1. Definition

Visualization of the [math]\displaystyle{ \wp }[/math]-function with invariants [math]\displaystyle{ g_2=1+i }[/math] and [math]\displaystyle{ g_3=2-3i }[/math] in which white corresponds to a pole, black to a zero. https://handwiki.org/wiki/index.php?curid=1280544

Let [math]\displaystyle{ \omega_1,\omega_2\in\mathbb{C} }[/math] be two complex numbers that are linear independent over [math]\displaystyle{ \mathbb{R} }[/math] and let [math]\displaystyle{ \Lambda:=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2:=\{m\omega_1+n\omega_2: m,n\in\mathbb{Z}\} }[/math] be the lattice generated by those numbers. Then the [math]\displaystyle{ \wp }[/math]-function is defined as follows:

[math]\displaystyle{ \weierp(z,\omega_1,\omega_2):=\weierp(z,\Lambda) := \frac{1}{z^2} + \sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac 1 {(z-\lambda)^2} - \frac 1 {\lambda^2}\right). }[/math]

This series converges locally uniformly absolutely in [math]\displaystyle{ \mathbb{C}\setminus\Lambda }[/math]. Oftentimes instead of [math]\displaystyle{ \wp(z,\omega_1,\omega_2) }[/math] only [math]\displaystyle{ \wp(z) }[/math] is written.

The Weierstrass [math]\displaystyle{ \wp }[/math]-function is constructed exactly in such a way that it has a pole of the order two at each lattice point.

Because the sum [math]\displaystyle{ \sum_{\lambda\in\Lambda}\frac 1{(z-\lambda)^2} }[/math] alone would not converge it is necessary to add the term [math]\displaystyle{ -\frac 1 {\lambda^2} }[/math].^[1]

It is common to use [math]\displaystyle{ 1 }[/math] and [math]\displaystyle{ \tau\in\mathbb{H}:=\{z\in\mathbb{C}:\operatorname{Im}(z)\gt 0\} }[/math] as generators of the lattice. Multiplying by [math]\displaystyle{ \frac 1{\omega_1} }[/math] maps the lattice [math]\displaystyle{ \mathbb{Z}\omega_1+\mathbb{Z}\omega_2 }[/math] isomorphically onto the lattice [math]\displaystyle{ \mathbb{Z}+\mathbb{Z}\tau }[/math] with [math]\displaystyle{ \tau=\frac{\omega_2}{\omega_1} }[/math]. By possibly substituting [math]\displaystyle{ \tau }[/math] by [math]\displaystyle{ -\tau }[/math] it can be assumed that [math]\displaystyle{ \tau\in\mathbb{H} }[/math]. One sets [math]\displaystyle{ \wp(z,\tau) := \wp(z, 1,\tau) }[/math].

2. Motivation

A cubic of the form [math]\displaystyle{ C_{g_2,g_3}^\mathbb{C}=\{(x,y)\in\mathbb{C}^2:y^2=4x^3-g_2x+g_3\} }[/math], where [math]\displaystyle{ g_2,g_3\in\mathbb{C} }[/math] are complex numbers with [math]\displaystyle{ g_2^3-27g_3^2\neq0 }[/math], can not be rationally parameterized.^[2] Yet one still wants to find a way to parameterize it.

For the quadric [math]\displaystyle{ K=\{(x,y)\in\mathbb{R}^2:x^2+y^2=1\} }[/math], the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function:

[math]\displaystyle{ \psi:\mathbb{R}/2\pi\mathbb{Z}\to K, \quad t\mapsto(\sin(t),\cos(t)) }[/math].

Because of the periodicity of the sine and cosine [math]\displaystyle{ \mathbb{R}/2\pi\mathbb{Z} }[/math] is chosen to be the domain, so the function is bijective.

In a similar way one can get a parameterization of [math]\displaystyle{ C_{g_2,g_3}^\mathbb{C} }[/math] by means of the doubly periodic [math]\displaystyle{ \wp }[/math]-function (see in the section "Relation to ellitpic curves"). This parameterization has the domain [math]\displaystyle{ \mathbb{C}/\Lambda }[/math], which is topologically equivalent to a torus.^[3]

There is another analogy to the trigonometric functions. Consider the integral function

[math]\displaystyle{ a(x)=\int_0^x\frac{dy}{\sqrt{(1-y^2)}} }[/math].

