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  "body": "\n## Description\n\nLet $\\mathbb{H} = \\set{\\tau \\in \\C : \\operatorname{Im}\\tau \u003e 0}$ be the [[upper half-plane|X6XGGH5FPW]], and for $\\tau \\in \\mathbb{H}$ let $q = e^{\\pi i \\tau}$.\nLet $\\theta_2$ and $\\theta_3$ be the [[theta functions|2F5CYQJVAF]]\n$$\\theta_2(\\tau) = \\sum_{n \\in \\Z} q^{(n + 1/2)^2}, \\qquad \\theta_3(\\tau) = \\sum_{n \\in \\Z} q^{n^2}.$$\nThe *modular lambda function* is\n$$\\lambda(\\tau) = \\frac{\\theta_2(\\tau)^4}{\\theta_3(\\tau)^4}, \\qquad \\tau \\in \\mathbb{H}.$$\n\n1. $\\lambda$ is holomorphic on $\\mathbb{H}$ and never takes the values $0$ and $1$.\n2. $\\lambda(\\tau + 1) = \\dfrac{\\lambda(\\tau)}{\\lambda(\\tau) - 1}$ and $\\lambda(-1/\\tau) = 1 - \\lambda(\\tau)$.\n3. $\\lambda$ is invariant under the principal [[congruence subgroup|1H6R856P3C]] $\\Gamma(2)$ of level $2$.\n4. $\\lambda(\\tau) = 16q + O(q^2)$ as $\\operatorname{Im}\\tau \\to \\infty$.\n\n## Remarks\n\nThe transformation rules in point 2 follow from those of the theta functions together with Jacobi's identity $\\theta_3^4 = \\theta_2^4 + \\theta_4^4$, where $\\theta_4(\\tau) = \\sum_{n} (-1)^n q^{n^2}$.\nFor example, $\\tau \\mapsto \\tau + 1$ changes $\\theta_2^4$ into $-\\theta_2^4$ and exchanges $\\theta_3$ and $\\theta_4$, which gives $-\\theta_2^4/\\theta_4^4 = \\lambda/(\\lambda - 1)$.\nThe map $\\lambda \\colon \\mathbb{H} \\to \\C \\setminus \\set{0, 1}$ is a holomorphic covering map.\nAn entire function that omits $0$ and $1$ lifts through this covering to an entire function with values in $\\mathbb{H}$.\nComposing with a Möbius map from $\\mathbb{H}$ onto the unit disc gives a bounded entire function, which is therefore constant; this is the classical proof of [[Picard's little theorem|R2PHM118NG]].\nThe [[j-invariant|FEC2XH822R]] is a rational function of $\\lambda$:\n$$j = 256\\,\\frac{(1 - \\lambda + \\lambda^2)^3}{\\lambda^2 (1 - \\lambda)^2}.$$\n\n## Explanations\n\n### L1 human\n\nThe lambda function is a holomorphic function on the upper half-plane with many symmetries.\nIt never takes the values $0$ and $1$, and every other complex value is taken infinitely often.\nThis makes it the tool that shows a nonconstant entire function can miss at most one complex value.\n",
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