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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?
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2 Answers 2

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The answer to the first question is easy: The hyperbolic metric defines a complex structure, and the Theta function only depends on the complex structure, so take that one.

The second question is much harder as it relates the algebraic/complex world (Jacobian, Theta function) with the trancendental/real-analytic world (the hyperbolic metric).

There are asymptotic results, however, relating pinching of curves to the periods of abelian differentials (starting the older work of Rauch, Masur and Royden). Check the keywords Rauch variation formulas, Satake-Baily-Borel compactification of moduli space, asymptotics of the L_2-Bergman metric, etc.

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  • $\begingroup$ ...Siegel (not Riemann-Siegel) theta function, ... $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 30, 2017 at 12:14
  • $\begingroup$ Dear @EluciusFTW, thanks for the answer. I am looking for a way to express the Riemann theta function in terms of Fenchel-Nielsen FN coordinates explicitly. The way that I am thinking is to first express the period matrix in terms of FN coordinates. This, I suppose, can be done by writing the holomorphic Abelian differential in terms of FN coordinates. Now the problem is to construct the $g$-dimensional space of holomorphic Abelian differential for a genus $g$ surface with $n$ puncture. $\endgroup$
    – QGravity
    Commented Jul 26, 2017 at 18:16
  • $\begingroup$ @EluciusFTW I was thinking about getting it from corresponding quantity, some holomorphic differential on the upper half plane, by using Poincare summation and then use the Maskit's representation of the generators of Fuchsian group in terms of FN coordinates, to express it in terms of FN coordinates. $\endgroup$
    – QGravity
    Commented Jul 26, 2017 at 18:19
  • $\begingroup$ Also, can you please mention a reference which explains the relation between accessory parameters and complex structure? $\endgroup$
    – QGravity
    Commented Jul 26, 2017 at 18:30
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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.

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  • $\begingroup$ Thank you for the answer and the references. I need the answer to my questions for some string theory project but for a different purpose and for the computation of determinants of Laplacian :). I am looking for a way to express the Riemann theta function in terms of Fenchel-Nielsen FN coordinates explicitly. The way that I am thinking is to first express the period matrix in terms of FN coordinates. This, I suppose, can be done by writing the holomorphic Abelian differential in terms of FN coordinates. $\endgroup$
    – QGravity
    Commented Jul 26, 2017 at 18:25
  • $\begingroup$ Now the problem is to construct the $g$-dimensional space of holomorphic Abelian differential for a genus $g$ surface with $n$ puncture. I was thinking about getting it from corresponding quantity, some holomorphic differential on the upper half plane, by using Poincare summation and then use the Maskit's representation of the generators of Fuchsian group in terms of FN coordinates, to express it in terms of FN coordinates. $\endgroup$
    – QGravity
    Commented Jul 26, 2017 at 18:26
  • $\begingroup$ Also, can you please mention a reference which explains the relation between accessory parameters and complex structure? $\endgroup$
    – QGravity
    Commented Jul 26, 2017 at 18:30
  • $\begingroup$ @QGravity check out this: books.google.com/… $\endgroup$
    – Igor Rivin
    Commented Jul 26, 2017 at 19:19

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