For a smooth projective variety X over \(F_{q}\), the eigenvalues of Frobenius on the i-th l-adic cohomology group \(H^{i}\)(X_{\bar{q}}, ℚ_ℓ) have absolute value \(q^{i/2}\). This is the Riemann Hypothesis analogue for zeta functions of varieties.
Conjectured by Weil (1949) after his wartime computations on curves. Rationality proven by Dwork (1960, p-adic analysis); functional equation by Grothendieck et al (1964-65, étale cohomology); the hard part — the Riemann hypothesis — required Grothendieck's full cohomological formalism plus Deligne's monodromy and Rankin-style estimates.
Proven by Pierre Deligne, 1974 (publication of the Weil I paper from 1968-74 work), using his strengthening of the Castelnuovo inequality and careful analysis of Lefschetz pencils. The proof built an entirely new cohomology theory and demonstrated that arithmetic problems over finite fields are governed by topological intuition about complex varieties — the foundational insight of modern motivic mathematics. Applications include the Ramanujan-tau bound and optimal coding theory constructions.