The Tate Conjecture

For a smooth projective variety X over a finitely generated field, the subspace of étale cohomology classes fixed by the Galois action on X(\bar{k}) is spanned by cycle classes (up to the usual Tate twist).

Proven for divisors in positive characteristic (Tate, 1960s) and for abelian varieties over number fields (Faltings, 1983 — part of his Fields-medal work); Kato–Trihan added divisor cases over function fields conditional on SW(NC). Beyond that, fragments only: K3 surfaces (Charles, Madapusi Pera, Maulik), some products of curves. It anchors the arithmetic side of the 'cycles are seen by cohomology' philosophy whose complex twin, the Hodge conjecture, already sits on our unsolved shelf.