They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
The generating function has coefficients derived from the numbers Nk of points over the extension field with qk elements.
The rationality was proved by Bernard Dwork (1960), the functional equation by Alexander Grothendieck (1965), and the analogue of the Riemann hypothesis by Pierre Deligne (1974).
The earliest antecedent of the Weil conjectures is by Carl Friedrich Gauss and appears in section VII of his Disquisitiones Arithmeticae (Mazur 1974), concerned with roots of unity and Gaussian periods.
In article 358, he moves on from the periods that build up towers of quadratic extensions, for the construction of regular polygons; and assumes that p is a prime number congruent to 1 modulo 3.
Gauss constructs the order-3 periods, corresponding to the cyclic group (Z/pZ)× of non-zero residues modulo p under multiplication and its unique subgroup of index three.
Gauss's determination of the coefficients of the products of the periods therefore counts the number of points on these elliptic curves, and as a byproduct he proves the analog of the Riemann hypothesis.
What was really eye-catching, from the point of view of other mathematical areas, was the proposed connection with algebraic topology.
Given that finite fields are discrete in nature, and topology speaks only about the continuous, the detailed formulation of Weil (based on working out some examples) was striking and novel.
It suggested that geometry over finite fields should fit into well-known patterns relating to Betti numbers, the Lefschetz fixed-point theorem and so on.
This took two decades (it was a central aim of the work and school of Alexander Grothendieck) building up on initial suggestions from Serre.
Deligne (1980) found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf.
Suppose that X is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements.
The zeta function ζ(X, s) of X is by definition where Nm is the number of points of X defined over the degree m extension Fqm of Fq.
The Weil conjectures state: The simplest example (other than a point) is to take X to be the projective line.
It is also easy to prove the Weil conjectures for other spaces, such as Grassmannians and flag varieties, which have the same "paving" property.
In algebraic topology the number of fixed points of an automorphism can be worked out using the Lefschetz fixed-point theorem, given as an alternating sum of traces on the cohomology groups.
So if there were similar cohomology groups for varieties over finite fields, then the zeta function could be expressed in terms of them.
To see this consider the case of a supersingular elliptic curve over a finite field of characteristic p. The endomorphism ring of this is an order in a quaternion algebra over the rationals, and should act on the first cohomology group, which should be a 2-dimensional vector space over the coefficient field by analogy with the case of a complex elliptic curve.
The functional equation for the zeta function follows from Poincaré duality for ℓ-adic cohomology, and the relation with complex Betti numbers of a lift follows from a comparison theorem between ℓ-adic and ordinary cohomology for complex varieties.
More generally, Grothendieck proved a similar formula for the zeta function (or "generalized L-function") of a sheaf F0: as a product over cohomology groups: The special case of the constant sheaf gives the usual zeta function.
Verdier (1974), Serre (1975), Katz (1976) and Freitag & Kiehl (1988) gave expository accounts of the first proof of Deligne (1974).
Deligne's first proof of the remaining third Weil conjecture (the "Riemann hypothesis conjecture") used the following steps: The heart of Deligne's proof is to show that the sheaf E over U is pure, in other words to find the absolute values of the eigenvalues of Frobenius on its stalks.
Langlands (1970, section 8) pointed out that a generalization of Rankin's result for higher even values of k would imply the Ramanujan conjecture, and Deligne realized that in the case of zeta functions of varieties, Grothendieck's theory of zeta functions of sheaves provided an analogue of this generalization.
The deduction of the Riemann hypothesis from this estimate is mostly a fairly straightforward use of standard techniques and is done as follows.
Deligne (1980) found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf.
In practice it is this generalization rather than the original Weil conjectures that is mostly used in applications, such as the hard Lefschetz theorem.
The main extra idea needed is an argument closely related to the theorem of Jacques Hadamard and Charles Jean de la Vallée Poussin, used by Deligne to show that various L-series do not have zeros with real part 1.
A constructible sheaf on a variety over a finite field is called pure of weight β if for all points x the eigenvalues of the Frobenius at x all have absolute value N(x)β/2, and is called mixed of weight ≤ β if it can be written as repeated extensions by pure sheaves with weights ≤ β. Deligne's theorem states that if f is a morphism of schemes of finite type over a finite field, then Rif!
Inspired by the work of Witten (1982) on Morse theory, Laumon (1987) found another proof, using Deligne's ℓ-adic Fourier transform, which allowed him to simplify Deligne's proof by avoiding the use of the method of Hadamard and de la Vallée Poussin.