Weil conjectures

In mathematics, the Weil conjectures were highly influential proposals by . 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 conjectures concern the generating functions (known as local zeta functions) derived from counting points on algebraic varieties over finite fields. A variety over a finite field with elements has a finite number of rational points (with coordinates in the original field), as well as points with coordinates in any finite extension of the original field. The generating function has coefficients derived from the numbers of points over the extension field with elements.
Weil conjectured that such "zeta functions" for smooth varieties are rational functions, satisfy a certain functional equation, and have their zeros in restricted places. The last two parts were quite consciously modeled on the Riemann zeta function, a kind of generating function for prime integers, which obeys a functional equation and (conjecturally) has its zeros restricted by the Riemann hypothesis. The rationality was proved by , the functional equation by , and the analogue of the Riemann hypothesis by .
Background and history.
The earliest antecedent of the Weil conjectures is by Carl Friedrich Gauss and appears in section VII of his "Disquisitiones Arithmeticae" , 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 is a prime number such that is divisible by 3. Then there is a cyclic cubic field inside the cyclotomic field of th roots of unity, and a normal integral basis of periods for the integers of this field (an instance of the Hilbert–Speiser theorem). Gauss constructs the order-3 periods, corresponding to the cyclic group of non-zero residues modulo under multiplication and its unique subgroup of index three. Gauss lets formula_1, formula_2, and formula_3 be its cosets. Taking the periods (sums of roots of unity) corresponding to these cosets applied to , he notes that these periods have a multiplication table that is accessible to calculation. Products are linear combinations of the periods, and he determines the coefficients. He sets, for example, formula_4 equal to the number of elements of which are in formula_1 and which, after being increased by one, are also in formula_1. He proves that this number and related ones are the coefficients of the products of the periods. To see the relation of these sets to the Weil conjectures, notice that if and are both in formula_1, then there exist and in such that and ; consequently, . Therefore formula_4 is the number of solutions to in the finite field . The other coefficients have similar interpretations. 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.
The Weil conjectures in the special case of algebraic curves were conjectured by . The case of curves over finite fields was proved by Weil, finishing the project started by Hasse's theorem on elliptic curves over finite fields. Their interest was obvious enough from within number theory: they implied upper bounds for exponential sums, a basic concern in analytic number theory .
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.
The analogy with topology suggested that a new homological theory be set up applying within algebraic geometry. This took two decades (it was a central aim of the work and school of Alexander Grothendieck) building up on initial suggestions from Serre. The rationality part of the conjectures was proved first by , using -adic methods. and his collaborators established the rationality conjecture, the functional equation and the link to Betti numbers by using the properties of étale cohomology, a new cohomology theory developed by Grothendieck and Michael Artin for attacking the Weil conjectures, as outlined in . 
Of the four conjectures the analogue of the Riemann hypothesis was the hardest to prove. Motivated by the proof of of an analogue of the Weil conjectures for Kähler manifolds, Grothendieck envisioned a proof based on his standard conjectures on algebraic cycles . However, Grothendieck's standard conjectures remain open (except for the hard Lefschetz theorem, which was proved by Deligne by extending his work on the Weil conjectures), and the analogue of the Riemann hypothesis was proved by , using the étale cohomology theory but circumventing the use of standard conjectures by an ingenious argument.
 found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf.
Statement of the Weil conjectures.
Suppose that is a non-singular -dimensional projective algebraic variety over the field with elements. The zeta function of is by definition
where is the number of points of defined over the degree extension of .
The Weil conjectures state:
Examples.
The projective line.
The simplest example (other than a point) is to take to be the projective line. The number of points of over a field with elements is just (where the "" comes from the "point at infinity"). The zeta function is just
It is easy to check all parts of the Weil conjectures directly. For example, the corresponding complex variety is the Riemann sphere and its initial Betti numbers are 1, 0, 1.
Projective space.
It is not much harder to do -dimensional projective space.
The number of points of over a field with elements is
just . The zeta function is just
It is again easy to check all parts of the Weil conjectures directly. (Complex projective space gives the relevant Betti numbers, which nearly determine the answer.)
