{1}(\tau )=\tfrac {\pi ^{2}}{3}(b^{4}+c^{4}); \quad e_{2}(\tau )=\tfrac {\pi ^{2}}{3}(-a^{4}-b^{4}); \quad e_{3}(\tau )=\tfrac {\pi ^{2}}{3}(a^{4}-c^{4}).\]

An example with purely imaginary input, producing real output:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; tau = '0.0 + 0.7j'
>>> dx1, dx2, dx3 = dec.elliptic_roots(tau); mx1, mx2, mx3 = mpm.elliptic_roots(tau);
>>> gx1, gx2, gx3  = gmp.elliptic_roots(tau)
>>> fx1, fx2, fx3 = fpm.elliptic_roots(tau); ax1, ax2, ax3 = apm.elliptic_roots(tau)
>>> print("lattice root e1")
>>> mpm.show([dx1, mx1, gx1, fx1, ax1], aligned=True)
lattice root e1
dec: 8.546997557335192078343940404355992077317E+0
mpm: 8.546997557335192078343940404355992077314e+0
gmp: 8.546997557335192078343940404355992077321E+00
fpm: 8.54699755733519E+00
apm: 8.546997557335192078343940404355992077315e+0 (4.298e-39%)

>>> print("lattice root e2")
>>> mpm.show([dx2, mx2, gx2, fx2, ax2], aligned=True)
lattice root e2
dec: -1.346877689428238019477524701740978037594E+1
mpm: -1.346877689428238019477524701740978037593e+1
gmp: -1.346877689428238019477524701740978037595E+01
fpm: -1.34687768942824E+01
apm: 4.921779336947188116431306613053788298616e+0 (1.493e-38%)

>>> print("lattice root e3")
>>> mpm.show([dx3, mx3, gx3, fx3, ax3], aligned=True)
lattice root e3
dec: 4.921779336947188116431306613053788298624E+0
mpm: 4.921779336947188116431306613053788298616e+0
gmp: 4.921779336947188116431306613053788298625E+00
fpm: 4.92177933694719E+00
apm: -1.346877689428238019477524701740978037593e+1 (-7.023e-38%)

!!! NEED TO FIX: Decimal complex negation and multiplication from left

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; tau = '0.4 + 0.7j'
>>> dx1, dx2, dx3 = dec.elliptic_roots(tau); mx1, mx2, mx3 = mpm.elliptic_roots(tau);
>>> gx1, gx2, gx3  = gmp.elliptic_roots(tau)
>>> fx1, fx2, fx3 = fpm.elliptic_roots(tau); ax1, ax2, ax3 = apm.elliptic_roots(tau)
>>> print("lattice root e1")
>>> mpm.show([dx1, mx1, gx1, fx1, ax1], aligned=True)
lattice root e1
dec: 5.0162095528741509613E+0              + 1.1199909932847896589E+0j
mpm: 5.0162095528741509605e+0              + 1.1199909932847896584e+0j
gmp: 5.0162095528741509611E+00             + 1.1199909932847896588E+00j
fpm: 5.01620955287415E+00                  + 1.11999099328479E+00j
apm: 5.0162095528741509605e+0 (4.525e-18%) + 1.1199909932847896584e+0 (1.989e-17%)j

>>> print("lattice root e2")
>>> mpm.show([dx2, mx2, gx2, fx2, ax2], aligned=True)
lattice root e2
dec: -3.2965087919479326553E+0               - 3.2515656088055451348E+0j
mpm: -4.8733168229696470033e+0               - 8.6347058331318457444e+0j
gmp: -4.8733168229696470039E+00              - 8.6347058331318457450E+00j
fpm: -4.87331682296965E+00                   - 8.63470583313184E+00j
apm: -1.4289272990450395721e-1 (-1.792e-16%) + 7.5147148398470560860e+0 (3.562e-18%)j

>>> print("lattice root e3")
>>> mpm.show([dx3, mx3, gx3, fx3, ax3], aligned=True)
lattice root e3
dec: -1.4289272990450395785E-1              + 7.5147148398470560861E+0j
mpm: -1.4289272990450395722e-1              + 7.5147148398470560860e+0j
gmp: -1.4289272990450395720E-01             + 7.5147148398470560862E+00j
fpm: -1.42892729904505E-01                  + 7.51471483984706E+00j
apm: -4.8733168229696470032e+0 (-6.82e-17%) - 8.6347058331318457441e+0 (-3.869e-17%)j