Weierstrass Elliptic Function

Basic construction

Weierstrass $\mathscr{p}$ function is the fundamental, and probably the simplest function on the complex torus $\mathbb{C}/L$, where $L$ is a lattice. It turns out that the Weierstrass $\mathscr{p}$

Def. A lattice $L$ in $\mathbb{C}$ is a set of complex numbers determined by linear independent $w_1, w_2\in\mathbb{C}$, such that

$$L=\{aw_1+bw_2:a,b\in\mathbb{Z}\}=\mathbb{Z}w_1\oplus\mathbb{Z}w_2.$$

For a given lattice $L$, define the complex torus to be the quotient space $\mathbb{C}/L$. Notice that a meromorphic function on complex torus $f:\mathbb{C}/L\rightarrow \mathbb{C}$​ can be lifted to a meromorphic function

$$\tilde{f}:\mathbb{C}\rightarrow\mathbb{C}$$

such that $\tilde{f}(z)=\tilde{f}(z+L)$. Such function $\tilde{f}$ is called a elliptic function. Define the Weierstrass function $\mathscr{p}(z)=\mathscr{p}(z,L)$ by

$$\wp(z):=\frac{1}{z^2}+\sum_{w\in L-0}\left(\frac{1}{(z-w)^2}-\frac{1}{w^2}\right)$$

The following propositions indicate the existence of $\mathscr{p}$.

Prop. The series

$$\sum_{w\in L-0} \frac{1}{|w|^s}$$

converges for $s>2$.

Proof. Suppose that $w_2/w_1=\tau$ has imaginary part $\mathrm{Im}(\tau)>0$. Let $\tau=u+iv$. Then

$$\begin{aligned} \sum_{w\in L-0}\frac{1}{|w|^s} &=|w_1|^s\mathop{\sum\sum}\limits_{a,b\in\mathbb{Z}, (a,b)\neq 0}\frac{1}{|a+b\tau|^s}\\ &=|w_1|^s\mathop{\sum\sum}\limits_{a,b\in\mathbb{Z}, (a,b)\neq 0}\left(\frac{1}{a^2+b^2u^2+2b^2uv+b^2v^2}\right)^{\frac{s}{2}}\\ &\leq|w_1|^s\mathop{\sum\sum}\limits_{a,b\in\mathbb{Z}, (a,b)\neq 0}\left(\frac{1}{a^2+b^2|\tau|^2}\right)^{\frac{s}{2}}\\ \end{aligned}$$

Since the summand converges, then the original series converges. $\square$

Prop. The Weierstrass function $\mathscr{p}(z)$ converges absolutely and uniformly in any compact set $K\subset\mathbb{C}/L$.

Proof. Since $z$ lies in a compact set, the for sufficiently large $a$ and $b$, we can assume that

$$|z|\leq R, \quad |w|\geq 2R.$$

for some sufficiently large $R>0$. Since

$$\sum_{w\in L-0}\left(\frac{1}{(z-w)^2}-\frac{1}{w^2}\right)=\sum_{w\in L-0}\frac{2wz-z^2}{(z-w)^2w^2},$$

and

$$\begin{aligned} \left|\frac{2wz-z^2}{(z-w)^2w^2}\right| &\leq \frac{2R|z|+|z|^2}{(|w|/2)^2|w|^2}\\ &\leq \frac{10R}{|w|^3} \end{aligned}$$

Thus the sequence converges uniformly, as by the previous property

$$\sum_{w\in L-0}\frac{10R}{|w|^3}<\infty.$$

This completes the proof. $\square$

The Weierstrass $\mathscr{p}$ function is almost the simplest elliptic function. It has the following properties

Prop. Suppose that $\mathscr{p}$ is the Weierstrass function for any lattice $L$. Then

1. $\mathscr{p}$ is elliptic;
2. $\mathscr{p}$ is even, whereas $\mathscr{p}'$ is even;
3. The set of all poles of $\mathscr{p}$ is $L$. Moreover, $\mathscr{p}$ has order $2$ at each pole.

Proof. Only by observing the expression of $\mathscr{p}$. $\square$

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Elliptic functions

Given a lattice $L\subset\mathbb{C}$. Define the set of all elliptic function by

$$\mathcal{E}_L:=\{f\in\mathcal{M}(\mathbb{C}):f(z)=f(z+w),\forall w\in L\}.$$

It can be verified that $\mathcal{E}_L$ is a field. Since $\mathscr{p}$ is an even function, we can sieve out those even elliptic functions

$$\mathcal{E}_L^+:=\{f\in\mathcal{E}_L:f(z)=f(-z),\forall z\in\mathbb{C}\}.$$

Prop. Let $f\in\mathcal{E}_L^+$ such that the poles of $f$ lie in $L$. Then $f\in\mathbb{C}[\mathscr{p}]$.

