# The Weil conjectures

## Zeta functions

Let ${\displaystyle X}$ be a variety defined over ${\displaystyle \mathbb {F} _{p}}$ and let ${\displaystyle |X|}$ be the set of closed points of ${\displaystyle X}$. For an ${\displaystyle x\in |X|}$ we define the degree

${\displaystyle d(x):=[k(x):\mathbb {F} _{p}].}$
The zeta function is the formal power series
${\displaystyle \zeta _{X}(s)=\prod _{x\in |X|}{\frac {1}{1-(p^{-s})^{d(x)}}}.}$
In what follows, we will perceive ${\displaystyle \zeta _{X}}$ as a series in the variable ${\displaystyle t=-p^{-s}}$.

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

Remark 12.1

The following equality holds

${\displaystyle \sum _{r\geq 0}\#X(\mathbb {F} _{p^{r}})t^{r}=t{\frac {d}{dt}}\mathrm {log} \zeta _{X}(t).}$

Example 12.1 (The zeta function of a projective space)

Let ${\displaystyle X=\mathbb {P} _{\mathbb {F} _{p}}^{n}}$. Then ${\displaystyle \#X(\mathbb {F} _{p^{r}})={\frac {(p^{r})^{n+1}-1}{p^{r}-1}}=1+p^{r}+\ldots +(p^{r})^{n}}$. By Proposition about zeta rational points, we have

${\displaystyle {\frac {d}{dt}}\mathrm {log} \zeta _{X}(t)={\frac {1}{1-t}}+{\frac {p}{1-pt}}+\ldots +{\frac {p^{n}}{1-p^{n}t}},}$
and so
${\displaystyle \zeta _{X}(t)=e^{-\log(1-t)-\log(1-pt)-\ldots -\log(1-p^{n}t)}=\prod _{i\geq 1}{\frac {1}{1-p^{i}t}}.}$

## The statement of the Weil conjectures

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

Let ${\displaystyle X}$ be a projective smooth variety of dimension ${\displaystyle n}$ defined over ${\displaystyle \mathbb {F} _{p}}$. Then

1. Rationality and integrality. The zeta function ${\displaystyle \zeta _{X}(t)}$ is a rational function of ${\displaystyle t}$. More precisely

${\displaystyle \zeta _{X}(t)={\frac {P_{1}(t)\cdot \ldots \cdot P_{2n-1}(t)}{P_{0}(t)\cdot \ldots \cdot P_{2n}(t)}},}$
with ${\displaystyle P_{i}(t)\in \mathbb {Z} [t]}$. Further,
{\displaystyle {\begin{aligned}P_{0}(t)&=1-t,\\P_{2n}(t)&=1-p^{n}t,{\text{ and}}\\P_{i}(t)&=\prod _{j}(1-\alpha _{ij}t),\end{aligned}}}
for some ${\displaystyle \alpha _{ij}\in \mathbb {C} }$. We denote ${\displaystyle a_{0,0}=1}$ and ${\displaystyle a_{2n,0}=p^{n}}$.

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

${\displaystyle \zeta _{X}(p^{-n}t^{-1})=\pm p^{\frac {\chi (X)n}{2}}t^{\chi (X)}\zeta _{X}(t),}$
where ${\displaystyle \chi (X)}$ is the topological Euler characteristic.

1. Betti numbers. If ${\displaystyle X}$ is a good reduction modulo ${\displaystyle p}$ of a smooth projective variety ${\displaystyle {\widetilde {X}}/K}$, where ${\displaystyle K\subseteq \mathbb {C} }$ is a number field, then

${\displaystyle \mathrm {deg} P_{i}(t)=\mathrm {dim} _{\mathbb {Q} }H^{i}({\widetilde {X}}(\mathbb {C} ),\mathbb {Q} ).}$

1. Riemann hypothesis. The zeroes ${\displaystyle \alpha _{ij}}$ of ${\displaystyle P_{i}(t)}$ for ${\displaystyle 1\leq i\leq 2n-1}$ are algebraic integers satisfying

${\displaystyle |\alpha _{ij}|=p^{\frac {i}{2}}.}$
In other words, those zeroes and poles of ${\displaystyle \zeta _{X}(s)}$ satisfy ${\displaystyle \mathrm {Re} (s)={\frac {i}{2}}}$.

