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  "body": "\n## Statement\n\nLet $q$ be a power of a prime $p$, and let $\\ell \\ne p$ be another prime.\nLet $X$ be a separated [[algebraic variety|HV6YBS0YDH]] over the finite field $\\F_q$.\nLet $\\bar{X} = X \\times_{\\F_q} \\bar{\\F}_q$, where $\\bar{\\F}_q$ is an algebraic closure of $\\F_q$.\nLet $H^i_c(\\bar{X}, \\Q_\\ell)$ be the [[ℓ-adic cohomology|E1DVHN06CR]] of $\\bar{X}$ with compact support.\nLet $F$ be the endomorphism of it induced by the [[Frobenius|0T3WVTD675]] morphism of $X$, which raises coordinates to the power $q$.\nThen the number of [[rational points|SWJYG87QQH]] of $X$ is\n$$\\# X(\\F_q) = \\sum_{i=0}^{2 \\dim X} (-1)^i \\, \\tr\\big(F \\mid H^i_c(\\bar{X}, \\Q_\\ell)\\big),$$\nwhere $\\tr$ is the [[trace|0RPDTWY2JS]] of a linear map.\n\n## Remarks\n\nThe points of $X(\\F_q)$ are the fixed points of the Frobenius morphism of $\\bar{X}$, so the formula is a Lefschetz fixed point formula, with the [[Lefschetz number|ZSYCA4EGEZ]] on the right.\nGrothendieck proved it in a more general form, for a constructible $\\ell$-adic sheaf $\\mathcal{F}$ on $X$.\nThen the sum over $x \\in X(\\F_q)$ of the traces of Frobenius on the stalks $\\mathcal{F}_{\\bar{x}}$ equals the alternating sum of the traces of $F$ on $H^i_c(\\bar{X}, \\mathcal{F})$.\nExample: for $X = \\mathbb{A}^1$, the only nonzero group is $H^2_c(\\bar{X}, \\Q_\\ell)$, of dimension $1$, on which $F$ acts by $q$.\nThe formula gives $\\# \\mathbb{A}^1(\\F_q) = q$.\nFor $X = \\mathbb{P}^1$, the groups $H^0$ and $H^2$ contribute $1$ and $q$, and the count is $q + 1$.\nCompact supports are needed when $X$ is not proper: the ordinary cohomology of $\\mathbb{A}^1_{\\bar{\\F}_q}$ is $\\Q_\\ell$ in degree $0$ only, which would give the wrong count $1$.\nApplied to $X \\times \\F_{q^m}$ for all $m$, the formula expresses the zeta function of $X$ as an alternating product of characteristic polynomials of $F$, which proves its rationality.\n\n## Explanations\n\n### L1 human\n\nCounting the solutions of polynomial equations over a finite field looks like pure arithmetic.\nThe formula says that this count can be read off from a geometric object: the cohomology of the variety, together with the action of the Frobenius map on it.\nThe points over $\\F_q$ are exactly the points that the Frobenius map does not move, so the count is a count of fixed points, as in topology.\n\n### L2 human\n\nThis is the main bridge between point counts over finite fields and $\\ell$-adic cohomology.\nIt gives the rationality and the cohomological form of the zeta function.\nDeligne's bounds on the eigenvalues of $F$ then give estimates such as $\\abs{\\# C(\\F_q) - q - 1} \\le 2g\\sqrt{q}$ for a smooth projective curve $C$ of genus $g$.\nIn the sheaf version it turns sums of characters, such as Gauss and Kloosterman sums, into traces on cohomology, which is how they are bounded.\n",
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