Weierstrass elliptic function

In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p. They play an important role in the theory of elliptic functions, i.e., meromorphic functions that are doubly periodic. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice. Symbol for Weierstrass $$\wp$$-function

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

For the quadric $$K=\left\{(x,y)\in\mathbb{R}^2:x^2+y^2=1\right\}$$; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function: $$\psi:\mathbb{R}/2\pi\mathbb{Z}\to K, \quad t\mapsto(\sin t,\cos t).$$ Because of the periodicity of the sine and cosine $$\mathbb{R}/2\pi\mathbb{Z}$$ is chosen to be the domain, so the function is bijective.

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

There is another analogy to the trigonometric functions. Consider the integral function $$a(x)=\int_0^x\frac{dy}{\sqrt{1-y^2}} .$$ It can be simplified by substituting $$y=\sin t $$ and $$s=\arcsin x $$: $$a(x)=\int_0^s dt = s = \arcsin x .$$ That means $$a^{-1}(x) = \sin x $$. So the sine function is an inverse function of an integral function.

Elliptic functions are the inverse functions of elliptic integrals. In particular, let: $$u(z)=-\int_z^\infin\frac{ds}{\sqrt{4s^3-g_2s-g_3}} .$$ Then the extension of $$u^{-1} $$ to the complex plane equals the $$\wp $$-function. This invertibility is used in complex analysis to provide a solution to certain nonlinear differential equations satisfying the Painlevé property, i.e., those equations that admit poles as their only movable singularities.

Definition


Let $$\omega_1,\omega_2\in\mathbb{C}$$ be two complex numbers that are linearly independent over $$\mathbb{R}$$ and let $$\Lambda:=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2:=\{m\omega_1+n\omega_2: m,n\in\mathbb{Z}\}$$ be the period lattice generated by those numbers. Then the $$\wp$$-function is defined as follows:
 * $$\weierp(z,\omega_1,\omega_2):=\wp(z) = \frac{1}{z^2} + \sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac 1 {(z-\lambda)^2} - \frac 1 {\lambda^2}\right).$$

This series converges locally uniformly absolutely in the complex torus $$\mathbb{C}\setminus\Lambda$$.

It is common to use $$1$$ and $$\tau$$ in the upper half-plane $$\mathbb{H}:=\{z\in\mathbb{C}:\operatorname{Im}(z) > 0\}$$ as generators of the lattice. Dividing by $\omega_1$ maps the lattice $$\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$$ isomorphically onto the lattice $$\mathbb{Z}+\mathbb{Z}\tau$$ with $\tau=\tfrac{\omega_2}{\omega_1}$. Because $$-\tau$$ can be substituted for $$\tau$$, without loss of generality we can assume $$\tau\in\mathbb{H}$$, and then define $$\wp(z,\tau) := \wp(z, 1,\tau)$$.

Properties

 * $$\wp$$ is a meromorphic function with a pole of order 2 at each period $$\lambda$$ in $$\Lambda$$.
 * $$\wp$$ is an even function. That means $$\wp(z)=\wp(-z)$$ for all $$z \in \mathbb{C} \setminus \Lambda$$, which can be seen in the following way:
 * $$\begin{align}

\wp(-z) & =\frac{1}{(-z)^2} + \sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac{1}{(-z-\lambda)^2}-\frac{1}{\lambda^2}\right) \\ & =\frac{1}{z^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac{1}{(z+\lambda)^2}-\frac{1}{\lambda^2}\right) \\ & =\frac{1}{z^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac{1}{(z-\lambda)^2}-\frac{1}{\lambda^2}\right)=\wp(z). \end{align}$$
 * The second last equality holds because $$\{-\lambda:\lambda \in \Lambda\}=\Lambda$$. Since the sum converges absolutely this rearrangement does not change the limit.

\wp(z+\omega_1) &= \wp(z) = \wp(z+\omega_2),\ \textrm{and} \\[3mu] \wp'(z+\omega_1) &= \wp'(z) = \wp'(z+\omega_2). \end{aligned}$$ It follows that $$\wp(z+\lambda)=\wp(z)$$ and $$\wp'(z+\lambda)=\wp'(z)$$ for all $$\lambda \in \Lambda$$.
 * The derivative of $$\wp$$ is given by: $$\wp'(z)=-2\sum_{\lambda \in \Lambda}\frac1{(z-\lambda)^3}.$$
 * $$\wp$$ and $$\wp'$$ are doubly periodic with the periods $$\omega_1 $$ and $$\omega_2$$. This means: $$\begin{aligned}

Laurent expansion
Let $$r:=\min\{{|\lambda}|:0\neq\lambda\in\Lambda\}$$. Then for $$0<|z|<r$$ the $$\wp$$-function has the following Laurent expansion $$\wp(z)=\frac1{z^2}+\sum_{n=1}^\infin (2n+1)G_{2n+2}z^{2n} $$ where $$G_n=\sum_{0\neq\lambda\in\Lambda}\lambda^{-n}$$ for $$n \geq 3$$ are so called Eisenstein series.