Remark 12.2

Note that Proposition about zeta rational points implies that

${\displaystyle \#X(\mathbb {F} _{p^{r}})=\sum _{i=0}^{2n}\sum _{j}(-1)^{i}\alpha _{ij}^{r}.}$
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:
${\displaystyle \#X(\mathbb {F} _{p^{r}})=p^{nr}+O(p^{\frac {nr}{2}}).}$

Example 12.2 (Curves)

Let ${\displaystyle X}$ be a smooth projective curve of genus ${\displaystyle g}$. Then, by the Weil conjectures, we have that

${\displaystyle \zeta _{X}(t)={\frac {\prod _{i=1}^{2g}(1-\omega _{i}t)}{(1-t)(1-tp)}},}$
where ${\displaystyle |\omega _{i}|=p^{i/2}}$.

## The philosophy of the proof

In the proof of Weil conjectures, a certain cohomology theory, the etale cohomology ${\displaystyle H_{\mathrm {et} }^{i}(X,\mathbb {Q} _{l})}$ for ${\displaystyle l}$-adic sheaves is used, where ${\displaystyle l}$ is prime and ${\displaystyle l\neq p}$. First, one constructs

${\displaystyle H_{\mathrm {et} }^{i}(X,\mathbb {Z} _{l^{k}})}$
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 ${\displaystyle X}$ is a good reduction modulo ${\displaystyle p}$ of a variety ${\displaystyle {\widetilde {X}}/\mathbb {C} }$, then
${\displaystyle H^{i}({\widetilde {X}},\mathbb {Q} )\otimes \mathbb {Q} _{l}=H_{\mathrm {et} }^{i}(X,\mathbb {Q} _{l}).}$

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 ${\displaystyle f\colon X\to X}$ is equal to

${\displaystyle \sum _{k\geq 0}(-1)^{k}\mathrm {Tr} (f^{*}\;|\;H^{k}(X,\mathbb {Q} )).}$
It is called the Lefschetz fixed-point formula.

Let ${\displaystyle {\overline {X}}}$ be the base change of ${\displaystyle X}$ to ${\displaystyle {\overline {\mathbb {F} }}_{p}}$. Let ${\displaystyle x=(x_{0}:\ldots :x_{m})\in \mathbb {P} _{{\overline {\mathbb {F} }}_{p}}^{m}}$ be a point on ${\displaystyle {\overline {X}}\subseteq \mathbb {P} _{{\overline {\mathbb {F} }}_{p}}^{m}}$. Then, ${\displaystyle x\in \mathbb {P} _{\mathbb {F} _{p^{r}}}^{m}}$ if and only if

${\displaystyle (x_{0}:\ldots :x_{m})=(x_{1}^{p^{r}}:\ldots :x_{m}^{p^{r}}),}$
that is, when ${\displaystyle x}$ is stable under ${\displaystyle F^{r}}$, the composition of ${\displaystyle r}$ Frobeniuses. Thus, ${\displaystyle \#X(\mathbb {F} _{p^{r}})}$ is the number of fixed points of ${\displaystyle F^{r}}$.

One can show, that the Lefschetz fixed-point formula holds for etale cohomologies, and applying it to ${\displaystyle F^{r}}$ yields:

${\displaystyle \#X(\mathbb {F} _{p^{r}})=\#X({\overline {\mathbb {F} }}_{p})^{F^{r}}=\sum _{i=0}^{2n}(-1)^{i}\mathrm {Tr} ((F^{r})^{*}\;|\;H_{\mathrm {et} }^{i}(X,\mathbb {Q} _{l})).}$
Let ${\displaystyle \alpha _{ij}}$ be eigenvalues of the action of ${\displaystyle F}$ on ${\displaystyle H_{\mathrm {et} }^{i}(X,\mathbb {Q} _{l})}$. Then
${\displaystyle \#X(\mathbb {F} _{p^{r}})=\sum _{i=0}^{2n}\sum _{j}(-1)^{i}\alpha _{ij}^{r},}$
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

