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  "body": "\n## Example\n\nLet the [[modular group|VEKV1EM77B]] $\\Gamma = \\mathrm{SL}_2(\\Z)$ act on the [[upper half-plane|X6XGGH5FPW]] $\\mathbb H$ by [[Möbius transformations|0SG74WBFA2]], generated by $T \\tau = \\tau + 1$ and $S \\tau = -1/\\tau$.\nLet\n$$\\mathcal F = \\set{\\tau \\in \\mathbb H : \\abs{\\operatorname{Re} \\tau} \\le \\tfrac12, \\ \\abs{\\tau} \\ge 1}.$$\nThen $\\mathcal F$ is a [[fundamental domain|KNJA2NEZ9F]] for $\\Gamma$:\n\n1. every [[orbit|0YQPZB5Q1G]] of $\\Gamma$ in $\\mathbb H$ meets $\\mathcal F$;\n2. two distinct points of $\\mathcal F$ lie in the same orbit only if they lie on the boundary of $\\mathcal F$.\n\nCheck of 1: for $\\gamma = \\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$, one has $\\operatorname{Im}(\\gamma \\tau) = \\operatorname{Im} \\tau / \\abs{c\\tau + d}^2$.\nOnly finitely many pairs $(c, d)$ of integers have $\\abs{c\\tau + d} \\le 1$, so some $\\gamma$ makes $\\operatorname{Im}(\\gamma \\tau)$ maximal.\nApply a power of $T$ to move $\\gamma \\tau$ into the strip $\\abs{\\operatorname{Re} \\tau} \\le \\tfrac12$; this does not change the imaginary part.\nIf the result $\\tau'$ had $\\abs{\\tau'} \u003c 1$, then $\\operatorname{Im}(S \\tau') = \\operatorname{Im} \\tau' / \\abs{\\tau'}^2 \u003e \\operatorname{Im} \\tau'$, against maximality, so $\\tau' \\in \\mathcal F$.\n\nCheck of 2: there are two boundary identifications.\nThe map $T$ sends the line $\\operatorname{Re} \\tau = -\\tfrac12$ to the line $\\operatorname{Re} \\tau = \\tfrac12$.\nThe map $S$ sends the arc $\\abs{\\tau} = 1$ to itself by $\\tau \\mapsto -\\bar\\tau$.\nA short computation with $\\abs{c\\tau + d} \\ge 1$ on $\\mathcal F$ shows there are no other identifications.\n\n## Remarks\n\nThe hyperbolic area of $\\mathcal F$ is\n$$\\int_{-1/2}^{1/2} \\int_{\\sqrt{1 - x^2}}^{\\infty} \\frac{dy \\, dx}{y^2} = \\int_{-1/2}^{1/2} \\frac{dx}{\\sqrt{1 - x^2}} = 2 \\arcsin \\tfrac12 = \\frac{\\pi}{3}.$$\nIn $\\mathcal F$, the points with nontrivial stabilizer in $\\mathrm{PSL}_2(\\Z)$ are the following.\nThe point $i$ is fixed by $S$ of order $2$, and the corners $e^{2\\pi i/3}$ and $e^{\\pi i/3}$ are fixed by elements of order $3$.\nGluing the sides of $\\mathcal F$ by $T$ and $S$ gives the modular curve $\\Gamma \\backslash \\mathbb H$, a sphere with one puncture at the cusp $i\\infty$.\n\n## Explanations\n\n### L1 human\n\nThe modular group moves points of the upper half-plane around in many ways.\nThe region $\\mathcal F$, a strip of width one above the unit circle, contains a copy of every point up to these moves, and only its edges are counted twice.\nIts images under the group tile the whole half-plane by curved triangles.\nA modular form is determined by its values on this one region.\n",
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