Selberg trace formula for compact hyperbolic surfaces

Statement

A compact hyperbolic surface $X$ can be written as the space of orbits

$$\Gamma \backslash \mathbf{H},$$

where $\Gamma$ is a subgroup of $\operatorname{PSL}(2, \mathbf{R})$, and $\mathbf{H}$ is the upper half plane, and $\Gamma$ acts on $\mathbf{H}$ by linear fractional transformations.

The Selberg trace formula for this case is easier than the general case because the surface is compact so there is no continuous spectrum, and the group $\Gamma$ has no parabolic or elliptic elements (other than the identity).

Then the spectrum for the Laplace–Beltrami operator on $X$ is discrete and real, since the Laplace operator is self adjoint with compact resolvent; that is

$$0 = \mu_0 < \mu_1 \leq \mu_2 \leq \cdots$$

where the eigenvalues $\mu_{n}$ correspond to $\Gamma$-invariant eigenfunctions $u$ in $C^{\infty}(\mathbf{H})$ of the Laplacian; in other words

$$\begin{cases} u(\gamma z) = u(z), \qquad \forall \gamma \in \Gamma \\ y^2 \left (u_{xx} + u_{yy} \right) + \mu_{n} u = 0. \end{cases}$$

Using the variable substitution

$$\mu = s(1-s), \qquad s=\tfrac{1}{2}+ir$$

the eigenvalues are labeled

$$r_{n}, n \geq 0.$$

Then the Selberg trace formula is given by

$$\sum_{n=0}^\infty h(r_n) = \frac{\mu(X)}{4 \pi } \int_{-\infty}^\infty r \, h(r) \tanh(\pi r)\,dr + \sum_{ \{T\} } \frac{ \log N(T_0) }{ N(T)^{\frac{1}{2}} - N(T)^{-\frac{1}{2}} } g(\log N(T)).$$

The right hand side is a sum over conjugacy classes of the group $\Gamma$, with the first term corresponding to the identity element and the remaining terms forming a sum over the other conjugacy classes $\{T\,\}$ (which are all hyperbolic in this case). The function $h$ has to satisfy the following:

• be analytic on $|\operatorname{Im}(r)| \le \tfrac{1}{2} + \delta$;
• $$h(-r) = h(r);$$
• there exist positive constants $\delta$ and $M$ such that: $$\vert h(r) \vert \leq M \left( 1+\left| \operatorname{Re}(r) \right| \right )^{-2-\delta}.$$

The function $g$ is the Fourier transform of $h$, that is,

$$h(r) = \int_{-\infty}^\infty g(u) e^{iru} \, du.$$