${\displaystyle H_{\mathrm {et} }^{i}(X,\mathbb {Q} _{l})\times H_{\mathrm {et} }^{2n-i}(X,\mathbb {Q} _{l})\to H_{\mathrm {et} }^{2n}(X,\mathbb {Q} _{l})}$
is compatible with the Frobenius action. Further, ${\displaystyle F}$ acts on ${\displaystyle H_{\mathrm {et} }^{2n}(X,\mathbb {Q} _{l})}$ as the multiplication by ${\displaystyle p^{n}}$, and so if ${\displaystyle \alpha _{ij}}$ is an eigenvalue of ${\displaystyle F}$ on ${\displaystyle H_{\mathrm {et} }^{i}(X,\mathbb {Q} _{l})}$, then ${\displaystyle {\frac {p^{n}}{\alpha _{ij}}}}$ is an eigenvalue of the action of ${\displaystyle F}$ on ${\displaystyle H_{\mathrm {et} }^{2n-i}(X,\mathbb {Q} _{l})}$. In particular
${\displaystyle P_{i}(p^{-n}t^{-1})=\prod _{j}(1-{\frac {\alpha _{ij}}{p^{n}t}})=\pm \prod _{j}{\frac {\alpha _{ij}}{p^{n}t}}(1-{\frac {p^{n}}{\alpha _{i,j}}}t)=\pm (p^{n})^{\frac {b_{i}n}{2}}t^{b_{i}}P_{2n-i}(t),}$
where ${\displaystyle b_{i}}$ is the ${\displaystyle i}$-th Betti number. Therefore
${\displaystyle \zeta _{X}(p^{-n}t^{-1})=\pm p^{\frac {\chi (X)n}{2}}t^{\chi (X)}\zeta _{X}(t),}$
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 ${\displaystyle X}$ be a smooth projective curve of genus ${\displaystyle g}$. It is not very difficult to prove, that the Riemann hypothesis for curves follows from the inequality

${\displaystyle |\#X(\mathbb {F} _{p^{r}})-p^{r}-1|\leq 2gp^{\frac {r}{2}},}$
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 ${\displaystyle {\overline {X}}}$ was the base change of ${\displaystyle X}$ to ${\displaystyle {\overline {\mathbb {F} }}_{p}}$. Consider a variety ${\displaystyle S:={\overline {X}}\times {\overline {X}}}$, its diagonal ${\displaystyle \Delta }$ and the graph ${\displaystyle \Gamma }$ of ${\displaystyle F^{r}}$, the composition of ${\displaystyle r}$ Frobenius morphisms. The graph ${\displaystyle \Gamma }$ consists of points

${\displaystyle ((x_{0}:\ldots :x_{m}),(x_{0}^{p^{r}}:\ldots :x_{m}^{p^{r}}))\in {\overline {X}}\times {\overline {X}},}$
where ${\displaystyle (x_{0}:\ldots x_{m})\in {\overline {X}}}$. One can easily show that ${\displaystyle \Delta }$ and ${\displaystyle \Gamma }$ intersect each other transversally, and so
${\displaystyle \#X(\mathbb {F} _{p^{r}})=\Delta \cdot \Gamma .}$
Let ${\displaystyle l_{1}={\overline {X}}\times \mathrm {point} }$ and ${\displaystyle l_{2}=\mathrm {point} \times {\overline {X}}}$. We have that ${\displaystyle l_{1}\cdot \Delta =l_{2}\cdot \Delta =l_{2}\cdot \Gamma =1}$ and ${\displaystyle l_{1}\cdot \Gamma =p^{r}}$.

First, we compute ${\displaystyle \Delta ^{2}}$ and ${\displaystyle \Gamma ^{2}}$ using the adjunction formula. Note that both curves are smooth of genus ${\displaystyle g}$. The canonical bundle ${\displaystyle K_{S}}$ of ${\displaystyle S}$ is equal to ${\displaystyle (2g-2)(l_{1}+l_{2})}$. Thus

{\displaystyle {\begin{aligned}2g-2&=\Delta \cdot (\Delta +K_{S})=\Delta ^{2}+2(2g-2),{\text{ and}}\\2g-2&=\Gamma \cdot (\Gamma +K_{S})=\Gamma ^{2}+(p^{r}+1)(2g-2).\end{aligned}}}
Using Hodge index theorem, one can show that for a divisor ${\displaystyle D}$ on ${\displaystyle S}$, the following inequality holds:
${\displaystyle D^{2}\leq 2(D\cdot l_{1})(D\cdot l_{2}).}$
Plugging ${\displaystyle D=a\Delta +b\Gamma }$ and working out the quadratic inequality, one gets the sought-for approximation for ${\displaystyle \#X(\mathbb {F} _{p^{r}})}$.