Proof. Consider the Laurent expansion of $f$ at $0$. Since $f$ is even, then the the coefficients are only nonzero at even terms. Namely, $f$ can be written as

$$f(z)=\frac{c_{2n}}{z^{2n}}+\frac{c_{2n-2}}{z^{2n-2}}+\cdots+\frac{c_{-2}}{z^{2}}+g(z)$$

where $g$ is a holomorphic function. By subtracting and comparing the coefficients,

$$f(z)-c_{2n}\wp(z)=\frac{b_{2n-2}}{z^{2n-2}}+\cdots+\frac{b_{-2}}{z^{2}}+h(z)$$

we can lower the highest order term. Since $n$ is finite, then doing the above procedure recursively give a holomorphic function in $\mathbb{C}$, and thus $f$ can be written as a polynomial of variable $\mathscr{p}$. $\square$

Prop. $\mathcal{E}_L^+\cong\mathbb{C}(\mathscr{p})$​.

Proof. Let $z_1,\ldots,z_m$ be the poles of $f\in\mathcal{E}_L^+$, where $z_i\notin L$ is the pole of order $n_i$. Then $f$ has finitely many poles in the fundamental domain. Consider

$$g(z)=f(z)\prod_{i=1}^n\left(\wp(z)-\wp(z_i)\right)^{n_i}$$

Then $g$ has only poles in $L$. By the above proposition, $g\in\mathbb{C}[\mathscr{p}]$. Namely, there exists a polynomial $P\in\mathbb{C}[z]$, such that $g(z)=P(\mathscr{p})$. Therefore,

$$f(z)=P(\wp)/\prod_{i=1}^n\left(\wp(z)-\wp(z_i)\right)^{n_i}\in\mathbb{C}(\wp),$$

and $\mathcal{E}_L^+\subset\mathbb{C}[\mathscr{p}]$. The other inclusion is obvious, which establishes the proposition. $\square$

Prop. $\mathcal{E}_L\cong\mathbb{C}(\mathscr{p},\mathscr{p}')$.

Proof. Using the fact that $\mathscr{p}'$ is an odd function, and every function $f$ can be written as the summand of its odd part and even part. The even part has been discussed, whereas the odd part can be written as

$$\wp'\cdot\text{even part}$$

This completes the proof. $\square$

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Algebraic properties

The Weierstrass $\mathscr{p}$ function also satisfies some interesting algebraic equation.

Prop. The Laurent expansion of $\mathscr{p}$ is

$$\wp(z)=\frac{1}{z^2}+\sum_{k=1}^\infty(2k+1)G_{2(k+1)}z^{2k}$$

where $G_k(L)=\sum\limits_{w\in L-0}\frac{1}{w^k}$.

Proof. Let $\tilde{\mathscr{p}}(z)=\mathscr{p}(z)-\frac{1}{z^2}$. Then

$$\tilde{\wp}(z)=\sum_{w\in L-0}\frac{-2}{(z-w)^3}$$

and the $n$-th derivative of $\tilde{p}$ is

$$\tilde{p}^{(n)}(z)=\sum_{w\in L-0}\frac{(-1)^n(n+1)!}{(z-w)^{n+2}}.$$

Therefore

$$\tilde{\wp}^{(2k)}(z)=(2k+1)!\sum_{w\in L-0}\frac{1}{w^{2k+2}}=(2k+1)!G_{2k+2}$$

Hence the Laurent expansion of $\mathscr{p}$ can be written as

$$\wp(z)=\frac{1}{z^2}+\sum_{k=1}^\infty(2k+1)G_{2(k+1)}z^{2k}.$$

by using the expansion of $\tilde{p}$. $\square$

Prop. $\mathscr{p}$ satisfies the algebraic differential equation:

$$(\wp'(z))^2=4\wp(z)^3-g_2\wp(z)-g_3$$

where $g_2=60G_4$ and $g_3=140G_6$. Therefore,

$$\mathcal{M}(\mathbb{C}/L)\cong\mathcal{E}_L\cong\mathbb{C}(\wp,\wp')\cong\mathbb{C}(x,y)/(y^2-4x^3+g_2x+g_3),$$

and $[\mathcal{M}(\mathbb{C}):\mathcal{M}(\mathbb{C/L})]=2$.

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Mapping behavior of $\mathscr{p}$