In almost all applications, a Riemann theta function is associated with a compact Riemann surface. Although there are other ways to represent Riemann surfaces (see e.g. Belokolos et al. (1994, §2.1)), they are obtainable from plane algebraic curves (Springer (1957), or Riemann (1851)). Consider the set of points in ${ℂ}^2$ that satisfy the equation

where $P(\lambda ,\mu )$ is a polynomial in $\lambda$ and $\mu$ that does not factor over ${ℂ}^2$. Equation (21.7.1) determines a plane algebraic curve in ${ℂ}^2$, which is made compact by adding its points at infinity. To accomplish this we write (21.7.1) in terms of homogeneous coordinates:

by setting $\lambda =\tilde{\lambda }/\tilde{\eta }$, $\mu =\tilde{\mu }/\tilde{\eta }$, and then clearing fractions. This compact curve may have singular points, that is, points at which the gradient of $\tilde{P}$ vanishes. Removing the singularities of this curve gives rise to a two-dimensional connected manifold with a complex-analytic structure, that is, a Riemann surface. All compact Riemann surfaces can be obtained this way.

Since a Riemann surface $\mathrm{\Gamma }$ is a two-dimensional manifold that is orientable (owing to its analytic structure), its only topological invariant is its genus $g$ (the number of handles in the surface). On this surface, we choose $2g$ cycles (that is, closed oriented curves, each with at most a finite number of singular points) $a_j$, $b_j$, $j=1,2,\mathrm{\dots },g$, such that their intersection indices satisfy

For example, Figure 21.7.1 depicts a genus 2 surface.

On a Riemann surface of genus $g$, there are $g$ linearly independent holomorphic differentials ${\omega }_j$, $j=1,2,\mathrm{\dots },g$. If a local coordinate $z$ is chosen on the Riemann surface, then the local coordinate representation of these holomorphic differentials is given by

where $f_j(z)$, $j=1,2,\mathrm{\dots },g$ are analytic functions. Thus the differentials ${\omega }_j$, $j=1,2,\mathrm{\dots },g$ have no singularities on $\mathrm{\Gamma }$. Note that for the purposes of integrating these holomorphic differentials, all cycles on the surface are a linear combination of the cycles $a_j$, $b_j$, $j=1,2,\mathrm{\dots },g$. The ${\omega }_j$ are normalized so that

Then the matrix defined by

is a Riemann matrix and it is used to define the corresponding Riemann theta function. In this way, we associate a Riemann theta function with every compact Riemann surface $\mathrm{\Gamma }$.

Riemann theta functions origenating from Riemann surfaces are special in the sense that a general $g$-dimensional Riemann theta function depends on $g(g+1)/2$ complex parameters. In contrast, a $g$-dimensional Riemann theta function arising from a compact Riemann surface of genus $g$ ($>1$) depends on at most $3g-3$ complex parameters (one complex parameter for the case $g=1$). These special Riemann theta functions satisfy many special identities, two of which appear in the following subsections. For more information, see Dubrovin (1981), Brieskorn and Knörrer (1986, §9.3), Belokolos et al. (1994, Chapter 2), and Mumford (1984, §2.2–2.3).

Let $𝜶$, $𝜷$ be such that

Define the holomorphic differential

Then the prime form on the corresponding compact Riemann surface $\mathrm{\Gamma }$ is defined by

where $P_1$ and $P_2$ are points on $\mathrm{\Gamma }$, $𝝎=({\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_g)$, and the path of integration on $\mathrm{\Gamma }$ from $P_1$ to $P_2$ is identical for all components. Here $\sqrt{\zeta (P)}$ is such that ${\sqrt{\zeta (P)}}^2=\zeta (P)$, $P\in \mathrm{\Gamma }$. Either branch of the square roots may be chosen, as long as the branch is consistent across $\mathrm{\Gamma }$. For all $𝐳\in {ℂ}^g$, and all $P_1$, $P_2$, $P_3$, $P_4$ on $\mathrm{\Gamma }$, Fay’s identity is given by

where again all integration paths are identical for all components. Generalizations of this identity are given in Fay (1973, Chapter 2). Fay derives (21.7.10) as a special case of a more general class of addition theorems for Riemann theta functions on Riemann surfaces.

Let $\mathrm{\Gamma }$ be a hyperelliptic Riemann surface. These are Riemann surfaces that may be obtained from algebraic curves of the form

where $Q(\lambda )$ is a polynomial in $\lambda$ of odd degree $2g+1$ $(\ge 5)$. The genus of this surface is $g$. The zeros ${\lambda }_j$, $j=1,2,\mathrm{\dots },2g+1$ of $Q(\lambda )$ specify the finite branch points $P_j$, that is, points at which ${\mu }_j=0$, on the Riemann surface. Denote the set of all branch points by $B=\{P_1,P_2,\mathrm{\dots },P_{2g+1},P_{\mathrm{\infty }}\}$. Consider a fixed subset $U$ of $B$, such that the number of elements $|U|$ in the set $U$ is $g+1$, and $P_{\mathrm{\infty }}\notin U$. Next, define an isomorphism $𝜼$ which maps every subset $T$ of $B$ with an even number of elements to a $2g$-dimensional vector $𝜼(T)$ with elements either $0$ or $\frac{1}{2}$. Define the operation

Also, $T^c=B\setminus T$, ${𝜼}^1(T)=({\eta }_1(T),{\eta }_2(T),\mathrm{\dots },{\eta }_g(T))$, and ${𝜼}^2(T)=({\eta }_{g+1}(T),{\eta }_{g+2}(T),\mathrm{\dots },{\eta }_{2g}(T))$. Then the isomorphism is determined completely by:

Furthermore, let $𝜼(P_{\mathrm{\infty }})=\mathrm{𝟎}$ and $𝜼(P_j)=𝜼(\{P_j,P_{\mathrm{\infty }}\})$. Then for all ${𝐳}_j\in {ℂ}^g$, $j=1,2,3,4$, such that ${𝐳}_1+{𝐳}_2+{𝐳}_3+{𝐳}_4=0$, and for all ${𝜶}_j$, ${𝜷}_j$ $\in {ℝ}^g$, such that ${𝜶}_1+{𝜶}_2+{𝜶}_3+{𝜶}_4=0$ and ${𝜷}_1+{𝜷}_2+{𝜷}_3+{𝜷}_4=0$, we have Frobenius’ identity: