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  "body": "\n## Description\n\nLet $\\omega_1, \\omega_2 \\in \\C$ be linearly independent over $\\R$, and let $\\Lambda = \\Z\\omega_1 + \\Z\\omega_2$ be the lattice they span.\nThe *Weierstrass elliptic function* of $\\Lambda$ is\n$$\\wp(z) = \\frac{1}{z^2} + \\sum_{\\omega \\in \\Lambda \\setminus \\set{0}} \\left( \\frac{1}{(z - \\omega)^2} - \\frac{1}{\\omega^2} \\right), \\qquad z \\in \\C \\setminus \\Lambda.$$\nThe series converges absolutely and uniformly on compact subsets of $\\C \\setminus \\Lambda$, because $\\sum_{\\omega \\ne 0} \\abs{\\omega}^{-3} \u003c \\infty$.\n\n1. $\\wp$ is a [[meromorphic function|0NSD1TPHR0]] on $\\C$ whose [[poles|0WPQYW8MTE]] are exactly the points of $\\Lambda$, each a pole of order $2$.\n2. $\\wp$ is even, and $\\wp(z + \\omega) = \\wp(z)$ for all $\\omega \\in \\Lambda$, so $\\wp$ is an [[elliptic function|FF9ZPCZ59K]] for $\\Lambda$.\n3. $\\wp$ satisfies the differential equation $(\\wp')^2 = 4\\wp^3 - g_2 \\wp - g_3$, where $g_2 = 60 \\sum_{\\omega \\ne 0} \\omega^{-4}$ and $g_3 = 140 \\sum_{\\omega \\ne 0} \\omega^{-6}$.\n\n## Remarks\n\nPeriodicity is checked on the derivative first: $\\wp'(z) = -2 \\sum_{\\omega \\in \\Lambda} (z - \\omega)^{-3}$ is visibly $\\Lambda$-periodic.\nSo $\\wp(z + \\omega_1) - \\wp(z)$ is a constant $c$; putting $z = -\\omega_1/2$ and using that $\\wp$ is even gives $c = 0$.\nThe differential equation follows because $(\\wp')^2 - 4\\wp^3 + g_2\\wp + g_3$ is elliptic without poles, hence constant, and its Laurent expansion at $0$ shows the constant is $0$.\nEvery elliptic function for $\\Lambda$ is a rational function of $\\wp$ and $\\wp'$.\nThe map $z \\mapsto (\\wp(z), \\wp'(z))$ identifies the torus $\\C / \\Lambda$ minus one point with the affine cubic curve $y^2 = 4x^3 - g_2 x - g_3$.\n\n## Explanations\n\n### L1 human\n\nThe Weierstrass function is the simplest nonconstant function on the plane that repeats itself along two independent directions.\nIt is built by placing a double pole at each point of a lattice and adding up, with correction terms that make the sum converge.\nIts square-derivative identity turns the torus into a cubic curve, which is the link between complex analysis and elliptic curves.\n",
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