Let $\tau$ be a fixed complex number with positive imaginary part. For $z \in \mathbb{C}$, define

$$v(z)=\sum_{n=-\infty}^{\infty} \exp \left(\pi i \tau n^{2}+2 \pi i n\left(z+\frac{1}{2}\right)\right) .$$

Prove the identities

$$v(z+1)=v(z), \quad v(-z)=v(z), \quad v(z+\tau)=-\exp (-\pi i \tau-2 \pi i z) \cdot v(z)$$

and deduce that $v(\tau / 2)=0$. Show further that $\tau / 2$ is the only zero of $v$ in the parallelogram $P$ with vertices $-1 / 2,1 / 2,1 / 2+\tau,-1 / 2+\tau$.

[You may assume that $v$ is holomorphic on $\mathbb{C}$.]

Now let $\left\{a_{1}, \ldots, a_{k}\right\}$ and $\left\{b_{1}, \ldots, b_{k}\right\}$ be two sets of complex numbers and

$$f(z)=\prod_{j=1}^{k} \frac{v\left(z-a_{j}\right)}{v\left(z-b_{j}\right)}$$

Prove that $f$ is a doubly-periodic meromorphic function, with periods 1 and $\tau$, if and only if $\sum_{j=1}^{k}\left(a_{j}-b_{j}\right)$ is an integer.