Étale Cohomology and the Weil Conjectures

We trace the development of étale cohomology from its origins \in Grothendieck's rewriting of algebraic geometry to its role \in Deligne's proof of the Riemann hypothesis for varieties over finite fields. The talk begins with the basic definitions of the étale site and sheaf cohomology, then develops the key comparison theorems relating étale and singular cohomology. We show how the Lefschetz trace formula, applied to the Frobenius endomorphism, yields rationality of the $\zeta$-function $Z(X, t) = exp\left(\sum_{n=1}^{\infty} \frac{|X(\mathbb{F}_{q^n})|}{n} t^n\right)$. The talk concludes with an outline of Deligne's argument for the analogue of the Riemann hypothesis.