The Riemann theta function for a genus $g$ closed Riemann surface with period matrix $\tau=[\tau_{ij}]$ is defined by

$$\theta(\{z_1,\cdots,z_g\}|\tau)=\Sigma_{n\in\mathbb{Z}^g}e^{\pi i(n\cdot\tau\cdot n + 2n\cdot z)}$$

The zeroth of this function is given by theta divisor, a divisor class $\Theta=W_{g-1}+\mathcal{K}$ in which $W_{g-1}$ is a degree $g-1$ divisor and $\mathcal{K}$ is the Riemann constant vector. I have two questions:

- How can Riemann theta function be defined for a hyperbolic Riemann surface?
- How can the theta divisor i.e. the locus of the zeroes of Riemann theta function be characterized in hyperbolic geometry? More specifically, Is there a way to characterize the theta divisor in terms of Fenchel-Nielsen coordinates on the Teichmuller space of hyperbolic Riemann surfaces?