Differential equation
Set $$g_2=60G_4$$ and $$g_3=140G_6$$. Then the $$\wp$$-function satisfies the differential equation $$ \wp'^2(z) = 4\wp ^3(z)-g_2\wp(z)-g_3.$$ This relation can be verified by forming a linear combination of powers of $$\wp$$ and $$\wp'$$ to eliminate the pole at $$z=0$$. This yields an entire elliptic function that has to be constant by Liouville's theorem.

Invariants


The coefficients of the above differential equation g2 and g3 are known as the invariants. Because they depend on the lattice $$\Lambda$$ they can be viewed as functions in $$\omega_1$$and $$\omega_2$$.

The series expansion suggests that g2 and g3 are homogeneous functions of degree −4 and −6. That is $$g_2(\lambda \omega_1, \lambda \omega_2) = \lambda^{-4} g_2(\omega_1, \omega_2)$$ $$g_3(\lambda \omega_1, \lambda \omega_2) = \lambda^{-6} g_3(\omega_1, \omega_2)$$ for $$\lambda \neq 0$$.

If $$\omega_1$$and $$\omega_2$$ are chosen in such a way that $$\operatorname{Im}\left( \tfrac{\omega_2}{\omega_1} \right)>0 $$, g2 and g3 can be interpreted as functions on the upper half-plane $$\mathbb{H}:=\{z\in\mathbb{C}:\operatorname{Im}(z)>0\}$$.

Let $$\tau=\tfrac{\omega_2}{\omega_1}$$. One has: $$g_2(1,\tau)=\omega_1^4g_2(\omega_1,\omega_2),$$ $$g_3(1,\tau)=\omega_1^6 g_3(\omega_1,\omega_2).$$ That means g2 and g3 are only scaled by doing this. Set $$g_2(\tau):=g_2(1,\tau) $$ and $$g_3(\tau):=g_3(1,\tau).$$ As functions of $$\tau\in\mathbb{H}$$ $$g_2,g_3$$ are so called modular forms.

The Fourier series for $$g_2$$ and $$g_3$$ are given as follows: $$g_2(\tau)=\frac43\pi^4 \left[ 1+ 240\sum_{k=1}^\infty \sigma_3(k) q^{2k} \right] $$ $$g_3(\tau)=\frac{8}{27}\pi^6 \left[ 1- 504\sum_{k=1}^\infty \sigma_5(k) q^{2k} \right] $$ where $$\sigma_a(k):=\sum_{d\mid{k}}d^\alpha$$ is the divisor function and $$q=e^{\pi i\tau}$$ is the nome.

Modular discriminant


The modular discriminant Δ is defined as the discriminant of the polynomial on the right-hand side of the above differential equation: $$ \Delta=g_2^3-27g_3^2. $$ The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as $$\Delta \left( \frac {a\tau+b} {c\tau+d}\right) = \left(c\tau+d\right)^{12} \Delta(\tau) $$ where $$a,b,d,c\in\mathbb{Z}$$ with ad − bc = 1.

Note that $$\Delta=(2\pi)^{12}\eta^{24}$$ where $$\eta$$ is the Dedekind eta function.

For the Fourier coefficients of $$\Delta$$, see Ramanujan tau function.

The constants e1, e2 and e3
$$e_1$$, $$e_2$$ and $$e_3$$ are usually used to denote the values of the $$\wp$$-function at the half-periods. $$e_1\equiv\wp\left(\frac{\omega_1}{2}\right)$$ $$e_2\equiv\wp\left(\frac{\omega_2}{2}\right)$$ $$e_3\equiv\wp\left(\frac{\omega_1+\omega_2}{2}\right)$$ They are pairwise distinct and only depend on the lattice $$\Lambda$$ and not on its generators.

$$e_1$$, $$e_2$$ and $$e_3$$ are the roots of the cubic polynomial $$4\wp(z)^3-g_2\wp(z)-g_3$$ and are related by the equation: $$e_1+e_2+e_3=0.$$ Because those roots are distinct the discriminant $$\Delta$$ does not vanish on the upper half plane. Now we can rewrite the differential equation: $$\wp'^2(z)=4(\wp(z)-e_1)(\wp(z)-e_2)(\wp(z)-e_3).$$ That means the half-periods are zeros of $$\wp'$$.

