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  "body": "\n## Statement\n\nLet $\\F_q$ be a [[finite field|0ZXZ82FTX2]] with $q$ elements, and let $E$ be an [[elliptic curve|427N6S6QQW]] over $\\F_q$.\nLet $N = \\# E(\\F_q)$ be the number of points of $E$ over $\\F_q$, including the point at infinity.\nThen\n$$\\abs{N - (q + 1)} \\le 2\\sqrt{q}.$$\n\n## Remarks\n\nOne proof uses the $q$-power Frobenius endomorphism $\\phi$ of $E$.\nThe points of $E(\\F_q)$ form the kernel of $1 - \\phi$, which is separable, so $N = \\deg(1 - \\phi)$.\nThe degree is a positive definite quadratic form on the endomorphism ring, with $\\deg \\phi = q$, and the Cauchy–Schwarz inequality for this form gives the bound.\nExample: $E \\colon y^2 = x^3 + 1$ over $\\F_5$.\nThe cubes modulo $5$ run through all residues, so $x \\mapsto x^3 + 1$ takes each value once.\nThe value $0$ gives one point, the squares $1$ and $4$ give two points each, and the non-squares give none.\nWith the point at infinity, $N = 1 + 2 + 2 + 1 = 6 = q + 1$.\nThe number $a_q = q + 1 - N$ is the trace of Frobenius, and the theorem says $\\abs{a_q} \\le 2\\sqrt{q}$.\n\n## Explanations\n\n### L1 human\n\nAn elliptic curve over a finite field with $q$ elements has about $q + 1$ points.\nHasse's theorem says the error is at most $2\\sqrt{q}$, which is small compared to $q$.\n\n### L2 human\n\nThe theorem is the Riemann hypothesis for the zeta function of an elliptic curve over a finite field: the roots of $1 - a_q T + q T^2$ have absolute value $q^{-1/2}$.\nWeil generalized it to curves of any genus $g$, with bound $2g\\sqrt{q}$, and Deligne to all smooth projective varieties.\nThe bound defines the angles in the [[Sato–Tate conjecture|TRGBJYPBXH]] and is used to estimate the running time of algorithms in elliptic curve cryptography.\n",
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