Let $\Lambda$ be a lattice in $\mathbb{C}$ generated by 1 and $\tau$, where $\tau$ is a fixed complex number with $\operatorname{Im} \tau>0$. The Weierstrass $\wp$-function is defined as a $\Lambda$-periodic meromorphic function such that

(1) the only poles of $\wp$ are at points of $\Lambda$, and

(2) there exist positive constants $\varepsilon$ and $M$ such that for all $|z|<\varepsilon$, we have

$$\left|\wp(z)-1 / z^{2}\right|<M|z|$$

Show that $\wp$ is uniquely determined by the above properties and that $\wp(-z)=\wp(z)$. By considering the valency of $\wp$ at $z=1 / 2$, show that $\wp^{\prime \prime}(1 / 2) \neq 0$.

Show that $\wp$ satisfies the differential equation

$$\wp^{\prime \prime}(z)=6 \wp^{2}(z)+A,$$

for some complex constant $A$.

[Standard theorems about doubly-periodic meromorphic functions may be used without proof provided they are accurately stated, but any properties of the $\wp$-function that you use must be deduced from first principles.]