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  "body": "\n## Definition\n\nLet $\\tau$ be a point of the [[upper half-plane|X6XGGH5FPW]].\nThe *nome* of $\\tau$ is $q = e^{\\pi i\\tau}$; it satisfies $0 \u003c \\abs{q} \u003c 1$.\n\nFor a real modulus $0 \u003c k \u003c 1$, let $K = K(k)$ and $K' = K(\\sqrt{1 - k^2})$ be [[complete elliptic integrals of the first kind|RM54CPPE12]], and let $\\tau = iK'/K$.\nThe *nome of the modulus* $k$ is\n$$q(k) = e^{\\pi i\\tau} = \\exp\\!\\left(-\\frac{\\pi K'}{K}\\right).$$\n\n## Remarks\n\nFor real $k \\in (0, 1)$ the nome is real with $0 \u003c q \u003c 1$, and it increases from $0$ to $1$ as $k$ goes from $0$ to $1$.\nExample: at $k = 1/\\sqrt2$ one has $k = \\sqrt{1 - k^2}$, so $K = K'$ and $q = e^{-\\pi} \\approx 0.0432$.\nAs $k \\to 0$, $K \\to \\pi/2$ and $K' = \\log(4/k) + o(1)$, so $q = k^2/16 + O(k^4)$.\nNon-example: $\\tau = 0$ or real $\\tau$ gives $\\abs{q} = 1$, which is not a nome, since the theta series in $q$ do not converge there.\nNotation varies: some authors, especially in the theory of [[modular forms|GWKN5VG78A]], use $q = e^{2\\pi i\\tau}$ instead and call it the nome as well.\nWith the nome $q = e^{\\pi i\\tau}$ defined above, the [[Jacobi theta function|2F5CYQJVAF]] is $\\vartheta(z, \\tau) = \\sum_{n \\in \\Z} q^{n^2} e^{2\\pi i n z}$.\n",
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    "name": "Nome",
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        "url": "https://en.wikipedia.org/wiki/Nome_(mathematics)"
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  "name": "Nome",
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