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  "body": "\n## Statement\n\nLet $E$ be an [[elliptic curve|427N6S6QQW]] over $\\Q$ without [[complex multiplication|D3HEJTVY7G]].\nFor each prime $p$ of good reduction for $E$, let $\\tilde{E}$ be the reduction of $E$ modulo $p$, a curve over the [[finite field|0ZXZ82FTX2]] $\\F_p$.\nLet $a_p = p + 1 - \\#\\tilde{E}(\\F_p)$.\nLet $\\theta_p \\in [0, \\pi]$ be the angle with $a_p = 2\\sqrt{p} \\cos \\theta_p$; it exists by [[Hasse's theorem|7Y26317GJ0]].\nThen for all $0 \\le \\alpha \\le \\beta \\le \\pi$,\n$$\\lim_{x \\to \\infty} \\frac{\\#\\set{p \\le x : p \\text{ good}, \\ \\alpha \\le \\theta_p \\le \\beta}}{\\#\\set{p \\le x : p \\text{ good}}} = \\frac{2}{\\pi} \\int_\\alpha^\\beta \\sin^2 \\theta \\, d\\theta.$$\n\n## Remarks\n\nIn words, the angles $\\theta_p$ are equidistributed with respect to the measure $\\frac{2}{\\pi} \\sin^2 \\theta \\, d\\theta$ on $[0, \\pi]$.\nThis is the image of the Haar measure of $\\mathrm{SU}(2)$ under the map that sends a matrix with eigenvalues $e^{\\pm i\\theta}$ to $\\theta$.\nThe conjecture was formulated independently by Mikio Sato and John Tate around 1960.\nTate showed that it follows from the meromorphic continuation and non-vanishing on $\\mathrm{Re}(s) \\ge 1$ of the symmetric power $L$-functions $L(\\mathrm{Sym}^m E, s)$ for all $m \\ge 1$.\nClozel, Harris, Shepherd-Barron and Taylor proved it for curves with multiplicative reduction at some prime, in work published in 2008.\nBarnet-Lamb, Geraghty, Harris and Taylor removed this condition in 2011, using potential automorphy of the symmetric powers.\nThe hypothesis of no complex multiplication is necessary.\nFor a CM curve, $a_p = 0$ for the good primes that are inert in the CM field, which are half of all primes.\nSo $\\theta_p = \\pi/2$ for these primes, and the limit distribution is different.\n\n## Explanations\n\n### L1 human\n\nFor each prime, the number of points of an elliptic curve modulo $p$ is close to $p + 1$, with an error that is at most $2\\sqrt{p}$ in size.\nRescaled to an angle between $0$ and $\\pi$, this error looks random as $p$ varies.\nThe theorem says exactly how it is distributed: angles near $\\pi/2$ are most common, and angles near $0$ and $\\pi$ are rare, with density proportional to $\\sin^2 \\theta$.\n\n### L2 human\n\nThe proof is a major application of automorphy lifting theorems in the [[Langlands program|J9T2M084W5]].\nThe statement generalises, through the notion of Sato–Tate group, to other motives and to modular forms.\nFor example, it is also known for non-CM holomorphic newforms of weight at least $2$.\n",
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