It can be simplified by substituting [math]\displaystyle{ y=\sin(t) }[/math] and [math]\displaystyle{ s=\arcsin(x) }[/math]:

[math]\displaystyle{ a(x)=\int_0^sdt=s=\arcsin(x) }[/math].

That means [math]\displaystyle{ a^{-1}(x)=\sin(x) }[/math]. So the sine function is an inverse function of an integral function.^[4]

Elliptic functions are also inverse functions of integral functions, namely of elliptic integrals. In particular the [math]\displaystyle{ \wp }[/math]-function is obtained in the following way:

Let

[math]\displaystyle{ u(z)=-\int_z^\infin\frac{ds}{\sqrt{4s^3-g_2s-g_3}} }[/math].

Then [math]\displaystyle{ u^{-1} }[/math] can be extended to the complex plane and this extension equals the [math]\displaystyle{ \wp }[/math]-function.^[5]

3. Properties

• ℘ is an even function. That means [math]\displaystyle{ \wp(z)=\wp(-z) }[/math] for all [math]\displaystyle{ z \in \mathbb{C} \setminus \Lambda }[/math], which can be seen in the following way:

[math]\displaystyle{ \weierp(-z)=\frac{1}{(-z)^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac1{(-z-\lambda)^2}-\frac1{\lambda^2}\right)=\frac{1}{z^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac1{(z+\lambda)^2}-\frac1{\lambda^2}\right)=\frac{1}{z^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac1{(z-\lambda)^2}-\frac1{\lambda^2}\right)=\wp(z) }[/math]

The second last equality holds because [math]\displaystyle{ \{-\lambda:\lambda \in \Lambda\}=\Lambda }[/math]. Since the sum converges absolutely this rearrangement does not change the limit.

• ℘ is meromorphic and its derivative is^[6]

[math]\displaystyle{ \wp'(z)=-2\sum_{\lambda \in \Lambda}\frac1{(z-\lambda)^3} }[/math].

• [math]\displaystyle{ \wp }[/math] and [math]\displaystyle{ \wp' }[/math] are doubly periodic with the periods [math]\displaystyle{ \omega_1 }[/math]und [math]\displaystyle{ \omega_2 }[/math].^[6] This means:

[math]\displaystyle{ \wp(z+\omega_1)=\wp(z)=\wp(z+\omega_2) }[/math] and [math]\displaystyle{ \wp'(z+\omega_1)=\wp'(z)=\wp'(z+\omega_2) }[/math].

It follows that [math]\displaystyle{ \wp(z+\lambda)=\wp(z) }[/math] and [math]\displaystyle{ \wp'(z+\lambda)=\wp'(z) }[/math] for all [math]\displaystyle{ \lambda \in \Lambda }[/math]. Functions which are meromorphic and doubly periodic are also called elliptic functions.

4. Laurent Expansion

Let [math]\displaystyle{ r:=\min\{{|\lambda}|:0\neq\lambda\in\Lambda\} }[/math]. Then for [math]\displaystyle{ 0\lt |z|\lt r }[/math] the [math]\displaystyle{ \wp }[/math]-function has the following Laurent expansion

[math]\displaystyle{ \wp(z)=\frac1{z^2}+\sum_{n=1}^\infin (2n+1)G_{2n+2}z^{2n} }[/math]

where

[math]\displaystyle{ G_n=\sum_{0\neq\lambda\in\Lambda}\lambda^{-n} }[/math] for [math]\displaystyle{ n\geq3 }[/math] are so called Eisenstein series.^[6]

5. Differential Equation

Set [math]\displaystyle{ g_2=60G_4 }[/math] and [math]\displaystyle{ g_3=140G_6 }[/math]. Then the [math]\displaystyle{ \wp }[/math]-function satisfies the differential equation^[6]

[math]\displaystyle{ \wp'^2(z) = 4\wp ^3(z)-g_2\wp(z)-g_3 }[/math].

