Grothendieck–Lefschetz trace formula

Remarks

The 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 on the right. Grothendieck proved it in a more general form, for a constructible $\ell$-adic sheaf $\mathcal{F}$ on $X$. Then 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 $H^i_c(\bar{X}, \mathcal{F})$. Example: 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 $q$. The formula gives $\# \mathbb{A}^1(\F_q) = q$. For $X = \mathbb{P}^1$, the groups $H^0$ and $H^2$ contribute $1$ and $q$, and the count is $q + 1$. Compact 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 $1$. Applied 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.

Going deeper

For a student

This is the main bridge between point counts over finite fields and $\ell$-adic cohomology. It gives the rationality and the cohomological form of the zeta function. Deligne'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 $g$. In the sheaf version it turns sums of characters, such as Gauss and Kloosterman sums, into traces on cohomology, which is how they are bounded.