Let $L$ be the lattice $\mathbb{Z} \omega_{1}+\mathbb{Z} \omega_{2}$ for two non-zero complex numbers $\omega_{1}, \omega_{2}$ whose ratio is not real. Recall that the Weierstrass function $\wp$ is given by the series

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

the function $\zeta$ is the (unique) odd anti-derivative of $-\wp$; and $\sigma$ is defined by the conditions

$$\sigma^{\prime}(u)=\zeta(u) \sigma(u) \quad \text { and } \quad \sigma^{\prime}(0)=1$$

(a) By writing a differential equation for $\sigma(-u)$, or otherwise, show that $\sigma$ is an odd function.

(b) Show that $\sigma\left(u+\omega_{i}\right)=-\sigma(u) \exp \left(a_{i}\left(u+b_{i}\right)\right)$ for some constants $a_{i}, b_{i}$. Use (a) to express $b_{i}$ in terms of $\omega_{i}$. [Do not attempt to express $a_{i}$ in terms of $\omega_{i}$.]

(c) Show that the function $f(u)=\sigma(2 u) / \sigma(u)^{4}$ is periodic with respect to the lattice $L$ and deduce that $f(u)=-\wp^{\prime}(u)$.