This relation can be verified by forming a linear combination of powers of [math]\displaystyle{ \wp }[/math] and [math]\displaystyle{ \wp' }[/math] to eliminate the pole at [math]\displaystyle{ z=0 }[/math]. This yields an entire elliptic function that has to be constant by Liouville's theorem .^[6]

6. Invariants

The real part of the invariant g₃ as a function of the nome q on the unit disk. https://handwiki.org/wiki/index.php?curid=1389931

The imaginary part of the invariant g₃ as a function of the nome q on the unit disk. https://handwiki.org/wiki/index.php?curid=1178918

The coefficients of the above differential equation g₂ and g₃ are known as the invariants. Because they depent on the lattice [math]\displaystyle{ \Lambda }[/math] they can be viewed as functions in [math]\displaystyle{ \omega_1 }[/math]and [math]\displaystyle{ \omega_2 }[/math].

The series expansion suggests that g₂ and g₃ are homogeneous functions of degree −4 and −6. That is^[7]

[math]\displaystyle{ g_2(\lambda \omega_1, \lambda \omega_2) = \lambda^{-4} g_2(\omega_1, \omega_2) }[/math]

[math]\displaystyle{ g_3(\lambda \omega_1, \lambda \omega_2) = \lambda^{-6} g_3(\omega_1, \omega_2) }[/math] for [math]\displaystyle{ \lambda\neq0 }[/math].

If [math]\displaystyle{ \omega_1 }[/math]and [math]\displaystyle{ \omega_2 }[/math] are chosen in such a way that [math]\displaystyle{ \operatorname{Im}\left( \frac{\omega_2}{\omega_1} \right)\gt 0 }[/math] g₂ and g₃ can be interpreted as functions on the upper half-plane [math]\displaystyle{ \mathbb{H}:=\{z\in\mathbb{C}:\operatorname{Im}(z)\gt 0\} }[/math].

Let [math]\displaystyle{ \tau=\frac{\omega_2}{\omega_1} }[/math]. One has:^[8]

[math]\displaystyle{ g_2(1,\tau)=\omega_1^4g_2(\omega_1,\omega_2) }[/math],

[math]\displaystyle{ g_3(1,\tau)=\omega_1^6 g_3(\omega_1,\omega_2) }[/math].

That means g₂ and g₃ are only scaled by doing this. Set

[math]\displaystyle{ g_2(\tau):=g_2(1,\tau) }[/math], [math]\displaystyle{ g_3(\tau):=g_3(1,\tau) }[/math].

As functions of [math]\displaystyle{ \tau\in\mathbb{H} }[/math] [math]\displaystyle{ g_2,g_3 }[/math] are so called modular forms.

The Fourier series for [math]\displaystyle{ g_2 }[/math] and [math]\displaystyle{ g_3 }[/math] are given as follows:^[9]

[math]\displaystyle{ g_2(\tau)=\frac{4}{3}\pi^4 \left[ 1+ 240\sum_{k=1}^\infty \sigma_3(k) q^{2k} \right] }[/math]

[math]\displaystyle{ g_3(\tau)=\frac{8}{27}\pi^6 \left[ 1- 504\sum_{k=1}^\infty \sigma_5(k) q^{2k} \right] }[/math]

where [math]\displaystyle{ \sigma_a(k):=\sum_{d\mid{k}}d^\alpha }[/math] is the divisor function and [math]\displaystyle{ q:=\exp(i\pi\tau) }[/math].

7. Modular Discriminant

The real part of the discriminant as a function of the nome q on the unit disk.

The modular discriminant Δ is defined as the discriminant of the polynomial at right-hand side of the above differential equation:

[math]\displaystyle{ \Delta=g_2^3-27g_3^2. \, }[/math]

The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as

[math]\displaystyle{ \Delta \left( \frac {a\tau+b} {c\tau+d}\right) = \left(c\tau+d\right)^{12} \Delta(\tau) }[/math]

where [math]\displaystyle{ a,b,d,c\in\mathbb{Z} }[/math] with ad − bc = 1.^[10]

Note that [math]\displaystyle{ \Delta=(2\pi)^{12}\eta^{24} }[/math] where [math]\displaystyle{ \eta }[/math] is the Dedekind eta function.^[11]

For the Fourier coefficients of [math]\displaystyle{ \Delta }[/math], see Ramanujan tau function.

