The purpose of this post is to describe an application of the general intersection theory machinery (for curves on surfaces) developed in the previous posts: the Weil bound on points on a curve over a finite field.

1. Statement of the Weil bound

Let ${C}$ be a smooth, projective, geometrically irreducible curve over ${\mathbb{F}_q}$ of genus $g$. Then the Weil bound states that:

$\displaystyle |C(\mathbb{F}_q) - q - 1 | \leq 2 g \sqrt{q}.$

Weil’s proof of this bound is based on intersection theory on the surface ${C \times C}$. More precisely, let

$\displaystyle \overline{C} = C \times_{\mathbb{F}_q} \overline{\mathbb{F}_q},$

so that ${\overline{C}}$ is a smooth, connected, projective curve. It comes with a Frobenius map

$\displaystyle F: \overline{C} \rightarrow \overline{C}$

of ${\overline{\mathbb{F}_q}}$-varieties: in projective coordinates the Frobenius runs

$\displaystyle [x_0: \dots : x_n] \mapsto [x_0^q: \dots : x_n^q].$

In particular, the map has degree ${q}$. One has

$\displaystyle C( \mathbb{F}_q) = \mathrm{Fix}(F, \overline{C}(\overline{\mathbb{F}}_q))$

representing the ${\mathbb{F}_q}$-valued points of ${C}$ as the fixed points of the Frobenius (Galois) action on the ${\overline{\mathbb{F}_q}}$-valued points. So the strategy is to count fixed points, using intersection theory.

Using the (later) theory of ${l}$-adic cohomology, one represents the number of fixed points of the Frobenius as the Lefschetz number of ${F}$: the action of ${F}$ on ${H^0}$ and ${H^2}$ give the terms ${q+1}$. The fact that (remaining) action of ${F}$ on the ${2g}$-dimensional vector space ${H^1}$ can be bounded is one of the Weil conjectures, proved by Deligne for general varieties: here it states that ${F}$ has eigenvalues which are algebraic integers all of whose conjugates have absolute value ${\sqrt{q}}$.

2. Some divisors on the square of $C$

Let’s consider the divisor ${\Gamma}$ given by the image of the closed immersion

$\displaystyle \overline{C} \rightarrow \overline{C} \times \overline{C}, \quad x \mapsto (x, Fx) .$

In particular, ${\Gamma}$ is a smooth curve. The set-theoretic intersection of ${\Gamma}$ with the diagonal divisor ${\Delta}$ clearly consists of the fixed points of ${F}$. Moreover,

$\displaystyle C( \mathbb{F}_q) = \Gamma. \Delta,$

because ${\Gamma}$ and ${\Delta}$ intersect transversely at all points where they intersect. The tangent space to ${\Gamma}$ is “horizontal” (the Frobenius has zero differential) while that to ${\Delta}$ is “diagonal.” So the main goal is to obtain bounds on the intersection pairing ${\Gamma.\Delta}$. The Hodge index theorem provides a powerful tool in this regard.

Namely, we know that ${\Gamma}$ has the following property:

• The intersection ${\Gamma. h}$ with the “horizontal” divisor ${C \times \left\{\ast\right\}}$ is given by ${q}$, the degree of the Frobenius.
• The intersection ${\Gamma. v}$ with the “vertical” divisor ${\left\{\ast\right\} \times C}$ is ${1}$. More generally, this is true for the graph of any morphism.

This raises a more general question: given a divisor ${D}$ on ${\overline{C} \times \overline{C}}$ with known “horizontal” and “vertical” intersection numbers, can we bound ${D. \Delta}$? This is something that we can answer using the Hodge index theorem proved in the previous post.

Namely, let’s work in the group ${\mathrm{Num}(X)\otimes_{\mathbb{Z}} \mathbb{Q}}$ of rational divisors modulo numerical equivalence. We have an orthogonal decomposition

$\displaystyle D = a h + bv + D', \quad \Delta = h + v + \Delta' .$

Here ${D', \Delta' }$ is perpendicular to ${\mathbb{Q} h \oplus \mathbb{Q} v}$. Using the relations ${h^2 = v^2 =0 , h.v = 1}$, ${a,b}$ can be computed from the knowledge of the “horizontal” and “vertical” intersection numbers of ${D}$ via

$\displaystyle a = D.v, \quad b = D.h.$

In particular, we get

$\displaystyle D. \Delta = ( ah + bv + D').( h + v + \Delta') = a + b + D'.\Delta'.$

But now the Hodge index theorem tells us that the intersection pairing is negative definite on ${(\mathbb{Q}h \oplus \mathbb{Q} v)^{\perp}}$—indeed, ${h+v}$ is an ample divisor, so that by Cauchy-Schwarz:

$\displaystyle |D' \Delta'| \leq \sqrt{ D'^2 \Delta'^2}. \ \ \ \ \ (1)$

Let’s now simplify this a bit. We had ${ \Delta^2 = 2-2g; }$ the self-intersection of the diagonal in ${\overline{C} \times \overline{C}}$ is the Euler characteristic. One can see this by observing that ${\Delta^2}$ is the degree of the normal bundle to the diagonal embedding ${\overline{C} \rightarrow \overline{C} \times \overline{C}}$; this is the tangent (or anticanonical) bundle to ${\overline{C}}$, which has degree ${2-2g}$.

A simple computation shows therefore

$\displaystyle \Delta'^2 = -2g, \quad D'^2 = D^2 - 2ab.$

Proposition 1 Let ${D}$ be any divisor on ${\overline{C} \times\overline{C}}$ with ${a = D.v, b = D.h}$. Then

$\displaystyle |D. \Delta - (a+b)| \leq \sqrt{ 2g(2ab - D^2)}. \ \ \ \ \ (2)$

3. The Weil bound

Let’s now apply (2) and prove the Weil bound. Let’s now apply this to the case when ${D}$ is the graph ${\Gamma}$ of the Frobenius. For ${\Gamma^2}$, we get ${q(2 - 2g)}$. The normal bundle to the imbedding

$\displaystyle \overline{C} \stackrel{x \mapsto (x, Fx)}{\rightarrow} \overline{C} \times \overline{C}$

is exactly ${F^* T_{\overline{C}}}$, which has degree ${q(2-2g)}$.

In this case,

$\displaystyle a = q, b = 1, \Gamma^2 = q(2 - 2g),$

and we conclude with:

Theorem 2 (Weil bound) For the smooth projective curve ${C/\mathbb{F}_q}$, one has

$\displaystyle |C(\mathbb{F}_q) - q - 1| \leq 2 g \sqrt{q}. \ \ \ \ \ (3)$