I was reading section 3.2.3 from riemannian surfaces by the way of complex analytic geometry,
Now we will construct all functions on the torus. We specialize to a lattice of the form $L=\mathbb Z+\tau\mathbb Z$ from some $\tau$ with $\operatorname{Im}(\tau)>0$. Fixing $\tau$ we define, $$\theta(z):=\sum_{n=-\infty}^{\infty}e^{i\pi(n^2\tau+2nz)}\tag1$$ Moreover, $$\theta(z+1)=\theta(z),\:\theta(z+\tau)=e^{-i\pi(\tau+2z)}\theta(z)$$ and $$\theta'(z+1)=\theta'(z),\: \theta'(z+\tau)=e^{-i\pi(\tau+2z)}\theta'(z)-2\pi ie^{-i\pi(\tau+2z)}\theta(z)$$
Question 1: I don't understand why the author define $\theta(z)$ in this way $(1)$? what's the motivation?
Later, again define, $$\theta^{(x)}(z):=\theta\left(z-\frac12-\frac\tau2-x\right)\tag2$$ And, $$\theta^{(x)}(z+\tau)=-e^{2\pi i(z-x)}\theta^{(x)}(z),\quad \theta^{(x)}(x)=0$$ Thus, for any finite collection $x_1,\cdots,x_N,y_1,\cdots,y_M\in\mathbb C$, perhaps with repetitions, the ratio $$\operatorname{R}(z):=\frac{\Pi_i\theta^{(x_i)}(z)}{\Pi_j\theta^{(y_j)}(z)}\tag3$$ is meromorphic in $\mathbb C$, has period $1$, and satisfies $$\operatorname{R}(z+\tau)=(-1)^{N-M}\exp\left(-2\pi i ((N-M)z+\sum_j y_j-\sum_i x_i)\right)R(z)$$ In particular, we see that if $N=M$ and $\sum_j y_j - \sum_i x_i\in\mathbb Z$, then $R$ descends to a meromorphic function on $\mathbb C/L$. Thus we have the following result.
Theorem $3.2.5:$ Any meromorphic function on a complex torus, whose zeros $[x_1],\cdots,[x_k]$ and poles $[y_1],\cdots,[y_k]$ (possibly with repetitions) satisfy $\sum (x_j-y_j)\in\mathbb Z$ is a quotient of two products of $\theta$ functions. The proof of Theorem 3.2.5 is left to the reader.
Question 2: I understand the whole section but couldn't relate the big picture to construct all functions on the torus. Can someone explain the big picture like why we need to define $\theta(z)$, then $\theta^{(x)}(z)$ and then $R(z)$?
