The Weil conjectures

Zeta functions

Let be a variety defined over and let be the set of closed points of . For an we define the degree

The zeta function is the formal power series
In what follows, we will perceive as a series in the variable .

The zeta function describes the number of rational points of a variety.

Remark 12.1

The following equality holds


Example 12.1 (The zeta function of a projective space)

Let . Then . By Proposition Template:Ref@fact:zetarationalpoints, we have

and so


The statement of the Weil conjectures

Theorem 12.1 (Weil, Grothendieck, Deligne, Dwork, \ldots)

Let be a projective smooth variety of dimension defined over . Then

  1. Rationality and integrality. The zeta function is a rational function of . More precisely

with . Further,
for some . We denote and .

  1. Functional equation and Poincare duality. The following functional equation holds

where is the topological Euler characteristic.

  1. Betti numbers. If is a good reduction modulo of a smooth projective variety , where is a number field, then

  1. Riemann hypothesis. The zeroes of for are algebraic integers satisfying

In other words, those zeroes and poles of satisfy .


Remark 12.2

Note that Proposition Failed to parse (unknown function "\ref"): {\displaystyle \ref{fact:zetarationalpoints}} implies that

In particular, using the last Weil conjecture, we get a beautiful approximation of the number of points on a variety, generalizing Hasse's result for elliptic curves:


Example 12.2 (Curves)

Let be a smooth projective curve of genus . Then, by the Weil conjectures, we have that

where .


The philosophy of the proof

In the proof of Weil conjectures, a certain cohomology theory, the etale cohomology for -adic sheaves is used, where is prime and . First, one constructs

using Grothendieck topologies and then defines the sought-for cohomology by taking a direct limit. This cohomology theory behaves similiarily to singular cohomology in characteristic zero. For example, if is a good reduction modulo of a variety , then

Weil conjecture 1 and 3: The proof of the Weil conjectures is based on a certain fixed point formula. For singular homology, the number of fixed points, counted with multiplicity, of an automorphism is equal to

It is called the Lefschetz fixed-point formula.

Let be the base change of to . Let be a point on . Then, if and only if

that is, when is stable under , the composition of Frobeniuses. Thus, is the number of fixed points of .

One can show, that the Lefschetz fixed-point formula holds for etale cohomologies, and applying it to yields:

Let be eigenvalues of the action of on . Then
and reversing the argument of Remark 12.2, we can conclude the first Weil conjecture. Further, by the comparison of cohomologies (see (The philosophy of the proof)), we obtain the third Weil conjecture.

Weil conjecture 2 In order to prove the second Weil conjecture, one uses Poincare duality for etale cohomologies. The perfect pairing

is compatible with the Frobenius action. Further, acts on as the multiplication by , and so if is an eigenvalue of on , then is an eigenvalue of the action of on . In particular
where is the -th Betti number. Therefore
and the second Weil conjecture is proved.

Weil conjecture 4 The last Weil conjecture is much more demanding. It was proved by Deligne using techniques of monodromy and Lefschetz pencils.

The Riemann hypothesis for curves

This section is based on Section 3.3 in . Let be a smooth projective curve of genus . It is not very difficult to prove, that the Riemann hypothesis for curves follows from the inequality

which generalizes the Hasse's result for elliptic curves. In this section, we will show that this inequality holds. The proof is based on the intersection theory on surfaces.

Recall that was the base change of to . Consider a variety , its diagonal and the graph of , the composition of Frobenius morphisms. The graph consists of points

where . One can easily show that and intersect each other transversally, and so
Let and . We have that and .

First, we compute and using the adjunction formula. Note that both curves are smooth of genus . The canonical bundle of is equal to . Thus

Using Hodge index theorem, one can show that for a divisor on , the following inequality holds:
Plugging and working out the quadratic inequality, one gets the sought-for approximation for .