8. The Constants e₁, e₂ and e₃

[math]\displaystyle{ e_1 }[/math], [math]\displaystyle{ e_2 }[/math] and [math]\displaystyle{ e_3 }[/math] are usually used to denote the values of the [math]\displaystyle{ \wp }[/math]-function at the half-periods.

[math]\displaystyle{ e_1\equiv\wp\left(\frac{\omega_1}{2}\right) }[/math]

[math]\displaystyle{ e_2\equiv\wp\left(\frac{\omega_2}{2}\right) }[/math]

[math]\displaystyle{ e_3\equiv\wp\left(\frac{\omega_1+\omega_2}{2}\right) }[/math]

They are pairwise distinct and only depend on the lattice [math]\displaystyle{ \Lambda }[/math] and not on its generators.^[12]

[math]\displaystyle{ e_1 }[/math], [math]\displaystyle{ e_2 }[/math] and [math]\displaystyle{ e_3 }[/math] are the roots of the cubic polynomial [math]\displaystyle{ 4\wp(z)^3-g_2\wp(z)-g_3 }[/math] and are related by the equation:

[math]\displaystyle{ e_1+e_2+e_3=0 }[/math].

Because those roots are distinct the discriminant [math]\displaystyle{ \Delta }[/math] does not vanish on the upper half plane.^[13] Now we can rewrite the differential equation:

[math]\displaystyle{ \wp'^2(z)=4(\wp(z)-e_1)(\wp(z)-e_2)(\wp(z)-e_3) }[/math].

That means the half-periods are zeros of [math]\displaystyle{ \wp' }[/math].

The invariants [math]\displaystyle{ g_2 }[/math] and [math]\displaystyle{ g_3 }[/math] can be expressed in terms of these constants in the following way:^[14]

[math]\displaystyle{ g_2=-4(e_1e_2+e_1e_3+e_2e_3) }[/math]

[math]\displaystyle{ g_3=4e_1e_2e_3 }[/math]

9. Relation to Elliptic Curves

Consider the projective cubic curve

[math]\displaystyle{ \bar C_{g_2,g_3}^\mathbb{C}=\{(x,y)\in\mathbb{C}^2:y^2=4x^3-g_2x+g_3\}\cup\{\infin\}\subset\mathbb{P}_\mathbb{C}^2 }[/math].

For this cubic, also called Weierstrass cubic, there exists no rational parameterization, if [math]\displaystyle{ \Delta\neq0 }[/math].^[2] In this case it is also called an elliptic curve. Nevertheless there is a parameterization that uses the [math]\displaystyle{ \wp }[/math]-function and its derivative [math]\displaystyle{ \wp' }[/math]:^[15]

[math]\displaystyle{ \varphi: \mathbb{C}/\Lambda\to\bar C_{g_2,g_3}^\mathbb{C}, \quad \bar{z}\mapsto \begin{cases} (\wp(z),\wp'(z),1) & \bar{z}\neq0\\ \infin \quad &\bar{z}=0 \end{cases} }[/math]

Now the map [math]\displaystyle{ \varphi }[/math] is bijective and parameterizes the elliptic curve [math]\displaystyle{ \bar C_{g_2,g_3}^\mathbb{C} }[/math].

[math]\displaystyle{ \mathbb{C}/\Lambda }[/math] is an abelian group and a topological space, equipped with the quotient topology.

