TITLE: Riemann Theta Function On Hyperbolic Riemann Surfaces
QUESTION [6 upvotes]: 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?

REPLY [2 votes]: To amplify on ElucisusFTW's answer, the relationship between the hyperbolic metric and the complex structure is the so-called "accessory parameters" problem, and very little is known about it, except in the very special setting of hyperbolic punctured spheres (and the related punctured tori), where there is a large number of papers by Zograf and Takhtajan (warning, the latter name is spelled in many different ways). This is closely related also to string theory (which I know next to nothing about), and the determinants of Laplacians, which CAN be computed in terms of hyperbolic invariants. See, for example, Pollicott-Rocha and Strohmaier-Uski (note that the latter paper says that the former paper is wrong...)
Pollicott, Mark; Rocha, Andr\'e C., A remarkable formula for the determinant of the Laplacian, Invent. Math. 130, No.2, 399-414 (1997). ZBL0896.58067.
Strohmaier, Alexander; Uski, Ville, An algorithm for the computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces, Commun. Math. Phys. 317, No. 3, 827-869 (2013). ZBL1261.65113.