The Weil Conjectures

We can finally state the conjectures for the zeta function ${\displaystyle Z_{X}(t)}$ formulated by André Weil in 1949 and proven completely by Deligne in 1974. Let ${\displaystyle \dim X=n}$.

Rationality
${\displaystyle Z_{X}(t)}$ is a rational function with integer coefficients of the form

${\displaystyle {\frac {P_{1}(t)P_{3}(t)\dots P_{2n-1}(t)}{P_{0}(t)P_{2}(t)\dots P_{2n}(t)}}}$
with all the ${\displaystyle P_{i}}$ being polynomials and ${\displaystyle P_{0}(t)=1-t}$, ${\displaystyle P_{2n}(t)=1-q^{n}t}$.

Functional equation
${\displaystyle Z_{X}(1/(q^{n}t))=\pm q^{nE/2}t^{E}Z(t)}$
with ${\displaystyle E}$ is the self-intersection number of the diagonal of ${\displaystyle X\times X}$.
Betti numbers
Setting ${\displaystyle B_{i}:=\deg P_{i}(t)}$ the ${\displaystyle i}$-th Betti number yields ${\displaystyle E=\sum (-1)^{i}B_{i}}$. If ${\displaystyle X}$ is the reduction of a variety ${\displaystyle Y}$ over a number ring ${\displaystyle R}$ modulo a prime ideal, then ${\displaystyle B_{i}=h^{i}((Y\times _{R}\mathbb {C} )_{h},\mathbb {Z} )}$ where ${\displaystyle _{h}}$ is the associated analytic space.
Riemann hypothesis
${\displaystyle P_{i}(t)}$ is a (unique) polynomial with integer coefficients looking like ${\displaystyle \prod (1-\alpha _{ij})}$ where ${\displaystyle a_{ij}}$ are algebraic integers satisfying ${\displaystyle |a_{ij}|=q^{i/2}}$ for any embedding ${\displaystyle {\overline {\mathbb {Q} }}\hookrightarrow \mathbb {C} }$. (Then the zeros of ${\displaystyle \zeta _{X}(s)}$ satisfy the usual Riemann hypothesis.)

Caveats

• The rationality does not imply that the ${\displaystyle P_{i}}$ have integer coefficients, only that after reducing, the whole fraction has integer coefficients.
• In the setting of ${\displaystyle X}$ having a model over a number ring like ${\displaystyle \mathbb {Z} }$, we do not require this model to be smooth. In fact, Fontaine showed that the only smooth projective curve over ${\displaystyle \operatorname {Spec} \,\mathbb {Z} }$ is ${\displaystyle \operatorname {Proj} \,^{1}}$. Relating the ${\displaystyle B_{i}}$ to the topological Betti numbers exhibits ${\displaystyle E}$ as the topological Euler characteristic.
• Algebraic numbers ${\displaystyle \alpha }$ for which all embeddings have the same absolute value ${\displaystyle |q|^{i/2}}$ are quite special. They are called ${\displaystyle q}$-Weil numbers of weight ${\displaystyle i}$.

Example

In our computed example of ${\displaystyle \operatorname {Proj} \,^{1}}$, the Betti numbers are ${\displaystyle 1,0,1}$ which coincides with the degrees of ${\displaystyle 1-t,1,1-qt}$.

Estimates

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

${\displaystyle \#X(\mathbb {F} _{q^{m}})=\sum _{i=0}^{2n}(-1)^{i}\sum _{j=1}^{B_{i}}\alpha _{ij}^{m}=1+q^{mn}+\sum _{i=1}^{2n-1}(-1)^{i}\sum _{j=1}^{B_{i}}\alpha _{ij}^{m}=1+q^{mn}+O(q^{m(n-1/2)})}$
In the case of a smooth projective curve of genus ${\displaystyle g}$, we find:
${\displaystyle |\#X(\mathbb {F} _{q})-(q+1)|\leq 2g{\sqrt {q}}}$
which is a direct generalisation of the Hasse-Weil for elliptic curves.

Personal side remark

Kapranov defines a motivic zeta function as

${\displaystyle Z'_{X}(t)=\sum _{m\geq 0}[X^{(n)}]t^{m}}$
where ${\displaystyle X^{(n)}}$ is the ${\displaystyle n}$-th symmetric power of ${\displaystyle X}$. In the topological setting, by a theorem of Macdonald (with an easy one-line-proof):
${\displaystyle Z'_{X}(t)=1/(1-t)^{\chi (X)}}$
On the other hand, in the algebraic setting after applying the counting measure, we recover the local zeta function appearing in the Weil conjectures. (How does one prove this?) So in two rather different situations, we have arrived at a rational function!

