Weil conjectures

Remarks

In 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. The Grothendieck–Lefschetz trace formula then gives rationality, Poincaré duality gives the functional equation, and $E$ is the Euler characteristic of $\bar{X}$. Dwork 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. Example: for an elliptic curve $X$ over $\F_q$, $Z(X, t) = \dfrac{1 - a t + q t^2}{(1 - t)(1 - q t)}$ with $a = q + 1 - N_1$. The 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 $\abs{a} \le 2\sqrt{q}$.

Going deeper

For a student

Point (3) is the analogue of the Riemann hypothesis for the zeta function of a variety over a finite field. It gives sharp estimates of point counts that are used throughout number theory. Weil proved the conjectures for curves and abelian varieties, and they motivated the creation of étale and $\ell$-adic cohomology. The local zeta functions $Z(X, t)$ at the primes of a number field are the factors of the Hasse–Weil zeta function. Grothendieck proposed his standard conjectures as a route to point (3), and they would give a second proof of it.