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 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 is a prime number congruent to 1 modulo 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 , , and 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, equal to the number of elements of 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 and are both in , then there exist and in such that and ; consequently, . Therefore is related to 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:
1. (Rationality) is a rational function of . More precisely, can be written as a finite alternating product
where each is an integral polynomial. Furthermore, , , and for , factors over as for some numbers . 2. (Functional equation and Poincaré duality) The zeta function satisfies
or equivalently
where is the Euler characteristic of . In particular, for each , the numbers , , ... equal the numbers , , ... in some order. 3. (Riemann hypothesis) for all and all . This implies that all zeros of lie on the "critical line" of complex numbers with real part . 4. (Betti numbers) If is a (good) "reduction mod " of a non-singular projective variety defined over a number field embedded in the field of complex numbers, then the degree of is the th Betti number of the space of complex points of .
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
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 curveLMFDB: Genus 2 curve 3125.a.3125.1
which is of genus and dimension . At first viewed as a curve defined over the rational numbers , this curve has good reduction at all primes . So, after reduction modulo , one obtains a hyperelliptic curve of genus 2, with . Taking as an example, the Weil polynomials , and the zeta function of assume the form
The values and can be determined by counting the numbers of solutions of over and , respectively, and adding 1 to each of these two numbers to allow for the point at infinity . This counting yields and . It follows:Chapter 6, Theorem 5.1 in
and
The zeros of 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 .
The non-singular, projective, complex manifold that belongs to has the Betti numbers .Chapter 7, Paragraph §7B in As described in part four of the Weil conjectures, the (topologically defined!) Betti numbers coincide with the degrees of the Weil polynomials , for all primes : .
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 LMFDB: Abelian variety isogeny class 2.41.aj_ct over F(41)
which was introduced in the section on hyperelliptic curves. The dimension of equals the genus of , so . There are algebraic integers such thatChapter V, Theorem 19.1 in the polynomial has coefficients in ; for all ; and for . The zeta-function of is given by
where , , and represents the complex variable of the zeta-function. The Weil polynomials have the following specific form :
for , and
is the same for the curve (see section above) and its Jacobian variety . This is, the inverse roots of are the products that consist of many, different inverse roots of . Hence, all coefficients of the polynomials can be expressed as polynomial functions of the parameters , and appearing in Calculating these polynomial functions for the coefficients of the shows that
Polynomial allows for calculating the numbers of elements of the Jacobian variety over the finite field and its field extension :Chapter 6, Theorem 5.1 in LMFDB: Abelian variety isogeny class 2.41.aj_ct over F(41)
The inverses of the zeros of do have the expected absolute value of (Riemann hypothesis). Moreover, the maps correlate the inverses of the zeros of and the inverses of the zeros of . A non-singular, complex, projective, algebraic variety with good reduction at the prime 41 to must have Betti numbers , since these are the degrees of the polynomials The Euler characteristic of is given by the alternating sum of these degrees/Betti numbers: .
By taking the logarithm of
it follows that
Aside from the values and already known, you can read off from this Taylor series all other numbers , , of -rational elements of the Jacobian variety, defined over , of the curve : for instance, and . In doing so, always implies since then, is a subgroup of .
Weil cohomology