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,
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
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
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.
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
2 Functional equation and Poincaré
3 Riemann hypothesis
4 Betti numbers
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