The invariants $$g_2$$ and $$g_3$$ can be expressed in terms of these constants in the following way: $$g_2 = -4 (e_1 e_2 + e_1 e_3 + e_2 e_3)$$ $$g_3 = 4 e_1 e_2 e_3$$ $$e_1$$, $$e_2$$ and $$e_3$$ are related to the modular lambda function: $$\lambda (\tau)=\frac{e_3-e_2}{e_1-e_2},\quad \tau=\frac{\omega_2}{\omega_1}.$$

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: $$ \wp(z) = e_3 + \frac{e_1 - e_3}{\operatorname{sn}^2 w} = e_2 + ( e_1 - e_3 ) \frac{\operatorname{dn}^2 w}{\operatorname{sn}^2 w} = e_1 + ( e_1 - e_3 ) \frac{\operatorname{cn}^2 w}{\operatorname{sn}^2 w} $$ where $$e_1,e_2$$and $$e_3$$ are the three roots described above and where the modulus k of the Jacobi functions equals $$k = \sqrt\frac{e_2 - e_3}{e_1 - e_3}$$ and their argument w equals $$w = z \sqrt{e_1 - e_3}.$$

Relation to Jacobi's theta functions
The function $$\wp (z,\tau)=\wp (z,1,\omega_2/\omega_1)$$ can be represented by Jacobi's theta functions: $$\wp (z,\tau)=\left(\pi \theta_2(0,q)\theta_3(0,q)\frac{\theta_4(\pi z,q)}{\theta_1(\pi z,q)}\right)^2-\frac{\pi^2}{3}\left(\theta_2^4(0,q)+\theta_3^4(0,q)\right)$$ where $$q=e^{\pi i\tau}$$ is the nome and $$\tau$$ is the period ratio $$(\tau\in\mathbb{H})$$. This also provides a very rapid algorithm for computing $$\wp (z,\tau)$$.

Relation to elliptic curves
Consider the embedding of the cubic curve in the complex projective plane
 * $$\bar C_{g_2,g_3}^\mathbb{C} = \{(x,y)\in\mathbb{C}^2:y^2=4x^3-g_2x-g_3\}\cup\{\infin\}\subset \mathbb{C}^{2} \cup \{\infty\} = \mathbb{P}_2(\mathbb{C}).$$

For this cubic there exists no rational parameterization, if $$\Delta \neq 0$$. In this case it is also called an elliptic curve. Nevertheless there is a parameterization in homogeneous coordinates that uses the $$\wp$$-function and its derivative $$\wp'$$:
 * $$ \varphi(\wp,\wp'): \mathbb{C}/\Lambda\to\bar C_{g_2,g_3}^\mathbb{C}, \quad

z \mapsto \begin{cases} \left[\wp(z):\wp'(z):1\right] & z \notin \Lambda \\ \left[0:1:0\right] \quad & z \in \Lambda \end{cases} $$

Now the map $$\varphi$$ is bijective and parameterizes the elliptic curve $$\bar C_{g_2,g_3}^\mathbb{C}$$.

$$\mathbb{C}/\Lambda $$ 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 $$g_2,g_3\in\mathbb{C}$$ with $$\Delta = g_2^3 - 27g_3^2 \neq 0 $$ there exists a lattice $$\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$$, such that

$$g_2=g_2(\omega_1,\omega_2) $$ and $$g_3=g_3(\omega_1,\omega_2) $$.

The statement that elliptic curves over $$\mathbb{Q}$$ can be parameterized over $$\mathbb{Q}$$, 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.

Addition theorems
Let $$z,w\in\mathbb{C}$$, so that $$z,w,z+w,z-w\notin\Lambda $$. Then one has: $$\wp(z+w)=\frac14 \left[\frac{\wp'(z)-\wp'(w)}{\wp(z)-\wp(w)}\right]^2-\wp(z)-\wp(w).$$

As well as the duplication formula: $$\wp(2z)=\frac14\left[\frac{\wp''(z)}{\wp'(z)}\right]^2-2\wp(z).$$

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

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

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

This is equivalent to: $$\det\left(\begin{array}{rrr} 1&\wp(u+v)&-\wp'(u+v)\\ 1&\wp(v)&\wp'(v)\\ 1&\wp(u)&\wp'(u)\\ \end{array}\right) =0 ,$$ where $$\wp(u) = a $$, $$\wp(v)=b $$ and $$u,v\notin\Lambda$$.

Typography
The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘, which was Weierstrass's own notation introduced in his lectures of 1862–1863.

In computing, the letter ℘ is available as  in TeX. In Unicode the code point is, with the more correct alias. In HTML, it can be escaped as.