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 V over a finite field with q 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 Nk of points over the extension field with qk 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 consciously modelled 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 p is a prime number congruent to 1 modulo 3. Then there is a cyclic cubic field inside the cyclotomic field of pth 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 (Z/pZ)× of non-zero residues modulo p under multiplication and its unique subgroup of index three. Gauss lets, , and be its cosets. Taking the periods (sums of roots of unity) corresponding to these cosets applied to exp(2πi/p) , 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, equal to the number of elements of Z/pZ which are in and which, after being increased by one, are also in. 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 α and α + 1 are both in, then there exist x and y in Z/pZ such that and x3 + 1 = y3 . Therefore is related to the number of solutions to x3 + 1 = y3 in the finite field Z/pZ . 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 p-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 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: ζ(X, s) is a rational function of . More precisely, ζ(X, s) can be written as a finite alternating product Pi(T) is an integral polynomial. Furthermore, , , and for 1 ≤ i ≤ 2n − 1 , Pi(T) factors over C as for some numbers αij . α2n−i,1 , α2n−i,2 , ... equal the numbers qn/αi,1 , qn/αi,2 , ... in some order. for all 1 ≤ i ≤ 2n − 1 and all j. This implies that all zeros of Pk(T) lie on the "critical line" of complex numbers s with real part k/2 . Pi is the ith Betti number of the space of complex points of Y.

Examples

The projective line

The simplest example (other than a point) is to take X to be the projective line. The number of points of X over a field with qm elements is just (where the "

• 1 " 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 n-dimensional projective space. The number of points of X over a field with qm 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 E is an elliptic curve over a finite field with q elements, then the number of points of E defined over the field with qm elements is 1 − αm − βm + qm , where α and β are complex conjugates with absolute value √q . The zeta function is The Betti numbers are given by the torus, 1,2,1, and the numerator is a quadratic.

Hyperelliptic curves

As an example, consider the hyperelliptic curve which is of genus g=2 and dimension n=1. At first viewed as a curve defined over the rational numbers \mathbb{Q}, this curve has good reduction at all primes. So, after reduction modulo q\ne 5, one obtains a hyperelliptic curve of genus 2, with. Taking q=41 as an example, the Weil polynomials P_i(T), i=0,1,2, and the zeta function of assume the form The values c_1=-9 and c_2=71 can be determined by counting the numbers of solutions (x,y) of y^2 + y=x^5 over and, respectively, and adding 1 to each of these two numbers to allow for the point at infinity \infty. This counting yields N_1=33 and N_2=1743. It follows: The zeros of P_1(T) are and (the decimal expansions of these real and imaginary parts are cut off after the fifth decimal place) together with their complex conjugates and. So, in the factorisation, we have. As stated in the third part (Riemann hypothesis) of the Weil conjectures, for j=1,2,3,4. The non-singular, projective, complex manifold that belongs to has the Betti numbers. As described in part four of the Weil conjectures, the (topologically defined!) Betti numbers coincide with the degrees of the Weil polynomials P_i(T), for all primes q\ne 5:.

Abelian surfaces

An Abelian surface is a two-dimensional Abelian variety. This is, they are projective varieties that also have the structure of a group, in a way that is compatible with the group composition and taking inverses. Elliptic curves represent one-dimensional Abelian varieties. As an example of an Abelian surface defined over a finite field, consider the Jacobian variety of the genus 2 curve which was introduced in the section on hyperelliptic curves. The dimension of X equals the genus of C, so n=2. There are algebraic integers such that The zeta-function of X is given by where q=41,, and s represents the complex variable of the zeta-function. The Weil polynomials P_i(T) have the following specific form : for, and is the same for the curve C (see section above) and its Jacobian variety X. This is, the inverse roots of P_i(T) are the products that consist of i many, different inverse roots of P_1(T). Hence, all coefficients of the polynomials P_i(T) can be expressed as polynomial functions of the parameters c_1=-9, c_2=71 and q=41 appearing in Calculating these polynomial functions for the coefficients of the P_i(T) shows that Polynomial P_1 allows for calculating the numbers of elements of the Jacobian variety over the finite field and its field extension : The inverses of the zeros of P_i(T) do have the expected absolute value of 41^{i/2} (Riemann hypothesis). Moreover, the maps correlate the inverses of the zeros of P_i(T) and the inverses of the zeros of P_{4-i}(T). A non-singular, complex, projective, algebraic variety Y with good reduction at the prime 41 to must have Betti numbers, since these are the degrees of the polynomials P_i(T). The Euler characteristic E of X is given by the alternating sum of these degrees/Betti numbers:. By taking the logarithm of it follows that Aside from the values M_1 and M_2 already known, you can read off from this Taylor series all other numbers M_m,, of -rational elements of the Jacobian variety, defined over , of the curve : for instance, and. In doing so, m_1|m_2 always implies since then, is a subgroup of.

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 F is the Frobenius automorphism over the finite field, then the number of points of the variety X over the field of order qm is the number of fixed points of Fm (acting on all points of the variety X 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 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. 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 p-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 ℓ-adic numbers for some prime ℓ ≠ p , 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 ℓ-adic numbers for each prime ℓ ≠ p , called ℓ-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 Pi is the determinant of I − TF on the ℓ-adic cohomology group Hi. 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 F0: 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:

Use of Lefschetz pencils

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 Ek 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 Rif! 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 Qℓ 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 Rif! 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.

Applications