Applications of Jacobi theta functions I've heard that the Jacobi theta function has wide applications, for instance in physics and proving elementary facts about the Fibonnaci sequence.
I'd be interested in knowing some of the most interesting applications: can anyone provide a reference?
 A: Disclaimer : I know nothing more about this that what I wrote. This application is from quantum field theory.
Let $M,V$ be manifold, the space of fields is $F = C^{\infty}(M,V)$. Classical field theory is basically solving Euler-Lagrange for some functional $S :F \to \Bbb R, \phi\mapsto \int_{M} L(\phi, \partial \phi) dx$, and understanding the geometry of the space of solutions. 
In quantum field theory, one needs also a measure space, so we can make sense of the partition function $Z = \int_F d\phi e^{i S(\phi)}$. We can then, for any observable $O : F \to \Bbb R$ define the expectation of $O$ as $E(O) = \frac{1}{Z} \int_F O(\phi)e^{i S(\phi)}$. It's usually divergent so one consider $Z(\hbar) = \int_F d\phi e^{i S(\phi)/\hbar}$ instead.
Now, we need to specialize a bit, we take $M = S^1$ and $V = S^1$. Every field can be written as a Fourier series + a term which contains its topological degree : $\phi(t) = \sum_{\Bbb Z} \phi_n e^{itn} + \omega Rt$ where $\omega$ is the topological degree and $R$ the radius of $V = S^1$ (we assume the radius of $M$ is $1$).
We have $$ Z(\hbar) = Z'(\hbar)(2 \pi R \sum_{w \in \Bbb Z} \exp(\frac{- \pi w^2 R^2}{\hbar}))$$ where $Z'(\hbar)$ does not depends on $R$, by direct computation. The $R$-dependant term is exactly $Z'' =  2 \pi R \theta (i R^2/\sqrt{\hbar})$, where $\theta$ is the Jacobi-theta function. 
Using the Poisson ressumation formula/Jacobi inversion we get $Z'' = 2 \pi \hbar^{1/2} \sum_{k \in \Bbb Z} \exp(-\frac{\pi \hbar k^2}{R^2})$
This last expression is better in physics, because the term in the exponential can be interpreted as a multiple of the "quantum energy of the field". As I said I'm just a poor mathematician so I don't really know more about the physical side of it, but there should be somewhere in the book "Quantum Fields and Strings: A Course for Mathematicians".
