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  "body": "\n## Statement\n\nLet $q$ be a prime power and let $X$ be a smooth, geometrically irreducible [[projective variety|A4H4CZ2V35]] of dimension $n$ over the [[finite field|04SGPNT8R9]] $\\F_q$.\nFor $m \\ge 1$ let $N_m$ be the number of [[rational points|SWJYG87QQH]] of $X$ over $\\F_{q^m}$.\nThe zeta function of $X$ is the power series\n$$Z(X, t) = \\exp\\Big(\\sum_{m \\ge 1} \\frac{N_m}{m} t^m\\Big),$$\nin the ring of formal power series over $\\Q$.\nThen the following hold.\n\n1. Rationality: $Z(X, t) = \\dfrac{P_1(t) P_3(t) \\cdots P_{2n-1}(t)}{P_0(t) P_2(t) \\cdots P_{2n}(t)}$ with $P_i(t) \\in \\Z[t]$, $P_0(t) = 1 - t$ and $P_{2n}(t) = 1 - q^n t$.\n2. Functional equation: $Z\\big(X, \\tfrac{1}{q^n t}\\big) = \\pm\\, q^{nE/2} \\, t^{E} \\, Z(X, t)$, where $E$ is the self-intersection number of the diagonal in $X \\times X$.\n3. Riemann hypothesis: for $1 \\le i \\le 2n - 1$, $P_i(t) = \\prod_j (1 - \\alpha_{ij} t)$, where the $\\alpha_{ij}$ are algebraic integers with $\\abs{\\alpha_{ij}} = q^{i/2}$ for every complex absolute value.\n4. Betti numbers: suppose $X$ is the good reduction modulo a prime of a smooth projective variety $Y$ over a number field $K \\subset \\C$.\n   Then $\\deg P_i$ is the $i$-th Betti number of the complex manifold $Y(\\C)$.\n\n## Remarks\n\nIn the proof one takes $P_i(t) = \\det(1 - tF \\mid H^i(\\bar{X}, \\Q_\\ell))$, where $F$ is the Frobenius acting on [[ℓ-adic cohomology|E1DVHN06CR]].\nThe [[Grothendieck–Lefschetz trace formula|P6QD6TKAGS]] then gives rationality, Poincaré duality gives the functional equation, and $E$ is the [[Euler characteristic|RNDDWDAFJ5]] of $\\bar{X}$.\nDwork proved rationality in 1960 by $p$-adic methods, Grothendieck proved rationality and the functional equation with étale cohomology, and Deligne proved the Riemann hypothesis in 1974.\nExample: for an elliptic curve $X$ over $\\F_q$, $Z(X, t) = \\dfrac{1 - a t + q t^2}{(1 - t)(1 - q t)}$ with $a = q + 1 - N_1$.\nThe Riemann hypothesis says that the two roots of $1 - a t + q t^2$ have inverses of absolute value $\\sqrt{q}$, which is [[Hasse's theorem|7Y26317GJ0]] $\\abs{a} \\le 2\\sqrt{q}$.\n\n## Explanations\n\n### L1 human\n\nTake a system of polynomial equations with integer coefficients and count its solutions in the finite fields with $q$, $q^2$, $q^3, \\ldots$ elements.\nWeil predicted that these counts are governed by finitely many numbers $\\alpha$, with $N_m$ a signed sum of their $m$-th powers.\nHe also predicted the sizes of these numbers and their link with the shape of the corresponding complex variety.\nAll four predictions are now theorems.\n\n### L2 human\n\nPoint (3) is the analogue of the [[Riemann hypothesis|VCGF9YR36X]] for the zeta function of a variety over a finite field.\nIt gives sharp estimates of point counts that are used throughout number theory.\nWeil proved the conjectures for curves and abelian varieties, and they motivated the creation of étale and $\\ell$-adic cohomology.\nThe local zeta functions $Z(X, t)$ at the primes of a number field are the factors of the [[Hasse–Weil zeta function|37TNNBQ4PK]].\nGrothendieck proposed his standard conjectures as a route to point (3), and they would give a second proof of it.\n",
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