Given a lattice $L \subseteq \mathbb{C}$, Alain Robert defines a theta function as a meromorphic function such that $\theta(z+\omega)=a(\omega) e^{\pi h(\omega)(z+\frac{\omega}{2})} \theta(z)$ for all $\omega \in L$ where $a:L \to \mathbb{C}^\times$ and $h:L \to \mathbb{C}$ are functions.
A trivial theta function is defined to be one of the form $e^{\alpha z^2+\beta z}$.
He then shows that if $\theta$ is a theta function whose divisor has positive degree $d$, then the vector space of holomorphic theta functions of the same type as $\theta$ has dimension $d$.
The way he proves it is by first letting $\tau = \frac{\omega_2}{\omega_1}$, where $\omega_j$ generate the lattice, then setting $T(u) = a(\omega_1)^{-u} e^{-\frac{1}{2} \pi h(\omega_1) \omega_1 u^2} \theta(\omega_1 u)$. Essentially, this amounts to modifying the function by a trivial theta function so that $a(\omega_1)=1$ and $h(\omega_1)=0$, then scaling the lattice. In particular, $T(u+1)=T(u)$, so we can write $T(u)=F(e^{2 \pi i u})$, where $F$ is entire. We then find that the transformation law in the $\tau$ direction gives us $a_{l+d} = C^{-1} q^{2 l + d} a_l$, where $C$ is a constant, $q=e^{\pi i \tau}$, and the $a_l$ are the Laurent coefficients of $F$ for $l \in \mathbb{Z}$. This is essentially a periodic relation on the coefficients of period $d$, so we choose a set of coefficients representing the equivalences class mod $d$, and this allows us to view it as a vector space of dimension $d$.
My first main question is, how is such a proof motivated? The definition of $T(u)$ seems so messy, so how did someone come up with it? It does make sense in that it gives us a function which is periodic, but that still doesn't answer why when we do that does the transformation law in the other direction, i.e. adding $\tau$, end up being so nice as to give us the kind of picture we want.
What's interesting is that we reduce a global problem about these theta functions to a problem that merely looks locally at a specific point (in this case, at u=$i\infty$, or $e^{2 \pi i u} = 0$ if you like). Knowing that the dimension we are trying to compute is the dimension of a cohomology group, it seems like the Laurent series we are considering have something to do with the cohomology of the disk with the origin taken out, maybe with the origin considered as a point of order $d$. Is there a deeper explanation for this? Can one generalize this idea? Even if what I just said doesn't make sense, is there some way to apply the ideas behind the proof to other contexts?
It's also interesting to in a sense get rid of one of the directions, the $\omega_1$ direction through a transformation, and then get rid of it by using the exponential function. This seems like a kind of reduction, potentially one which could be repeated. Are there other examples of it?
