Let $\Lambda=\mathbb{Z}+\mathbb{Z} \tau$ be a lattice in $\mathbb{C}$, where $\tau$ is a fixed complex number with positive imaginary part. The Weierstrass $\wp$-function is the unique meromorphic $\Lambda$-periodic function on $\mathbb{C}$ such that $\wp$ is holomorphic on $\mathbb{C} \backslash \Lambda$, and $\wp(z)-1 / z^{2} \rightarrow 0$ as $z \rightarrow 0$.

Show that $\wp(-z)=\wp(z)$ and find all the zeros of $\wp^{\prime}$ in $\mathbb{C}$.

Show that $\wp$ satisfies a differential equation

$$\wp^{\prime}(z)^{2}=Q(\wp(z))$$

for some cubic polynomial $Q(w)$. Further show that

$$Q(w)=4\left(w-\wp\left(\frac{1}{2}\right)\right)\left(w-\wp\left(\frac{1}{2} \tau\right)\right)\left(w-\wp\left(\frac{1}{2}(1+\tau)\right)\right)$$

and that the three roots of $Q$ are distinct.

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