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  "body": "\n## Statement\n\nLet $\\Gamma \\subset \\mathrm{PSL}_2(\\R)$ be a torsion-free [[Fuchsian group|VNJWSCVP6V]] such that $X = \\Gamma \\backslash \\mathbb H$ is compact, where $\\mathbb H$ is the [[upper half-plane|X6XGGH5FPW]] with the hyperbolic metric $(dx^2 + dy^2)/y^2$.\nLet $0 = \\lambda_0 \u003c \\lambda_1 \\le \\lambda_2 \\le \\cdots$ be the [[eigenvalues|0BQAA2BK3Q]] of the [[Laplace–Beltrami operator|F4NWZNTRMD]] $\\Delta = -y^2 (\\partial_x^2 + \\partial_y^2)$ on $X$, repeated with multiplicity, and write $\\lambda_j = \\tfrac14 + r_j^2$ with $r_j \\in \\C$.\nLet $h$ be an even function, holomorphic on a strip $\\abs{\\operatorname{Im} r} \\le \\tfrac12 + \\varepsilon$ with $\\varepsilon \u003e 0$, such that $\\abs{h(r)} \\le C (1 + \\abs{r})^{-2-\\varepsilon}$ there.\nLet $g(u) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} h(r) e^{-iru} \\, dr$.\nThen\n$$\\sum_{j=0}^{\\infty} h(r_j) = \\frac{\\operatorname{area}(X)}{4\\pi} \\int_{-\\infty}^{\\infty} r \\, h(r) \\tanh(\\pi r) \\, dr + \\sum_{\\gamma} \\sum_{k=1}^{\\infty} \\frac{\\ell(\\gamma)}{2 \\sinh(k \\ell(\\gamma)/2)} \\, g(k \\ell(\\gamma)),$$\nwhere $\\gamma$ runs over the primitive oriented closed [[geodesics|NSYMQMHPDN]] of $X$ and $\\ell(\\gamma)$ is the length of $\\gamma$.\nAll sums and integrals converge absolutely.\n\n## Remarks\n\nThe closed geodesics of $X$ correspond to the nontrivial [[conjugacy classes|0DEBXAJ5SA]] of $\\Gamma$, and $k \\ell(\\gamma)$ is the length of the $k$-th power.\nThe proof computes the trace of an integral operator on $L^2(X)$ in two ways.\nThe kernel $\\sum_{\\sigma \\in \\Gamma} k(z, \\sigma w)$ is built from a point-pair invariant $k$ whose spectral transform is $h$.\nExpanding in eigenfunctions gives the left side; integrating the kernel over the diagonal and grouping $\\sigma$ by conjugacy classes gives the right side, with the identity giving the area term.\nExample: $h(r) = e^{-t(r^2 + 1/4)}$ with $t \u003e 0$ gives the trace of the heat kernel $\\sum_j e^{-t \\lambda_j}$, and the area term behaves like $\\operatorname{area}(X)/(4\\pi t)$ as $t \\to 0$.\nThis yields Weyl's law: the number of $\\lambda_j \\le T$ is asymptotic to $\\operatorname{area}(X) \\, T / (4\\pi)$.\nFor non-compact $X$ of finite area, such as $\\mathrm{SL}_2(\\Z) \\backslash \\mathbb H$, the formula needs extra terms from the cusps and from the continuous spectrum.\n\n## Explanations\n\n### L1 human\n\nOn a compact hyperbolic surface there are two natural lists of numbers.\nOne list is the frequencies at which the surface can vibrate, the eigenvalues of the Laplacian.\nThe other is the lengths of the closed geodesics, the loops that a light ray could travel forever.\nThe Selberg trace formula is an exact identity that links the two lists.\n\n### L2 human\n\nThe formula is a non-abelian analogue of the Poisson summation formula, which links a lattice in $\\R$ with its dual lattice.\nIt gives the prime geodesic theorem, the analytic properties of the Selberg zeta function, and Weyl's law for $X$.\nApplied to congruence subgroups with Hecke operators inserted, it links Maass forms with arithmetic, and it is the prototype of the Arthur–Selberg trace formula.\n",
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