(Elliptic function)
Fix a lattice \(\Lambda = \gens{1, \tau}_\ZZ \subseteq \CC\) with \(\tau \notin \RR\). An elliptic function for \(\Lambda\) is a meromorphic \(f \colon \CC \to \PP^1\) with \(f(z + \omega) = f(z)\) for every \(\omega \in \Lambda\), that is, a meromorphic function on the torus \(\CC/\Lambda\). These form a field.
(The Weierstrass functions)
Write \(\Lambda' \da \Lambda \smz\). \[ \wp(z) \da \frac{1}{z^2} + \sum_{\omega \in \Lambda'} \qty{ \frac{1}{(z-\omega)^2} - \frac{1}{\omega^2} }, \qquad \wp'(z) = \sum_{\omega \in \Lambda} \frac{-2}{(z-\omega)^3} . \] The series converges locally uniformly on \(\CC \setminus \Lambda\), so \(\wp\) is elliptic with a double pole at each lattice point and no other poles; \(\wp\) is even and \(\wp'\) is odd. Set the Eisenstein series of the lattice, \[ g_2 \da 60 \sum_{\omega \in \Lambda'} \frac{1}{\omega^4}, \qquad g_3 \da 140 \sum_{\omega \in \Lambda'} \frac{1}{\omega^6}, \qquad \Delta \da g_2^3 - 27 g_3^2 . \]
\[ (\wp')^2 = 4\wp^3 - g_2 \wp - g_3 , \] and the field of elliptic functions for \(\Lambda\) is \(\CC(\wp, \wp')\).
For \(z\) in a fixed compact set, the bracket \(\frac{1}{(z-\omega)^2} - \frac{1}{\omega^2}\) is \(O(\abs\omega^{-3})\), and \(\sum_{\omega\in\Lambda'} \abs{\omega}^{-3}\) converges, so the series for \(\wp\) converges; \(\sum_{\omega\in\Lambda'}\abs{\omega}^{-2}\) diverges. For the same reason the Eisenstein series \(G_{2k}\da\sum_{\omega \in \Lambda'} \omega^{-2k}\) converge absolutely for \(k\ge2\), and the odd-weight sums vanish because \(\Lambda = -\Lambda\).
Integrating an elliptic function \(f\) around the boundary of a fundamental parallelogram gives \(\sum \operatorname{Res} f = 0\) on \(\CC/\Lambda\), so no elliptic function has exactly one simple pole in a fundamental parallelogram. The function \(\wp\) has a single double pole there. Counting poles gives the degree of the induced map \(\CC/\Lambda\to\PP^1\): \(\wp\) has degree \(2\) and \(\wp'\) degree \(3\).
The differential equation is proved by subtraction: expand \[ \wp(z) = z^{-2} + \sum_{k \geq 1} (2k+1) G_{2k+2} z^{2k}, \] form \((\wp')^2 - 4\wp^3 + g_2\wp + g_3\), and check that its polar terms and its constant term at \(0\) vanish. The difference is an elliptic function with no poles, hence constant by Liouville on the compact torus, hence \(0\).
Every elliptic function lies in \(\CC(\wp,\wp')\): an even elliptic function is a rational function of \(\wp\) (match its zeros and poles in \(\wp\)-values), and a general \(f\) splits as \(\tfrac{1}{2}(f(z)+f(-z)) + \wp'(z) \cdot \tfrac{f(z)-f(-z)}{2\wp'(z)}\) with both brackets even.