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  "body": "\n## Statement\n\nA compact hyperbolic surface $X$ can be written as the space of orbits\n\n$$\\Gamma \\backslash \\mathbf{H},$$\n\nwhere $\\Gamma$ 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.\n\nThe 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).\n\nThen 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\n\n$$0 = \\mu_0 \u003c \\mu_1 \\leq \\mu_2 \\leq \\cdots$$\n\nwhere the eigenvalues $\\mu_{n}$ correspond to $\\Gamma$-invariant eigenfunctions $u$ in $C^{\\infty}(\\mathbf{H})$ of the Laplacian; in other words\n\n$$\\begin{cases} u(\\gamma z) = u(z), \\qquad \\forall \\gamma \\in \\Gamma \\\\ y^2 \\left (u_{xx} + u_{yy} \\right) + \\mu_{n} u = 0. \\end{cases}$$\n\nUsing the variable substitution\n\n$$\\mu = s(1-s), \\qquad s=\\tfrac{1}{2}+ir$$\n\nthe eigenvalues are labeled\n\n$$r_{n}, n \\geq 0.$$\n\nThen the **Selberg trace formula** is given by\n\n$$\\sum_{n=0}^\\infty h(r_n) = \\frac{\\mu(X)}{4 \\pi } \\int_{-\\infty}^\\infty r \\, h(r) \\tanh(\\pi r)\\,dr + \\sum_{ \\{T\\} } \\frac{ \\log N(T_0) }{ N(T)^{\\frac{1}{2}} - N(T)^{-\\frac{1}{2}} } g(\\log N(T)).$$\n\nThe 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:\n\n- be analytic on $|\\operatorname{Im}(r)| \\le \\tfrac{1}{2} + \\delta$;\n- $$h(-r) = h(r);$$\n- there exist positive constants $\\delta$ and $M$ such that: $$\\vert h(r) \\vert \\leq M \\left( 1+\\left| \\operatorname{Re}(r) \\right| \\right )^{-2-\\delta}.$$\n\nThe function $g$ is the Fourier transform of $h$, that is,\n\n$$h(r) = \\int_{-\\infty}^\\infty g(u) e^{iru} \\, du.$$\n",
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