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  "body": "\n## Description\n\nThe upper half-plane is the set\n$$\\mathbb{H} = \\set{\\tau \\in \\C : \\operatorname{Im} \\tau \u003e 0}$$\nof [[complex numbers|09FTAET5Z4]] with positive imaginary part.\nIt is an open subset of $\\C$, so it is a [[Riemann surface|5K0N8QMV6Q]].\nThe group $\\mathrm{SL}_2(\\R)$ acts on $\\mathbb{H}$ by [[Möbius transformations|0SG74WBFA2]] $\\tau \\mapsto (a\\tau + b)/(c\\tau + d)$.\n\n## Remarks\n\nThe action is well defined because\n$$\\operatorname{Im}\\left(\\frac{a\\tau + b}{c\\tau + d}\\right) = \\frac{(ad - bc) \\operatorname{Im} \\tau}{\\abs{c\\tau + d}^2},$$\nwhich is positive when $ad - bc = 1$ and $\\operatorname{Im} \\tau \u003e 0$.\nThe holomorphic bijections of $\\mathbb{H}$ onto itself are exactly these Möbius transformations, and they form the group $\\mathrm{PSL}_2(\\R)$.\nThe map $\\tau \\mapsto (\\tau - i)/(\\tau + i)$ is a biholomorphic map of $\\mathbb{H}$ onto the open unit disc.\nWith the metric $ds^2 = (dx^2 + dy^2)/y^2$, where $\\tau = x + iy$, the upper half-plane is a model of the hyperbolic plane, and $\\mathrm{PSL}_2(\\R)$ acts by isometries.\n[[Modular forms|GWKN5VG78A]] are functions on $\\mathbb{H}$.\n\n## Explanations\n\n### L1 human\n\nThe upper half-plane is the part of the complex plane above the real axis.\nIt is the stage on which modular forms live.\nThe [[modular group|VEKV1EM77B]] moves its points around, for example by $\\tau \\mapsto \\tau + 1$ and $\\tau \\mapsto -1/\\tau$, and never pushes them below the axis.\nWith a suitable way of measuring length, it is a picture of non-Euclidean, hyperbolic geometry.\n",
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