Let $\Lambda$ be a lattice in $\mathbb{C}$ generated by 1 and $\tau$, where $\tau$ is a fixed complex number with non-zero imaginary part. Suppose that $f$ is a meromorphic function on $\mathbb{C}$ for which the poles of $f$ are precisely the points in $\Lambda$, and for which $f(z)-1 / z^{2} \rightarrow 0$ as $z \rightarrow 0$. Assume moreover that $f^{\prime}(z)$ determines a doubly periodic function with respect to $\Lambda$ with $f^{\prime}(-z)=-f^{\prime}(z)$ for all $z \in \mathbb{C} \backslash \Lambda$. Prove that:

(i) $f(-z)=f(z)$ for all $z \in \mathbb{C} \backslash \Lambda$.

(ii) $f$ is doubly periodic with respect to $\Lambda$.

(iii) If it exists, $f$ is uniquely determined by the above properties.

(iv) For some complex number $A, f$ satisfies the differential equation $f^{\prime \prime}(z)=6 f(z)^{2}+A$.