Weil Conjectures

The Weil Conjectures were highly influential proposals made by Andre Weil in 1949 describing deep regularities in the number of solutions a system of polynomial equations has over finite fields, encoded in a generating function analogous to the Riemann zeta function. They predicted that this generating function is rational, satisfies a precise functional equation, and obeys an analogue of the Riemann hypothesis. The conjectures led to a successful multi-decade program in which many leading researchers, including Alexander Grothendieck, developed much of the framework of modern algebraic geometry, and Pierre Deligne completed the proof of the final and hardest conjecture in 1974.