The Weil Conjectures

By Edward Frenkel
Scientific American guest blog, May 21, 2013

Edited by Andy Ross

The 2013 Abel Prize goes to Pierre Deligne, at the Institute for Advanced Study in Princeton. Deligne worked on the interface of number theory and geometry. He proved the last and deepest Weil conjecture.

André Weil had suggested from prison in 1940 that sentences written in the language of number theory could be translated into the language of geometry, and vice versa. Weil saw that given an algebraic equation, such as

x2 + y2 = 1,

we can look for its solutions in different domains, such as real or complex numbers, or in natural numbers modulo N. For example, solutions of the equation in real numbers form a circle, but solutions in complex numbers form a sphere. The same equation has many avatars. The avatars of algebraic equations in complex numbers give us geometric shapes like the sphere or the torus. Solutions in natural numbers modulo N give us more elusive avatars.

Weil organized the solutions modulo N in a way that made them look like geometric shapes. The Weil conjectures did for mathematics what quantum mechanics and relativity theory did for physics.

Weil Conjectures

The Weil conjectures concern the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields.

André Weil conjectured in a 1949 paper that such zeta-functions should be rational functions, should satisfy a form of functional equation, and should have their zeroes in restricted places.

There are four Weil conjectures:

1 Rationality
2 Functional equation and Poincaré duality
3 Riemann hypothesis
4 Betti numbers

Their statements are very technical.

Weil proved the conjectures for the special case of curves over finite fields. The proposed connection with algebraic topology was the great novelty. Given that finite fields are discrete and topology is continuous, his detailed formulation using examples was striking. It suggested that geometry over finite fields should fit into known patterns relating to Betti numbers, the Lefschetz fixed-point theorem, and so on.

The analogy with topology suggested creating a new homological theory to apply within algebraic geometry. This took two decades. Conjecture 1 was proved first by Bernard Dwork in 1960, using
p-adic methods. In 1965, Alexander Grothendieck and his collaborators proved conjectures 1, 2, and 4 using (their new) étale cohomology. Conjecture 3 was the hardest to prove. Pierre Deligne first proved it in 1974, using the étale cohomology theory. Then, in 1980, he found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf.

AR I abandoned my attempt to study sheaf theory in 1975.