It can be shown that every Weierstrass cubic is given in such a way. That is to say that for every pair [math]\displaystyle{ g_2,g_3\in\mathbb{C} }[/math] with [math]\displaystyle{ \Delta=g_2^3-27g_3^2\neq0 }[/math] there exists a lattice [math]\displaystyle{ \mathbb{Z}\omega_1+\mathbb{Z}\omega_2 }[/math], such that

[math]\displaystyle{ g_2=g_2(\omega_1,\omega_2) }[/math] and [math]\displaystyle{ g_3=g_3(\omega_1,\omega_2) }[/math].^[16]

The statement that elliptic curves over [math]\displaystyle{ \mathbb{Q} }[/math] can be parameterized over [math]\displaystyle{ \mathbb{Q} }[/math], is known as the modularity theorem. This is an important theorem in number theory. It was part of Andrew Wiles' proof (1995) of Fermat's Last Theorem.

10. Addition Theorems

Let [math]\displaystyle{ z,w\in\mathbb{C} }[/math], so that [math]\displaystyle{ z,w,z+w,z-w\notin\Lambda }[/math]. Then one has:^[17]

[math]\displaystyle{ \wp(z+w)=\frac1{4} \left[\frac{\wp'(z)-\wp'(w)}{\wp(z)-\wp(w)}\right]^2-\wp(z)-\wp(w) }[/math].

As well as the duplication formula:^[17]

[math]\displaystyle{ \wp(2z)=\frac1{4}\left[\frac{\wp''(z)}{\wp'(z)}\right]^2-2\wp(z) }[/math].

These formulas also have a geometric interpretation, if one looks at the elliptic curve [math]\displaystyle{ \bar C_{g_2,g_3}^\mathbb{C} }[/math] together with the mapping [math]\displaystyle{ {\varphi}:\mathbb{C}/\Lambda\to\bar C_{g_2,g_3}^\mathbb{C} }[/math] as in the previous section.

The group structure of [math]\displaystyle{ (\mathbb{C}/\Lambda,+) }[/math] translates to the curve [math]\displaystyle{ \bar C_{g_2,g_3}^\mathbb{C} }[/math]and can be geometrically interpreted there:

The sum of three pairwise different points [math]\displaystyle{ a,b,c\in\bar C_{g_2,g_3}^\mathbb{C} }[/math]is zero if and only if they lie on the same line in [math]\displaystyle{ \mathbb{P}_\mathbb{C}^2 }[/math].^[18]

This is equivalent to:

[math]\displaystyle{ \det\left(\begin{array}{rrr} 1&\wp(u+v)&-\wp'(u+v)\\ 1&\wp(v)&\wp'(v)\\ 1&\wp(u)&\wp'(u)\\ \end{array}\right) =0 }[/math],

where [math]\displaystyle{ \wp(u)=a }[/math], [math]\displaystyle{ \wp(v)=b }[/math] and [math]\displaystyle{ u,v\notin\Lambda }[/math].^[19]

11. Relation to Jacobi's Elliptic Functions

For numerical work, it is often convenient to calculate the Weierstrass elliptic function in terms of Jacobi's elliptic functions.

The basic relations are:^[20]

[math]\displaystyle{ \wp(z) = e_3 + \frac{e_1 - e_3}{\operatorname{sn}^2 w} = e_2 + ( e_1 - e_3 ) \frac{\operatorname{dn}^2 w}{\operatorname{sn}^2 w} = e_1 + ( e_1 - e_3 ) \frac{\operatorname{cn}^2 w}{\operatorname{sn}^2 w} }[/math]

where [math]\displaystyle{ e_1,e_2 }[/math]and [math]\displaystyle{ e_3 }[/math] are the three roots described above and where the modulus k of the Jacobi functions equals

[math]\displaystyle{ k =\sqrt{\frac{e_2 - e_3}{e_1 - e_3}} }[/math]

and their argument w equals

[math]\displaystyle{ w =z \sqrt{e_1 - e_3}. }[/math]

12. Typography

The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘.^[21]

In computing, the letter ℘ is available as \wp in TeX. In Unicode the code point is U+2118 ℘ SCRIPT CAPITAL P (HTML ℘ · ℘), with the more correct alias weierstrass elliptic function.^[22] In HTML, it can be escaped as ℘.

The content is sourced from: https://handwiki.org/wiki/Weierstrass%27s_elliptic_functions