The number of points on the projective line and projective space are so easy to calculate because they can be written as disjoint unions of a finite number of copies of affine spaces. It is also easy to prove the Weil conjectures for other spaces, such as Grassmannians and flag varieties, which have the same "paving" property.
Elliptic curves.
These give the first non-trivial cases of the Weil conjectures (proved by Hasse).
If is an elliptic curve over a finite field with elements, then the number of points of defined over the field with elements is , 
where and are complex conjugates with absolute value .
The zeta function is
Weil cohomology.
Weil suggested that the conjectures would follow from the existence of a suitable "Weil cohomology theory" for varieties over finite fields, similar to the usual cohomology with rational coefficients for complex varieties.
His idea was that if is the Frobenius automorphism over the finite field, then the number of points of the variety over the field of order is the number of fixed points of (acting on all points of the variety defined over the algebraic closure). 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.
The first problem with this is that the coefficient field for a Weil cohomology theory cannot be the rational numbers. To see this consider the case of a supersingular elliptic curve over a finite field of characteristic . 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. However a quaternion algebra over the rationals cannot act on a 2-dimensional vector space over the rationals. The same argument eliminates the possibility of the coefficient field being the reals or the -adic numbers, because the quaternion algebra is still a division algebra over these fields. However it does not eliminate the possibility that the coefficient field is the field of -adic numbers for some prime , because over these fields the division algebra splits and becomes a matrix algebra, which can act on a 2-dimensional vector space. Grothendieck and Michael Artin managed to construct suitable cohomology theories over the field of -adic numbers for each prime , called -adic cohomology.
Grothendieck's proofs of three of the four conjectures.
By the end of 1964 Grothendieck together with Artin and Jean-Louis Verdier (and the earlier 1960 work by Dwork) proved the Weil conjectures apart from the most difficult third conjecture above (the "Riemann hypothesis" conjecture) (Grothendieck 1965). The general theorems about étale cohomology allowed Grothendieck to prove an analogue of the Lefschetz fixed point formula for the "l"-adic cohomology theory, and by applying it to the Frobenius automorphism "F" he was able to prove the conjectured formula for the zeta function:
where each polynomial "P""i" is the determinant of "I − TF" on the "l"-adic cohomology group "H""i".
The rationality of the zeta function follows immediately. 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 "F"0:
as a product over cohomology groups:
The special case of the constant sheaf gives the usual zeta function.
Deligne's first proof of the Riemann hypothesis conjecture.
, , and gave expository accounts of the first proof of . Much of the background in "l"-adic cohomology is described in .
Deligne's first proof of the remaining third Weil conjecture (the "Riemann hypothesis conjecture") used the following steps:
The key estimate.
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. This is done by studying the zeta functions of the even powers "E""k" of "E" and applying Grothendieck's formula for the zeta functions as alternating products over cohomology groups. The crucial idea of considering even "k" powers of "E" was inspired by the paper , who used a similar idea with "k"=2 for bounding the Ramanujan tau function. 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.
Completion of the proof.
The deduction of the Riemann hypothesis from this estimate is mostly a fairly straightforward use of standard techniques and is done as follows.
Deligne's second proof.
 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. Much of the second proof is a rearrangement of the ideas of his first proof. 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 "R""i""f"! takes mixed sheaves of weight ≤"β" to mixed sheaves of weight ≤"β"+"i".
The original Weil conjectures follow by taking "f" to be a morphism from a smooth projective variety to a point and considering the constant sheaf Q"l" on the variety. This gives an upper bound on the absolute values of the eigenvalues of Frobenius, and Poincaré duality then shows that this is also a lower bound.
In general "R""i""f"! does not take pure sheaves to pure sheaves. However it does when a suitable form of Poincaré duality holds, for example if "f" is smooth and proper, or if one works with perverse sheaves rather than sheaves as in .
Inspired by the work of on Morse theory, 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. His proof generalizes the classical calculation of the absolute value of Gauss sums using the fact that the norm of a Fourier transform has a simple relation to the norm of the original function. used Laumon's proof as the basis for their exposition of Deligne's theorem. gave a further simplification of Laumon's proof, using monodromy in the spirit of Deligne's first proof. gave another proof using the Fourier transform, replacing etale cohomology with rigid cohomology.