Weil conjectures
In 1949, André Weil posed the landmark Weil conjectures about the local zeta-functions of algebraic varieties over finite fields. These conjectures offered a framework between algebraic geometry and number theory that propelled Alexander Grothendieck to recast the foundations making use of sheaf theory (together with Jean-Pierre Serre), and later scheme theory, in the 1950s and 1960s. Bernard Dwork proved one of the four Weil conjectures (rationality of the local zeta function) in 1960. Grothendieck developed étale cohomology theory to prove two of the Weil conjectures (together with Michael Artin and Jean-Louis Verdier) by 1965. The last of the Weil conjectures (an analogue of the Riemann hypothesis) would be finally proven in 1974 by Pierre Deligne.