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 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 $\mathfrak$, $\mathfrak'$, and $\mathfrak''$ 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, $(\mathfrak\mathfrak)$ equal to the number of elements of which are in $\mathfrak$ and which, after being increased by one, are also in $\mathfrak$. 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 $\mathfrak$, then there exist and in such that and ; consequently, . Therefore $(\mathfrak\mathfrak)$ 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 Artins, 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

:$\zeta(X, s) = \exp\left(\sum_^\infty \frac q^\right)$

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
:::$\prod_^ P_i(q^)^ = \frac,$
::where each is an integral polynomial. Furthermore, , , and for , factors over as $\textstyle\prod_j (1 - \alpha_T)$ for some numbers .
:2. (Functional equation and Poincaré duality) The zeta function satisfies
:::$\zeta(X,n-s)=\pm q^\zeta(X,s)$
::or equivalently
:::$\zeta(X,q^T^)=\pm q^T^E\zeta(X,T)$
::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

:$\frac$.

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

:$\frac$.

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

:$\frac$.

The Betti numbers are given by the torus, 1,2,1, and the numerator is a quadratic.

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

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