13. Theorem 2.13.5 elliptic differential equation

Let $℘ (z)$ be a weierstrauss $℘$ -function. Then it satisfies a differntial equation:

(℘^{″} (z))^{2} = 4 ℘ (z)^{3} - g_{2} ℘ (z) - g_{3}

Where $g_{2}, g_{3}$ are determined 1;; by the lattice of $℘$ , $L$

Remark

(General statement) Let $f, g$ be two elliptic functions on the same lattice $L$ , with orders $m, n$ respectively. Then there exists a polynomial in two variables $P (f, g) = 0$ , with the highest degree of $f$ being $n$ , and the highest degree of $g$ being $m$ .

Remark

In general, the idea of these problems is to show that the relation is holomorphic, and then by Louiville the relation must be constant.

PROOF:

We also used geometric series/absolute convergence, since eventually the periods have a greater absolute value than a fixed value. That is, $\frac{| z_{0} |}{| ω |} < 1$ eventually.