# The Weil Conjectures

We can finally state the conjectures for the zeta function formulated by André Weil in 1949 and proven completely by Deligne in 1974. Let .

- Rationality
- is a rational function with integer coefficients of the form

- Functional equation
- with is the self-intersection number of the diagonal of .

- Betti numbers
- Setting the -th Betti number yields . If is the reduction of a variety over a number ring modulo a prime ideal, then where is the associated analytic space.

- Riemann hypothesis
- is a (unique) polynomial with integer coefficients looking like where are algebraic integers satisfying for any embedding . (Then the zeros of satisfy the usual Riemann hypothesis.)

## Caveats[edit | edit source]

- The rationality does not imply that the have integer coefficients, only that after reducing, the whole fraction has integer coefficients.
- In the setting of having a model over a number ring like , we do not require this model to be smooth. In fact, Fontaine showed that the only smooth projective curve over is . Relating the to the topological Betti numbers exhibits as the topological Euler characteristic.
- Algebraic numbers for which all embeddings have the same absolute value are quite special. They are called
*-Weil numbers of weight*.

## Example[edit | edit source]

In our computed example of , the Betti numbers are which coincides with the degrees of .

## Estimates[edit | edit source]

Knowing the zeta function (and its logarithm), we can recover a formula for the point counts:

## Personal side remark[edit | edit source]

Kapranov defines a *motivic zeta function* as

Where does this phenomenon come from? Is Kapranov's zeta function always rational? The answer to the second question is *No*, Larsen-Lunts gave a counter-example over with a specially constructed motivic measure. This measure vanishes on , in consequence rationality is still an open question if we localise by .^{[1]}

## Weil cohomology theories[edit | edit source]

The way to prove the Weil conjectures is to give them a cohomological interpretation. Much of the algebraic geometry initiated Grothendieck was devoted to finding the right cohomology theory, from which the conjectures could be deduced. This so-called *Weil cohomology theory* should be modelled after singular cohomology for smooth, projective complex varieties.

Let be any field and a field of characteristic . We have the following wish list:

- A Weil cohomology theory over with coefficients in should be a contravariant functor

- Poincaré duality: Multiplication induces perfect pairings:

- Künneth formula:
- Cycle map: There exists a for where denotes the group of algebraic cycles of codimension .
- Lefschetz trace formula: For any endomorphism over with graph , we can compute the number of fixed-points:

- Hard Lefschetz Theorem: If is a hyperplane section, then multiplication by induces an isomorphism .
- Comparison theorem: For and , .

There are different known Weil cohomology theories apart from the traditional singular and de Rham, which do not work over finite fields. These are -adic cohomology building upon étale cohomology with ( prime) and crystalline cohomology with (the Witt vectors of ). Rigid cohomology is a p-adic cohomology theory that extends crystalline cohomology.

The Lefschetz formula allows us to count points cohomologically, using the fact that the fixed-points under the Frobenius map (raising coordinates to the -th power) are the -points of :

The functional equation can be derived from Poincaré duality while the Riemann hypothesis is a deeper fact and amounts to the fact of proving that the eigenvalues of the Frobenius map acting on cohomology are Weil numbers.^{[2]}