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 ${\displaystyle \mathbb {C} }$ with a specially constructed motivic measure. This measure vanishes on ${\displaystyle [\mathbb {A} ^{1}]}$, in consequence rationality is still an open question if we localise by ${\displaystyle [\mathbb {A} ^{1}]}$.[1]

Weil cohomology theories

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 ${\displaystyle k}$ be any field and ${\displaystyle K}$ a field of characteristic ${\displaystyle 0}$. We have the following wish list:

• A Weil cohomology theory over ${\displaystyle k}$ with coefficients in ${\displaystyle K}$ should be a contravariant functor

{\displaystyle {\begin{aligned}\{{\text{smooth, projective varieties}}/k\}&\to &\{{\text{graded }}K{\text{-algebras}}\}\\X&\mapsto &H^{*}(X,K)=\bigoplus H^{i}(X,K)\end{aligned}}}
where the graded components are finite dimensional ${\displaystyle K}$-vector spaces and zero outside ${\displaystyle 0\leq i\leq 2\dim X}$.

• Poincaré duality: Multiplication induces perfect pairings:

${\displaystyle H^{i}(X,K)\times H^{2d-i}(X,K)\to H^{2d}(X,K)\cong K}$

• Künneth formula:
${\displaystyle H^{*}(X\times Y,K)=H^{*}(X,K)\otimes H^{*}(Y,K)}$
• Cycle map: There exists a ${\displaystyle \mathrm {(} cl):Z^{i}(X)\to H^{2i}(X,K)}$ for where ${\displaystyle Z_{i}}$ denotes the group of algebraic cycles of codimension ${\displaystyle i}$.
• Lefschetz trace formula: For any endomorphism ${\displaystyle f:X\to X}$ over ${\displaystyle k={\overline {k}}}$ with graph ${\displaystyle \Gamma _{f}}$, we can compute the number of fixed-points:

${\displaystyle (\Gamma _{f}\cdot \Delta _{X})=\sum _{i=0}^{2d}(-1)^{i}\operatorname {Tr} }$

• Hard Lefschetz Theorem: If ${\displaystyle H\subset X}$ is a hyperplane section, then multiplication by ${\displaystyle \mathrm {cl} (H)^{i}}$ induces an isomorphism ${\displaystyle H^{i}(X,K)\to H^{2d-i}(X,K)}$.
• Comparison theorem: For ${\displaystyle X/\mathbb {C} }$ and ${\displaystyle K\subset \mathbb {C} }$, ${\displaystyle H^{*}(X,K)\otimes _{K}\mathbb {C} \cong H_{\text{sing}}^{*}(X,\mathbb {C} )}$.

There are different known Weil cohomology theories apart from the traditional singular and de Rham, which do not work over finite fields. These are ${\displaystyle l}$-adic cohomology building upon étale cohomology with ${\displaystyle K=\mathbb {Q} _{l}}$ (${\displaystyle l\neq \mathrm {char} k}$ prime) and crystalline cohomology with ${\displaystyle K=W(k)}$ (the Witt vectors of ${\displaystyle k}$). 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 ${\displaystyle F}$ (raising coordinates to the ${\displaystyle q}$-th power) are the ${\displaystyle \mathbb {F} _{q}}$-points of ${\displaystyle X}$:

${\displaystyle \#X(\mathbb {F} _{q^{m}})=(\Gamma _{F^{m}}\cdot \Delta _{X})=\sum _{i=0}^{2d}(-1)^{i}\operatorname {Tr} }$
Using the lemma that
${\displaystyle -\log(\operatorname {charpoly} )=\sum _{m\geq 1}\operatorname {Tr} {\frac {t^{m}}{m}}}$
for general vector space endomorphisms ${\displaystyle \phi }$, we can deduce the required rationality with
${\displaystyle P_{i}=\operatorname {charpoly} }$
(Integrality follows once we see that ${\displaystyle Z_{X}}$ is a power series with coefficients in ${\displaystyle \mathbb {Z} }$ and by the above a rational function in ${\displaystyle K}$.)

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]

1. I found these things in notes by Vakil () which contain more intriguing facts and questions about the Grothendieck ring of varieties.
2. For a reference, try Deligne's original Weil papers or Freitag-Kiehl's book/Milne's notes on Étale Cohomology.